tag:theconversation.com,2011:/ca/topics/mathematicians-22694/articlesMathematicians – The Conversation2023-08-09T12:31:58Ztag:theconversation.com,2011:article/2106122023-08-09T12:31:58Z2023-08-09T12:31:58ZA brief illustrated guide to ‘scissors congruence’ − an ancient geometric idea that’s still fueling cutting-edge mathematical research<figure><img src="https://images.theconversation.com/files/541530/original/file-20230807-29-khkb7x.jpg?ixlib=rb-1.1.0&rect=188%2C25%2C1746%2C997&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">While scissors congruence accurately captures the modern algebraic notion of 2D area, things get more complicated in higher dimensions.</span> <span class="attribution"><span class="source">Maxine Calle</span>, <a class="license" href="http://creativecommons.org/licenses/by-nd/4.0/">CC BY-ND</a></span></figcaption></figure><p>In math class, you probably learned how to <a href="https://www.cuemath.com/measurement/area/">compute the area</a> of lots of different shapes by memorizing algebraic formulas. Remember “base x height” for rectangles and “½ base x height” for triangles? Or “𝜋 x radius²” for circles?</p>
<p>But if you were in math class in ancient Greece, you might have learned something very different. Ancient Greek mathematicians, <a href="https://www.britannica.com/biography/Euclid-Greek-mathematician">such as Euclid</a>, thought of area as something geometric, not algebraic. Euclid’s geometric perspective, recorded in his foundational work “<a href="https://www.claymath.org/library/historical/euclid/">Elements</a>,” has influenced research programs across centuries – even the work of mathematicians today, like <a href="https://scholar.google.com/citations?user=mzqTDjwAAAAJ&hl=en">the two</a> <a href="https://web.sas.upenn.edu/callem/">of us</a>.</p>
<figure class="align-right zoomable">
<a href="https://images.theconversation.com/files/541055/original/file-20230803-25-o1740b.gif?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="A gif of a blue blob bouncing between a triangle, circle and square, along with formulas for their areas." src="https://images.theconversation.com/files/541055/original/file-20230803-25-o1740b.gif?ixlib=rb-1.1.0&q=45&auto=format&w=237&fit=clip" srcset="https://images.theconversation.com/files/541055/original/file-20230803-25-o1740b.gif?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=600&fit=crop&dpr=1 600w, https://images.theconversation.com/files/541055/original/file-20230803-25-o1740b.gif?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=600&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/541055/original/file-20230803-25-o1740b.gif?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=600&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/541055/original/file-20230803-25-o1740b.gif?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=754&fit=crop&dpr=1 754w, https://images.theconversation.com/files/541055/original/file-20230803-25-o1740b.gif?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=754&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/541055/original/file-20230803-25-o1740b.gif?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=754&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">Math today treats area algebraically, using formulas for simple shapes and calculus for more complicated regions.</span>
<span class="attribution"><span class="source">Maxine Calle</span>, <a class="license" href="http://creativecommons.org/licenses/by-nd/4.0/">CC BY-ND</a></span>
</figcaption>
</figure>
<p>Modern mathematicians refer to Euclid’s concept of “having equal area” as “<a href="http://dx.doi.org/10.1090/noti898">being scissors congruent</a>.” This idea, based on cutting up shapes and pasting them back together in different ways, has inspired interesting mathematics beyond just computing areas of triangles and squares. The story of scissors congruence demonstrates how classical problems in geometry can find new life in the strange world of abstract modern math.</p>
<h2>Euclid’s notion of area</h2>
<p>Today, people think of the area of a shape as a single number that can be computed using algebraic formulas or calculus. So what does it mean to think of area as something geometric the way the ancient Greeks did?</p>
<p>Imagine you’re back in math class and you have a pair of scissors, some tape and a piece of construction paper. Your teacher instructs you to make a new flat, two-dimensional shape using all of the construction paper and only straight-line cuts. Using your scissors, you cut the paper into a bunch of pieces. You start moving these pieces around – maybe you rotate them or flip them over – and you tape them back together to form a new shape. </p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/541117/original/file-20230803-15-7hhb5e.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="A drawing of a student doodling on some paper. Above them, a pentagon is cut into 3 triangles, and those triangles are reassembled into a new polygon." src="https://images.theconversation.com/files/541117/original/file-20230803-15-7hhb5e.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/541117/original/file-20230803-15-7hhb5e.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=342&fit=crop&dpr=1 600w, https://images.theconversation.com/files/541117/original/file-20230803-15-7hhb5e.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=342&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/541117/original/file-20230803-15-7hhb5e.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=342&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/541117/original/file-20230803-15-7hhb5e.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=430&fit=crop&dpr=1 754w, https://images.theconversation.com/files/541117/original/file-20230803-15-7hhb5e.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=430&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/541117/original/file-20230803-15-7hhb5e.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=430&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">What kind of shapes can you make from a pentagon using only scissors and tape?</span>
<span class="attribution"><span class="source">Maxine Calle</span>, <a class="license" href="http://creativecommons.org/licenses/by-nd/4.0/">CC BY-ND</a></span>
</figcaption>
</figure>
<p>Using your algebraic formulas for area, you could check that the area of your new shape is equal to the original area of the construction paper. No matter how a 2D shape is cut up – as long as all the pieces are taped back together without overlap – the area of the old and the new shape will always be equal.</p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/541118/original/file-20230803-17-ji9w80.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="A blue pentagon with a central eyeball is cut up into pieces." src="https://images.theconversation.com/files/541118/original/file-20230803-17-ji9w80.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/541118/original/file-20230803-17-ji9w80.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=339&fit=crop&dpr=1 600w, https://images.theconversation.com/files/541118/original/file-20230803-17-ji9w80.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=339&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/541118/original/file-20230803-17-ji9w80.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=339&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/541118/original/file-20230803-17-ji9w80.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=426&fit=crop&dpr=1 754w, https://images.theconversation.com/files/541118/original/file-20230803-17-ji9w80.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=426&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/541118/original/file-20230803-17-ji9w80.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=426&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">You’re not allowed to make curved cuts or tape overlapping pieces.</span>
<span class="attribution"><span class="source">Maxine Calle</span>, <a class="license" href="http://creativecommons.org/licenses/by-nd/4.0/">CC BY-ND</a></span>
</figcaption>
</figure>
<p>For Euclid, area is the measurement that is preserved by this geometric “cutting-and-pasting.” <a href="http://aleph0.clarku.edu/%7Edjoyce/elements/bookII/bookII.html">He would say</a> that the new shape you made is “equal” to the original piece of construction paper – mathematicians today would say the two are “scissors congruent.”</p>
<p>What can your new shape look like? Because you’re only allowed to make straight-line cuts, it has to be a polygon, meaning none of the sides can be curved.</p>
<figure class="align-left zoomable">
<a href="https://images.theconversation.com/files/541060/original/file-20230803-27-21x5oo.gif?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="A gif of a pentagon splitting up into five triangles" src="https://images.theconversation.com/files/541060/original/file-20230803-27-21x5oo.gif?ixlib=rb-1.1.0&q=45&auto=format&w=237&fit=clip" srcset="https://images.theconversation.com/files/541060/original/file-20230803-27-21x5oo.gif?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=600&fit=crop&dpr=1 600w, https://images.theconversation.com/files/541060/original/file-20230803-27-21x5oo.gif?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=600&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/541060/original/file-20230803-27-21x5oo.gif?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=600&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/541060/original/file-20230803-27-21x5oo.gif?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=754&fit=crop&dpr=1 754w, https://images.theconversation.com/files/541060/original/file-20230803-27-21x5oo.gif?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=754&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/541060/original/file-20230803-27-21x5oo.gif?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=754&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">When a polygon is cut up into pieces, its area doesn’t change.</span>
<span class="attribution"><span class="source">Maxine Calle</span>, <a class="license" href="http://creativecommons.org/licenses/by-nd/4.0/">CC BY-ND</a></span>
</figcaption>
</figure>
<p>Could you have made any possible polygon with the same area as your original piece of paper? The answer, amazingly, is yes – there’s even a <a href="https://dmsm.github.io/scissors-congruence/">step-by-step guide</a> from the 1800s that tells you exactly how to do it. </p>
<p>In other words, for polygons, Euclid’s notion of area is exactly the same as the modern one. In fact, you may have even used Euclid’s idea of area before in computations without knowing it.</p>
<p>For example, you can use scissors congruence to compute the area of a pentagon. Since area is preserved if you cut the pentagon up into smaller triangles, you can instead find the area of these triangles (using “½ base x height”) and add them up to get the answer.</p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/541116/original/file-20230803-15-otnpws.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="A pentagon, the same pentagon split up into five triangles, and the five separate triangles" src="https://images.theconversation.com/files/541116/original/file-20230803-15-otnpws.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/541116/original/file-20230803-15-otnpws.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=344&fit=crop&dpr=1 600w, https://images.theconversation.com/files/541116/original/file-20230803-15-otnpws.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=344&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/541116/original/file-20230803-15-otnpws.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=344&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/541116/original/file-20230803-15-otnpws.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=432&fit=crop&dpr=1 754w, https://images.theconversation.com/files/541116/original/file-20230803-15-otnpws.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=432&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/541116/original/file-20230803-15-otnpws.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=432&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">It can be easier to compute the area of a pentagon by chopping it up into smaller triangles.</span>
<span class="attribution"><span class="source">Maxine Calle</span>, <a class="license" href="http://creativecommons.org/licenses/by-nd/4.0/">CC BY-ND</a></span>
</figcaption>
</figure>
<h2>Hilbert’s third problem</h2>
<p>Perhaps the most infamous appearance of scissors congruence is on the famous German mathematician <a href="https://www.britannica.com/biography/David-Hilbert">David Hilbert</a>’s <a href="http://aleph0.clarku.edu/%7Edjoyce/hilbert/problems.html">list of problems</a>, which consisted of some of the most important mathematical questions of the 1900s. <a href="https://www.simonsfoundation.org/2020/05/06/hilberts-problems-23-and-math/">Of the original 23 problems</a> Hilbert proposed, some have been solved, some have been shown to be unsolvable and others are still unresolved. The third problem on the list, and the first to be resolved, is about scissors congruence.</p>
<p>Instead of two-dimensional polygons, Hilbert asked about their three-dimensional cousins: <a href="https://www.mathsisfun.com/geometry/polyhedron.html">polyhedra</a>. Euclid’s notion of scissors congruence was known to be an accurate description of two-dimensional area, but could it be a good notion of three-dimensional volume?</p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/541526/original/file-20230807-24-n0k98j.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="An illustration of the five platonic solids" src="https://images.theconversation.com/files/541526/original/file-20230807-24-n0k98j.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/541526/original/file-20230807-24-n0k98j.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=342&fit=crop&dpr=1 600w, https://images.theconversation.com/files/541526/original/file-20230807-24-n0k98j.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=342&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/541526/original/file-20230807-24-n0k98j.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=342&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/541526/original/file-20230807-24-n0k98j.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=430&fit=crop&dpr=1 754w, https://images.theconversation.com/files/541526/original/file-20230807-24-n0k98j.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=430&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/541526/original/file-20230807-24-n0k98j.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=430&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">There are just five Platonic solids – polyhedra, whose faces are all the same polygon.</span>
<span class="attribution"><span class="source">Maxine Calle</span>, <a class="license" href="http://creativecommons.org/licenses/by-nd/4.0/">CC BY-ND</a></span>
</figcaption>
</figure>
<p>The answer came within a year, provided by one of Hilbert’s students, <a href="https://www.britannica.com/biography/Max-Dehn">Max Dehn</a>. <a href="https://doi.org/10.1007/BF01448001">Dehn’s solution</a> to the problem was very different from the two-dimensional case. He showed that when polyhedra are cut up, volume is not the only thing that is preserved. There is another preserved measurement, now called <a href="https://www.youtube.com/watch?v=eYfpSAxGakI">the Dehn invariant</a>, which is constructed from the lengths of edges and the angles between the faces of the polyhedron. </p>
<p>If two polyhedra are scissors congruent, then they have to have the same Dehn invariant. So, if Dehn could find two polyhedra with the same volume but different values of this invariant, that would prove the answer to Hilbert’s third problem is no – scissors congruence doesn’t precisely capture 3D volume.</p>
<figure class="align-right zoomable">
<a href="https://images.theconversation.com/files/541133/original/file-20230804-29-at8e7l.gif?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="A tetrahedron splits into pieces. In its thought bubble is a cube, also splitting into pieces." src="https://images.theconversation.com/files/541133/original/file-20230804-29-at8e7l.gif?ixlib=rb-1.1.0&q=45&auto=format&w=237&fit=clip" srcset="https://images.theconversation.com/files/541133/original/file-20230804-29-at8e7l.gif?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=600&fit=crop&dpr=1 600w, https://images.theconversation.com/files/541133/original/file-20230804-29-at8e7l.gif?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=600&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/541133/original/file-20230804-29-at8e7l.gif?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=600&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/541133/original/file-20230804-29-at8e7l.gif?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=754&fit=crop&dpr=1 754w, https://images.theconversation.com/files/541133/original/file-20230804-29-at8e7l.gif?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=754&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/541133/original/file-20230804-29-at8e7l.gif?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=754&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">There is no way to cut a tetrahedron into pieces and glue them back together to make a cube with the same volume.</span>
<span class="attribution"><span class="source">Maxine Calle</span>, <a class="license" href="http://creativecommons.org/licenses/by-nd/4.0/">CC BY-ND</a></span>
</figcaption>
</figure>
<p>This is exactly what Dehn did, showing that the invariants associated to a cube and tetrahedron with the same volume are different. This means that there’s no possible way to cut up a tetrahedron into a finite number of pieces and reassemble them back into a cube with the same volume.</p>
<p>Are volume and the Dehn invariant all we need to know? If two polyhedra have the same volume and the same Dehn invariant, does that tell us they’re scissors congruent? It took mathematicians another 60 years to answer this question. In 1965, <a href="https://doi.org/10.1007/BF02564364">Jean-Pierre Sydler confirmed</a> that the answer is yes, closing this chapter on scissors congruence.</p>
<h2>Strange shapes and stranger connections</h2>
<p>But the story doesn’t end there. Mathematics is <a href="https://theconversation.com/the-weird-world-of-one-sided-objects-101936">full of shapes</a> living in higher dimensions – like 4D, 100D, 3,485D or any dimension you can imagine – which are impossible to visualize. An active new research area called <a href="https://www.nsf.gov/awardsearch/showAward?AWD_ID=1654522">generalized scissors congruence</a> seeks to uncover whether Hilbert’s question about scissors congruence can also be stated – and maybe even solved – for these strange shapes. </p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/541523/original/file-20230807-19-hbqdqs.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="An illustration of various shapes lining a landscape, with arrows pointing between them. Some resemble waves." src="https://images.theconversation.com/files/541523/original/file-20230807-19-hbqdqs.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/541523/original/file-20230807-19-hbqdqs.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=341&fit=crop&dpr=1 600w, https://images.theconversation.com/files/541523/original/file-20230807-19-hbqdqs.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=341&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/541523/original/file-20230807-19-hbqdqs.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=341&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/541523/original/file-20230807-19-hbqdqs.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=429&fit=crop&dpr=1 754w, https://images.theconversation.com/files/541523/original/file-20230807-19-hbqdqs.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=429&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/541523/original/file-20230807-19-hbqdqs.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=429&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">Mathematicians study other strange kinds of geometries, like hyperbolic and spherical geometry.</span>
<span class="attribution"><span class="source">Maxine Calle</span>, <a class="license" href="http://creativecommons.org/licenses/by-nd/4.0/">CC BY-ND</a></span>
</figcaption>
</figure>
<p>However, what it means for two things to be scissors congruent is now far more complicated. While Hilbert and Dehn cared about things like volume and angles, other mathematicians could exchange these physical traits for something <a href="https://doi.org/10.1016/j.topol.2022.108105">far less tangible</a>. </p>
<p>A recent research program pioneered by mathematicians <a href="http://www.jonathanacampbell.com/">Jonathan Campbell</a> and <a href="https://pi.math.cornell.edu/%7Ezakh/">Inna Zakharevich</a> proposes a <a href="http://dx.doi.org/10.1090/bull/1527">unifying framework</a> for generalized scissors congruence. This framework is built using a very abstract, seemingly unrelated mathematical toolkit called <a href="https://bookstore.ams.org/gsm-145">algebraic K-theory</a>. </p>
<figure class="align-right zoomable">
<a href="https://images.theconversation.com/files/541135/original/file-20230804-15-qs4z18.gif?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="A triangle jumps into a machine and splits into three triangles." src="https://images.theconversation.com/files/541135/original/file-20230804-15-qs4z18.gif?ixlib=rb-1.1.0&q=45&auto=format&w=237&fit=clip" srcset="https://images.theconversation.com/files/541135/original/file-20230804-15-qs4z18.gif?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=600&fit=crop&dpr=1 600w, https://images.theconversation.com/files/541135/original/file-20230804-15-qs4z18.gif?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=600&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/541135/original/file-20230804-15-qs4z18.gif?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=600&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/541135/original/file-20230804-15-qs4z18.gif?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=754&fit=crop&dpr=1 754w, https://images.theconversation.com/files/541135/original/file-20230804-15-qs4z18.gif?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=754&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/541135/original/file-20230804-15-qs4z18.gif?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=754&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">Algebraic K-theory was developed in the late 20th century in the field of abstract math known as algebraic topology.</span>
<span class="attribution"><span class="source">Maxine Calle</span>, <a class="license" href="http://creativecommons.org/licenses/by-nd/4.0/">CC BY-ND</a></span>
</figcaption>
</figure>
<p>The big idea of K-theory is that mathematical objects can be understood by how they decompose into fundamental building blocks – much like molecules are broken up into atoms. With a little bit of adjustment, mathematicians can harness the machinery of K-theory and <a href="https://www.quantamagazine.org/mathematicians-cut-apart-shapes-to-find-pieces-of-equations-20191031/">apply it</a> to generalized scissors congruence problems.</p>
<p>This use of K-theory reimagines the problem of scissors congruence and opens the doors for future research. But at the end of the day, scissors congruence is a concrete idea that you don’t need fancy math to understand – just some patience, creativity, a pair of scissors and a lot of tape.</p><img src="https://counter.theconversation.com/content/210612/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Maxine Calle is the 2023 AAAS Mass Media Fellow at The Conversation U.S. and she receives funding from the National Science Foundation. </span></em></p><p class="fine-print"><em><span>Mona Merling receives funding from the National Science Foundation.</span></em></p>This is a story about geometry, algebra and many different dimensions, best read with construction paper, scissors and tape on hand.Maxine Calle, Ph.D. Candidate in Mathematics, University of PennsylvaniaMona Merling, Assistant Professor of Mathematics, University of PennsylvaniaLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/1972032023-01-10T17:15:46Z2023-01-10T17:15:46ZRichard Price: how one of the 18th century’s most influential thinkers was forgotten<figure><img src="https://images.theconversation.com/files/503384/original/file-20230106-23-db9yxn.png?ixlib=rb-1.1.0&rect=2%2C2%2C1914%2C1069&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">Richard Price reading a letter dated 1784 from his friend, Benjamin Franklin.</span> <span class="attribution"><span class="source">Benjamin West, National Library of Wales & Shutterstock</span></span></figcaption></figure><p>According to the eulogies and <a href="https://archive.org/details/sim_gentlemans-magazine_1791-04_61_4/page/388/mode/2up?q=price">obituaries</a> written at the time of his death in 1791, <a href="https://richardpricesociety.org.uk/">Richard Price’s</a> name would be remembered alongside figures such as Benjamin Franklin, John Locke, George Washington and Thomas Paine. </p>
<p>Three hundred years on from his birth in the village of Llangeinor, near Bridgend in south Wales, why has he therefore been lost from our popular memory? </p>
<p>After all, here was a polymath whose lasting contributions ranged across a number of disciplines, including moral philosophy, <a href="https://rss.onlinelibrary.wiley.com/doi/10.1111/j.1740-9713.2013.00638.x">mathematics</a> and theology. Moreover, Price’s contribution as a public intellectual made a huge impact, not least in international politics. </p>
<p>A useful starting point are the parallels with his friend <a href="https://www.theguardian.com/lifeandstyle/womens-blog/2015/oct/05/original-suffragette-mary-wollstonecraft?CMP=share_btn_link">Mary Wollstonecraft</a>. She was a philosopher, a women’s rights advocate and the mother of <a href="https://www.bl.uk/people/mary-shelley">Mary Shelley</a>. </p>
<p>Wollstonecraft was both inspired by Price and indebted to him. Indeed, her most influential texts are directly linked to Price and the pamphlet war known as the <a href="https://en.wikipedia.org/wiki/Revolution_Controversy">Revolution controversy</a>. </p>
<p>In these texts, influential thinkers discussed the political issues arising from the <a href="https://www.britannica.com/event/French-Revolution">French Revolution</a>. It has subsequently been recognised as a <a href="https://www.jstor.org/stable/26213839">formative debate in terms of modern political ideas. </a></p>
<p>It was Price who sparked the controversy with a sermon in 1789 entitled <a href="https://www.google.co.uk/books/edition/A_Discourse_on_the_Love_of_Our_Country/92QNAAAAIAAJ?hl=en&gbpv=1&printsec=frontcover">A Discourse on the Love of Our Country</a>, in which he supported the opening events of the revolution in France. </p>
<p>He declared it to be a continuation of the spreading of enlightened values and ideas introduced by the <a href="https://www.parliament.uk/about/living-heritage/evolutionofparliament/parliamentaryauthority/revolution/">Glorious Revolution of 1688</a> in England. </p>
<figure>
<iframe width="440" height="260" src="https://www.youtube.com/embed/XHjtIO0ZFs4?wmode=transparent&start=0" frameborder="0" allowfullscreen=""></iframe>
<figcaption><span class="caption">Richard Price’s sermon to the Revolution Society in 1789.</span></figcaption>
</figure>
<p>This provoked a response from the philosopher and Anglo-Irish Whig MP <a href="https://www.britannica.com/biography/Edmund-Burke-British-philosopher-and-statesman">Edmund Burke</a>, with his famous text, <a href="https://www.bl.uk/collection-items/reflections-on-the-revolution-in-france-by-edmund-burke">Reflections on the Revolution in France</a>. </p>
<p>This is regarded as a formative text of modern conservative thought. It defended the importance of the traditional institutions of state and society while warning of the excesses of revolution. </p>
<p>In response, Wollstonecraft published <a href="https://oll.libertyfund.org/title/wollstonecraft-a-vindication-of-the-rights-of-men">A Vindication of the Rights of Men</a> in 1790. It was both a critique of Burke and a defence of Price, who died a year later. </p>
<p>Then in 1792, she wrote her profoundly influential <a href="https://www.bl.uk/collection-items/mary-wollstonecraft-a-vindication-of-the-rights-of-woman">A Vindication of the Rights of Woman</a>, explicitly extending dissenting ideals to women, with a searing social critique. </p>
<p>Both Price and Wollstonecraft would subsequently be written out of history. </p>
<p>Price’s biographer, <a href="https://www.uwp.co.uk/author/Paul-Frame-663/">Paul Frame</a>, suggests this can be partly accounted for by events in France and the <a href="https://www.britannica.com/event/Reign-of-Terror">violent turn to terror during the French Revolution</a>. </p>
<p>As a result, <a href="https://www.uwp.co.uk/book/libertys-apostle-richard-price-his-life-and-times/">Frame suggests</a> Burke was “the man who had accurately predicted the direction of the Revolution”. This “undermined the more optimistic faith in rationalism and natural rights” of Price and others. </p>
<p>They both also suffered in terms of their personal reputation. Price became a caricature of the picture painted by Burke, captured in the cartoons of the day. </p>
<figure class="align-center ">
<img alt="Satirical cartoon of Richard Price at his writing desk overlooked by a large nose and eyes surrounded by haze representing Edmund Burke, carrying a crown, a cross and a copy of his pamphlet." src="https://images.theconversation.com/files/503100/original/file-20230104-70338-pvtb8n.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/503100/original/file-20230104-70338-pvtb8n.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=427&fit=crop&dpr=1 600w, https://images.theconversation.com/files/503100/original/file-20230104-70338-pvtb8n.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=427&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/503100/original/file-20230104-70338-pvtb8n.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=427&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/503100/original/file-20230104-70338-pvtb8n.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=537&fit=crop&dpr=1 754w, https://images.theconversation.com/files/503100/original/file-20230104-70338-pvtb8n.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=537&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/503100/original/file-20230104-70338-pvtb8n.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=537&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">A caricature of Richard Price with a vision of Edmund Burke looking over his shoulder, by James Gillray.</span>
<span class="attribution"><span class="source">Library of Congress</span></span>
</figcaption>
</figure>
<p>Wollstonecraft was posthumously <a href="https://lithub.com/how-a-husbands-loving-biography-ruined-his-wifes-reputation/">undone by the candid biography of her widower</a>, its contents deployed maliciously by those who sought to undermine her. Thankfully, <a href="https://theconversation.com/mary-wollstonecraft-statue-a-provocative-tribute-for-a-radical-woman-149888">her works and good name were recovered by the feminist movement</a>. </p>
<p>As Frame suggests however, there were deeper, structural factors at play. </p>
<p>Price was the embodiment of a reformism the British establishment had a material interest in thwarting. He represented a dissenting community whose <a href="https://welshchapels.wales/nonconformity/">nonconformist Christian denominations</a> were in opposition to the established church and discriminated against. </p>
<p>Price spoke out against the crown, slavery and chauvinistic nationalism. He advocated equality, democratic principles and civic nationalism. </p>
<p>The hostility towards the progressive forces he embodied was symbolised by the <a href="https://www.oxfordreference.com/display/10.1093/oi/authority.20110803100452318;jsessionid=7677A3EB1D19321A218678801F2EDCD1">Seditious Meetings Act</a> introduced in 1795 to stifle the reform movement. </p>
<figure class="align-center ">
<img alt="An illustration from 1790 showing three men speaking from a church pulpit to a group of others reading and tearing up documents." src="https://images.theconversation.com/files/503390/original/file-20230106-6729-mq16ci.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/503390/original/file-20230106-6729-mq16ci.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=841&fit=crop&dpr=1 600w, https://images.theconversation.com/files/503390/original/file-20230106-6729-mq16ci.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=841&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/503390/original/file-20230106-6729-mq16ci.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=841&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/503390/original/file-20230106-6729-mq16ci.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=1056&fit=crop&dpr=1 754w, https://images.theconversation.com/files/503390/original/file-20230106-6729-mq16ci.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=1056&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/503390/original/file-20230106-6729-mq16ci.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=1056&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">Richard Price, Joseph Priestley and Theophilus Lindsay in a 1790 engraving satirising the campaign to have the Test Act repealed.</span>
<span class="attribution"><span class="source">James Sayers</span></span>
</figcaption>
</figure>
<p>There would have been very real consequences had it been Price and his ilk – and not Burke – who were lionised as the spirit of Britain (a state less than a century old at the time). Arguably, we still live with the ramifications today. </p>
<p>Price’s politics eventually had their day as the social tumult of the 19th century meant the tide of reform could not be stemmed. </p>
<p>Burke’s conservatism, however, conceivably still symbolises where the balance of power sits in terms of the UK’s political culture. The Tory party is often <a href="https://go.gale.com/ps/i.do?id=GALE%7CA271975015&sid=googleScholar&v=2.1&it=r&linkaccess=abs&issn=15555623&p=AONE&sw=w&userGroupName=anon%7E26847d25">still regarded as the natural party of power</a>, and deference towards the ruling classes remains. </p>
<figure class="align-center ">
<img alt="A memorial stone dedicated to Richard Price in Newington Green Unitarian Church" src="https://images.theconversation.com/files/503386/original/file-20230106-24-s9fgwp.JPG?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/503386/original/file-20230106-24-s9fgwp.JPG?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=450&fit=crop&dpr=1 600w, https://images.theconversation.com/files/503386/original/file-20230106-24-s9fgwp.JPG?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=450&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/503386/original/file-20230106-24-s9fgwp.JPG?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=450&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/503386/original/file-20230106-24-s9fgwp.JPG?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=566&fit=crop&dpr=1 754w, https://images.theconversation.com/files/503386/original/file-20230106-24-s9fgwp.JPG?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=566&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/503386/original/file-20230106-24-s9fgwp.JPG?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=566&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">Memorial to Richard Price in Newington Green Unitarian Church in North London.</span>
<span class="attribution"><span class="source">Jonathan Cardy</span>, <a class="license" href="http://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA</a></span>
</figcaption>
</figure>
<p>Given the collective amnesia towards him within Britain, it is perhaps apt that celebrations of Price’s life and works should begin this month with a talk at <a href="https://www.amphilsoc.org/events/electrifying-thinkers">the American Philosophical Association</a> in Philadelphia. </p>
<p>There will, however, be <a href="https://www.facebook.com/profile.php?id=100089200358334">a programme of events at home</a> to reflect on his contribution and contemporary relevance. </p>
<p><div data-react-class="Tweet" data-react-props="{"tweetId":"1599865290761785344"}"></div></p>
<p>This will include a birthday celebration in Llangeinor, an academic conference, and <a href="https://contemporancient.org/">a play</a>. </p>
<p>If he has not been celebrated by a British culture, for which he had such high hopes, then it is high time it happened in Wales, at the very least.</p><img src="https://counter.theconversation.com/content/197203/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Huw L Williams works for Cardiff University who are a lead partner in the 'Price 300' project celebrating Richard Price's tercentenary in 2023. His work as a philosopher is part-funded by the Coleg Cymraeg Cenedlaethol, a government-funded body responsible for promoting academic activity and teaching through the medium of Welsh. He is the President of the Adran Athroniaeth Cymdeithas Cynfyfyrwyr Prifysgol Cymru that promotes philosophy through the medium of Welsh and Welsh-language philosophy.</span></em></p>He was an important philosopher, mathematician and social reformer of his time. But Richard Price was subsequently written out of history.Huw L Williams, Reader in Political Philosophy, Cardiff UniversityLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/1905292022-12-28T21:09:29Z2022-12-28T21:09:29ZThe history and mystery of Tangram, the children’s puzzle game that harbours a mathematical paradox or two<figure><img src="https://images.theconversation.com/files/498377/original/file-20221201-20-cw8scj.jpg?ixlib=rb-1.1.0&rect=0%2C39%2C6630%2C4337&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">
</span> <span class="attribution"><span class="source">Shutterstock</span></span></figcaption></figure><p>Have you played the puzzle game Tangram? </p>
<p>I remember, as a child, being fascinated by how just seven simple wooden triangles and other shapes could offer endless entertainment. Unlike LEGO, the Tangram pieces do not snap together, and unlike the pieces of a jigsaw puzzle, they do not form a painted picture.</p>
<p>Instead, Tangram invites you to fit all the pieces together to form countless varieties of shapes. You can make your own shapes or you can try to form shapes that others have created. For instance, here’s one way to form a swan shape using Tangram pieces:</p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/498376/original/file-20221201-26-syova6.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="A swan shape in Tangram." src="https://images.theconversation.com/files/498376/original/file-20221201-26-syova6.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/498376/original/file-20221201-26-syova6.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=400&fit=crop&dpr=1 600w, https://images.theconversation.com/files/498376/original/file-20221201-26-syova6.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=400&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/498376/original/file-20221201-26-syova6.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=400&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/498376/original/file-20221201-26-syova6.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=503&fit=crop&dpr=1 754w, https://images.theconversation.com/files/498376/original/file-20221201-26-syova6.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=503&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/498376/original/file-20221201-26-syova6.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=503&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">This is one of several ways to make a swan shape using Tangram. Can you find another?</span>
<span class="attribution"><span class="source">Shutterstock</span></span>
</figcaption>
</figure>
<p>But it’s not the only way to make a swan. Can you find others? If you do not have the physical puzzle at hand, you can <a href="https://toytheater.com/tangram/">use</a> a virtual version of Tangram.</p>
<p>Tangram is accessible and yet challenging, and an excellent <a href="https://link.springer.com/article/10.1007/BF02354839">educational tool</a>. It’s still <a href="https://education.nsw.gov.au/teaching-and-learning/curriculum/mathematics/mathematics-curriculum-resources-k-12/mathematics-k-6-resources/how-to-make-a-tangram">used</a> in <a href="https://education.nsw.gov.au/teaching-and-learning/curriculum/mathematics/mathematics-curriculum-resources-k-12/mathematics-k-6-resources/how-to-make-a-tangram">schools</a> today to help illustrate mathematical concepts and develop mathematical thinking skills. It even features a paradox or two.</p>
<hr>
<p>
<em>
<strong>
Read more:
<a href="https://theconversation.com/5-math-skills-your-child-needs-to-get-ready-for-kindergarten-103194">5 math skills your child needs to get ready for kindergarten</a>
</strong>
</em>
</p>
<hr>
<h2>A long history of rearrangement puzzles</h2>
<p>Tangram is one of many rearrangement puzzles that have appeared throughout the ages. The earliest known rearrangement puzzle, the <a href="https://mathworld.wolfram.com/Stomachion.html">Stomachion</a>, was invented by Greek mathematician Archimedes 2,200 years ago and was popular for centuries among Greeks and Romans. </p>
<p>It consists of 14 puzzle pieces that can fit together in the form of many different shapes. There are <a href="https://mathweb.ucsd.edu/%7Efan/stomach/tour/stomach.html">536 different ways</a> to fit the pieces together as a square. </p>
<p>Then there’s the <a href="https://www.mathpuzzle.com/eternity.html">Eternity Puzzle</a>, released in 1999, which consists of 209 blue puzzle pieces that together form a big circle-like shape. It was very popular and sold <a href="https://en.wikipedia.org/wiki/Eternity_puzzle">500,000 copies</a> worldwide, perhaps due to the 1 million British pounds promised to whoever first solved it. </p>
<p>Less than a year later, the mathematicians Alex Selby and Oliver Riordan <a href="https://plus.maths.org/content/prize-specimens">solved the puzzle</a> and claimed the prize. The creator of the puzzle, the <a href="https://en.wikipedia.org/wiki/Christopher_Monckton,_3rd_Viscount_Monckton_of_Brenchley">controversial</a> Christopher Monckton, said at the time he had to <a href="http://news.bbc.co.uk/1/hi/entertainment/992393.stm">sell his house</a> to raise the prize money. </p>
<p>The origins of Tangram stretch back to the third century Chinese mathematician <a href="https://en.wikipedia.org/wiki/Liu_Hui">Liu Hui</a>. Among many other <a href="https://www.jstor.org/stable/2691200">accomplishments</a>, Liu Hui used rearrangements of geometrical shapes to elegantly explain mathematical facts such as the <a href="https://en.wikipedia.org/wiki/Pythagorean_theorem">Gougu Rule</a>, also known as Pythagoras’ Theorem. </p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/498587/original/file-20221201-16-l7rcjr.gif?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="Rearrangement proof of Pythagorean theorem" src="https://images.theconversation.com/files/498587/original/file-20221201-16-l7rcjr.gif?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/498587/original/file-20221201-16-l7rcjr.gif?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=345&fit=crop&dpr=1 600w, https://images.theconversation.com/files/498587/original/file-20221201-16-l7rcjr.gif?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=345&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/498587/original/file-20221201-16-l7rcjr.gif?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=345&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/498587/original/file-20221201-16-l7rcjr.gif?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=433&fit=crop&dpr=1 754w, https://images.theconversation.com/files/498587/original/file-20221201-16-l7rcjr.gif?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=433&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/498587/original/file-20221201-16-l7rcjr.gif?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=433&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">Shapes can be rearranged to explain the Gougu Rule, also known as Pythagoras’ Theorem.</span>
<span class="attribution"><a class="source" href="https://commons.wikimedia.org/wiki/File:Animated_gif_version_of_SVG_of_rearrangement_proof_of_Pythagorean_theorem.gif">Animation by William B. Faulk, CC BY-SA 4.0, via Wikimedia Commons</a>, <a class="license" href="http://creativecommons.org/licenses/by/4.0/">CC BY</a></span>
</figcaption>
</figure>
<p>This rearrangement approach to geometry was later evident in the creation of 12th century <a href="https://www.wired.com/2011/01/games-for-the-hands-and-mind-chinese-puzzles-at-the-moca/">Chinese banquet tables</a> (rectangular tables designed to be arranged into patterns that might please or entertain dinner guests).</p>
<p>A different version, known as a <a href="https://www.logicagiochi.com/en/the-history/">butterfly table</a>, was popularised in the early 17th century and featured a broader variety of shapes. A surviving table set can be seen in the <a href="https://www.chinadiscovery.com/jiangsu/suzhou/lingering-garden.html">Lingering Garden (Liu Yuan)</a> which is part of a <a href="https://whc.unesco.org/en/list/813/">UNESCO World Cultural Heritage</a> site in Suzhou.</p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/484166/original/file-20220913-1734-sepc5v.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="A Tangram puzzle lies on a table." src="https://images.theconversation.com/files/484166/original/file-20220913-1734-sepc5v.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/484166/original/file-20220913-1734-sepc5v.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=400&fit=crop&dpr=1 600w, https://images.theconversation.com/files/484166/original/file-20220913-1734-sepc5v.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=400&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/484166/original/file-20220913-1734-sepc5v.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=400&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/484166/original/file-20220913-1734-sepc5v.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=503&fit=crop&dpr=1 754w, https://images.theconversation.com/files/484166/original/file-20220913-1734-sepc5v.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=503&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/484166/original/file-20220913-1734-sepc5v.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=503&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">The Tangram was popularised as a puzzle game around the year 1800.</span>
<span class="attribution"><span class="source">Shutterstock</span></span>
</figcaption>
</figure>
<h2>The Tangram craze</h2>
<p>According to <a href="https://www.amazon.com/Tangram-Book-Jerry-Slocum/dp/1402704135">The Tangram Book</a> by Jerry Slocum and other authors, the Tangram was popularised as a puzzle game around the year 1800. </p>
<p>They report the inventor, an unknown Chinese person using the pen name Yang-Cho-Chu-Shih (“Dimwitted recluse”), published Ch'i chi'iao t'u (“Pictures Using Seven Clever Pieces”), a book containing hundreds of Tangram puzzle shapes. </p>
<figure class="align-right ">
<img alt="Patterns from a Tangram puzzle and solution books, China c. 1815 (British Library 15257.d.5, 15257.d.14)" src="https://images.theconversation.com/files/497605/original/file-20221128-18-d9qelw.png?ixlib=rb-1.1.0&q=45&auto=format&w=237&fit=clip" srcset="https://images.theconversation.com/files/497605/original/file-20221128-18-d9qelw.png?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=968&fit=crop&dpr=1 600w, https://images.theconversation.com/files/497605/original/file-20221128-18-d9qelw.png?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=968&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/497605/original/file-20221128-18-d9qelw.png?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=968&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/497605/original/file-20221128-18-d9qelw.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=1216&fit=crop&dpr=1 754w, https://images.theconversation.com/files/497605/original/file-20221128-18-d9qelw.png?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=1216&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/497605/original/file-20221128-18-d9qelw.png?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=1216&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">Patterns from a Tangram puzzle and solution books, China c. 1815 (British Library 15257.d.5, 15257.d.14)</span>
<span class="attribution"><span class="source">British Library</span></span>
</figcaption>
</figure>
<p>This sparked a craze for the game in China. Other Tangram puzzle books were soon published, with some eventually making their way to Japan, the United States and England, where they were translated and extended. </p>
<p>During 1817-18, the Tangram <a href="https://collections.libraries.indiana.edu/lilly/exhibitions_legacy/collections/overview/puzzle_docs/Tangram-Worlds_First_Puzz_Craze.pdf">craze</a> spread like <a href="https://www.puzzlemuseum.com/month/picm09/2009-03-early-tangram.htm">wildfire</a> to France, Denmark and other European countries. Worldwide interest in Tangram has endured ever since. </p>
<h2>An educational tool harbouring a paradox or two</h2>
<p>The lasting popularity of Tangram might partly be due to it allowing so many shapes with so few pieces. </p>
<p>Researchers have found that Tangram can help students’ <a href="https://journaljesbs.com/index.php/JESBS/article/view/765">visual and geometric thinking</a> and even their <a href="https://www.tandfonline.com/doi/abs/10.1080/15248372.2012.725186">arithmetic skills</a>.</p>
<p>Tangram may help in the assessment of children’s learning of <a href="https://journals.sagepub.com/doi/10.1177/2332858419829723">written languages</a> and of their <a href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6798005/">emotional regulation skills</a>.</p>
<p>For most people, though, Tangram is just a fun and creative challenge.</p>
<p>There are also some Tangram “paradox” puzzles discussed in <a href="https://www.amazon.com/Tangram-Book-Jerry-Slocum/dp/1402704135">The Tangram Book</a> and elsewhere online, where Tangram pieces are arranged to make two seeming identical shapes (but where one appears to have a leftover piece). </p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/498373/original/file-20221201-14-oyfja5.png?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="The Monk puzzle" src="https://images.theconversation.com/files/498373/original/file-20221201-14-oyfja5.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/498373/original/file-20221201-14-oyfja5.png?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=505&fit=crop&dpr=1 600w, https://images.theconversation.com/files/498373/original/file-20221201-14-oyfja5.png?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=505&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/498373/original/file-20221201-14-oyfja5.png?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=505&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/498373/original/file-20221201-14-oyfja5.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=634&fit=crop&dpr=1 754w, https://images.theconversation.com/files/498373/original/file-20221201-14-oyfja5.png?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=634&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/498373/original/file-20221201-14-oyfja5.png?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=634&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">The two monks Tangram paradox.</span>
<span class="attribution"><a class="source" href="https://commons.wikimedia.org/wiki/File:Two_monks_tangram_paradox.svg">AlphaZeta, CC0, via Wikimedia Commons</a>, <a class="license" href="http://creativecommons.org/licenses/by/4.0/">CC BY</a></span>
</figcaption>
</figure>
<p>Can you explain the “paradox” – why one has a triangular “foot” and the other does not, even though both images use all seven pieces? </p>
<p>As a bonus challenge, perhaps you can you solve the similar infinite chocolate bar “paradox” popularised on Instagram and TikTok.</p>
<p><iframe id="tc-infographic-795" class="tc-infographic" height="400px" src="https://cdn.theconversation.com/infographics/795/af484f026421a1a75b5436ba26c883774684659d/site/index.html" width="100%" style="border: none" frameborder="0"></iframe></p>
<p>Good luck and happy puzzling!</p>
<hr>
<p>
<em>
<strong>
Read more:
<a href="https://theconversation.com/learn-how-to-make-a-sonobe-unit-in-origami-and-unlock-a-world-of-mathematical-wonder-171390">Learn how to make a sonobe unit in origami – and unlock a world of mathematical wonder</a>
</strong>
</em>
</p>
<hr>
<img src="https://counter.theconversation.com/content/190529/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>The Conversation bought the author a Tangram set to play with so he could write this article.</span></em></p>Tangram is accessible yet challenging, and an excellent educational tool. It’s still used in schools today to help illustrate mathematical concepts and develop mathematical thinking skills.Thomas Britz, Senior Lecturer, UNSW SydneyLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/1416812020-07-27T12:13:50Z2020-07-27T12:13:50ZThe mystery of the missing portrait of Robert Hooke, 17th-century scientist extraordinaire<figure><img src="https://images.theconversation.com/files/346709/original/file-20200709-62-m6gz0v.jpg?ixlib=rb-1.1.0&rect=0%2C2%2C1613%2C1995&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">Known as Mary Beale's 'Portrait of a Mathematician,' could the circa 1680 painting depict Hooke?</span> <span class="attribution"><a class="source" href="https://research.tamu.edu/2019/10/02/has-an-am-biologist-found-one-of-the-holy-grails-of-science-history/">Mary Beale</a>, <a class="license" href="http://creativecommons.org/licenses/by/4.0/">CC BY</a></span></figcaption></figure><p>Groundbreaking discoveries in science often come with two iconic images, one representing the breakthrough and the other, the discoverer. For example, the page from Darwin’s notebook sketching <a href="https://en.wikipedia.org/wiki/Charles_Darwin#/media/File:Darwin_Tree_1837.png">the branching pattern of evolution</a> often accompanies a portrait of Darwin in <a href="https://en.wikipedia.org/wiki/Charles_Darwin#/media/File:Charles_Darwin_by_G._Richmond.png">his early years when the notebook was written</a>. Likewise the drawing of the <a href="https://upload.wikimedia.org/wikipedia/commons/d/d0/Sidereus_Nuncius_Medicean_Stars.jpg">orbits of the moons of Jupiter</a> often accompanies a <a href="https://en.wikipedia.org/wiki/Galileo_Galilei#/media/File:Justus_Sustermans_-_Portrait_of_Galileo_Galilei,_1636.jpg">portrait of Galileo</a>.</p>
<figure class="align-right zoomable">
<a href="https://images.theconversation.com/files/349267/original/file-20200723-33-1crssot.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="Original etching of cells from a piece of cork" src="https://images.theconversation.com/files/349267/original/file-20200723-33-1crssot.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=237&fit=clip" srcset="https://images.theconversation.com/files/349267/original/file-20200723-33-1crssot.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=800&fit=crop&dpr=1 600w, https://images.theconversation.com/files/349267/original/file-20200723-33-1crssot.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=800&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/349267/original/file-20200723-33-1crssot.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=800&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/349267/original/file-20200723-33-1crssot.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=1005&fit=crop&dpr=1 754w, https://images.theconversation.com/files/349267/original/file-20200723-33-1crssot.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=1005&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/349267/original/file-20200723-33-1crssot.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=1005&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">Hooke’s famous etching of the tiny magnified cells he saw in a piece of cork.</span>
<span class="attribution"><a class="source" href="https://commons.wikimedia.org/wiki/File:RobertHookeMicrographia1665.jpg">Robert Hooke, Micrographia, 1665/Wikimedia Commons</a>, <a class="license" href="http://creativecommons.org/licenses/by/4.0/">CC BY</a></span>
</figcaption>
</figure>
<p>Another groundbreaking discovery in science was the discovery of the cell by Robert Hooke (1635-1703). The iconic image of the breakthrough, published in the first scientific bestseller, 1665’s “Micrographia,” is <a href="https://www.archive.org/download/mobot31753000817897/page/n157_w523">an etching of the cells that make up a piece of cork</a>. It’s sliced two ways – across the grain and along the grain, showing not only the cells but also their polarity. However, there is no image of Hooke himself.</p>
<p>The absence of any contemporary portrait of Hooke stands out because he was a founding member, fellow, curator and secretary of the <a href="https://royalsociety.org/about-us/history/">Royal Society of London</a>, a group fundamental to the establishment of our current notion of experimental science and its reporting, which continues to the present day.</p>
<p>As an admirer of Hooke, I couldn’t resist putting aside my day job as a <a href="https://scholar.google.com/citations?user=Vdag4_4AAAAJ&hl=en&oi=sra">plant cell biology professor</a> to investigate what could be called the mystery of the missing portrait. And without even setting foot in an art gallery, I think <a href="https://doi.org/10.1111/jmi.12828">I’ve cracked the case</a>.</p>
<p>I started by following up a <a href="https://blogs.royalsociety.org/history-of-science/2010/12/03/hooke-newton-missing-portrait/">rumor behind its absence</a>, that none other than Isaac Newton was somehow involved in its suppression.</p>
<h2>What’s within the frame</h2>
<p>My hypothesis was that the portrait should show someone illustrating a mathematical principle for which Newton claimed credit – that could hint at a motive for why Newton might have suppressed a painting of a scientific rival.</p>
<p>The best candidate for the artist was the well-known portraitist <a href="https://en.wikipedia.org/wiki/Mary_Beale">Mary Beale</a>, whom Hooke knew and visited, although there’s no explicit record of him sitting for her. Amazingly, when I entered the search terms “Mary Beale mathematician” online, the first link that appeared was (and still is) her “<a href="http://www.historicalportraits.com/Gallery.asp?Page=Item&ItemID=126&Desc=Portrait-of-a-mathematician-%7C-Mary-Beale">Portrait of a Mathematician</a>.”</p>
<p>It matched the <a href="http://www.roberthooke.org.uk/leonardo.htm">physical description of Hooke</a> from contemporary sources: He was known to have gray eyes and natural brown hair that had “an excellent moist curl” and hung down over his forehead. The absence of a periwig indicates that the sitter is not nobility or of high social consequence; indeed, Hooke was one of the first professional scientists. Although he was known to have a disability, spinal curvature, the large mantle worn by the man in the painting would have covered it.</p>
<figure class="align-right zoomable">
<a href="https://images.theconversation.com/files/349269/original/file-20200723-25-cbj19a.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="Painted bust of a 17th century Dutch man" src="https://images.theconversation.com/files/349269/original/file-20200723-25-cbj19a.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=237&fit=clip" srcset="https://images.theconversation.com/files/349269/original/file-20200723-25-cbj19a.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=755&fit=crop&dpr=1 600w, https://images.theconversation.com/files/349269/original/file-20200723-25-cbj19a.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=755&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/349269/original/file-20200723-25-cbj19a.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=755&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/349269/original/file-20200723-25-cbj19a.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=949&fit=crop&dpr=1 754w, https://images.theconversation.com/files/349269/original/file-20200723-25-cbj19a.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=949&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/349269/original/file-20200723-25-cbj19a.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=949&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">Portrait by Mary Beale, believed to be chemist Jan Baptist van Helmont, not Hooke.</span>
<span class="attribution"><a class="source" href="https://commons.wikimedia.org/wiki/File:Jan_Baptist_van_Helmont_portrait.jpg">Mary Beale/Wikimedia Commons</a>, <a class="license" href="http://creativecommons.org/licenses/by/4.0/">CC BY</a></span>
</figcaption>
</figure>
<p>Art historians, however, believe that matching physical descriptions is insufficient to identify the sitter. This blunder was made by historian Lisa Jardine when in 2004 <a href="https://www.harpercollins.com/products/the-curious-life-of-robert-hooke-lisa-jardine">she misidentified a portrait</a> of 17th-century chemist Jan Baptist Van Helmont <a href="https://doi.org/10.1179/amb.2004.51.3.263">as being Hooke</a>.</p>
<p>So is there other evidence in the Beale painting besides the appearance of the sitter to support the idea that it depicts Hooke?</p>
<p>The sitter openly engages his audience and points to his drawing of elliptical motion. By digitally enhancing the online image, I found that the major lines match those of an <a href="http://physics.ucsc.edu/%7Emichael/hooke5.pdf">unpublished 1685 manuscript by Hooke</a> in which he geometrically proved that a central force that is a constant, or linear, function of the distance between two bodies produces an elliptical orbit.</p>
<p>In his 1687 “Principia Mathematica,” Newton proved the converse and claimed priority. The two men were at odds. Only Hooke possessed the drawing of his version of how things worked. It was starting to look like this painting indeed included visualizations of physics principles important to Newton and that he might not be eager to have on public display.</p>
<h2>Foregrounding clues from the background</h2>
<p>Beale painted a partial view of a device on the table to the man’s left. Completing the model reveals that it is an orrery – a mechanical model of the solar system – depicting Mercury, Venus and Earth elliptically orbiting the Sun. It’s a physical version of the drawing of elliptical motion also displayed on the table. To me, it provides further supporting evidence for the nature of the drawing and that this man is Hooke.</p>
<p>That Beale included the device is interesting in its own right because she painted this portrait decades before the <a href="https://en.wikipedia.org/wiki/Orrery">first modern orrery</a> was constructed in 1704 by an instrument maker and close collaborator of Hooke, Thomas Tompion. The instrument got its name from the 4th Earl of Orrery, a relative of <a href="https://en.wikipedia.org/wiki/Robert_Boyle">Robert Boyle</a> <a href="https://en.wikipedia.org/wiki/Robert_Boyle#/media/File:Boyle-hooke.jpg">for whom Hooke had worked</a> prior to his employment in the Royal Society. I believe she’s painted Hooke’s prototype of an orrery here. </p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/349462/original/file-20200725-25-io84zk.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="Metal model of the solar system from 1767" src="https://images.theconversation.com/files/349462/original/file-20200725-25-io84zk.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/349462/original/file-20200725-25-io84zk.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=400&fit=crop&dpr=1 600w, https://images.theconversation.com/files/349462/original/file-20200725-25-io84zk.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=400&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/349462/original/file-20200725-25-io84zk.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=400&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/349462/original/file-20200725-25-io84zk.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=503&fit=crop&dpr=1 754w, https://images.theconversation.com/files/349462/original/file-20200725-25-io84zk.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=503&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/349462/original/file-20200725-25-io84zk.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=503&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">Orreries became more common throughout the 18th century as models of the solar system, but none had been built yet at the time the portrait was painted.</span>
<span class="attribution"><a class="source" href="https://www.gettyimages.com/detail/news-photo/harvard-university-campus-the-department-of-the-history-of-news-photo/125328759">Dina Rudick/The Boston Globe via Getty Images</a></span>
</figcaption>
</figure>
<p>The landscape background, rare for Beale, presents a final clue. I hypothesized that Hooke, the city architect of London, had designed the buildings pictured in the painting. Consulting a list of Hooke’s architectural commissions from 1675-1685, the closest visual match was Lowther Castle and its Church of St. Michael. And indeed Hooke had redesigned the latter, with renovations completed in 1686.</p>
<p>The question then became whether Mary Beale could have sketched the castle and church. I was astonished to learn that she had received a remarkable commission for <a href="https://everything2.com/title/Mary+Beale">30 portraits from the Lowther family</a>, so indeed probably knew and sketched the castle and its grounds. </p>
<h2>A visual makeover for a 17th-century scientist</h2>
<p><a href="https://doi.org/10.1111/jmi.12828">If this is indeed Hooke</a>, the portrait provides an iconic image.</p>
<p>So where has it been for more than 300 years?</p>
<p>I turned to the rumor that Newton could have been involved in the portrait’s disappearance. The two scientists did have a quarrelsome history.</p>
<p>One big clash was over the nature of light. Hooke explained his experiments on color as light traveling in waves through thin sheets of the mineral mica. Newton explained his experiments on color as light traveling through prisms as corpuscles or particles. They argued – was light a wave or was it particles?</p>
<p>Newton claimed victory, but admitted, <a href="https://digitallibrary.hsp.org/index.php/Detail/objects/9792">“If I have seen further it is by standing on the sholders [sic] of Giants”</a> – an unfortunate turn of phrase, given Hooke’s pronounced curvature of the spine. At at any rate, they were both at least partially right: Physicists today appreciate the wave-particle duality of light.</p>
<p>[<em>Deep knowledge, daily.</em> <a href="https://theconversation.com/us/newsletters/the-daily-3?utm_source=TCUS&utm_medium=inline-link&utm_campaign=newsletter-text&utm_content=deepknowledge">Sign up for The Conversation’s newsletter</a>.]</p>
<p>Then there was the dispute perhaps alluded to in the portrait, about the elliptical orbits of the planets. Hooke claimed in 1684 that he could mathematically demonstrate what’s known as Kepler’s first law, which Newton published in his famous “Principia Mathematica” (1687). The upshot was that Newton removed mention of Hooke’s important contributions from his book – and they never got along again.</p>
<p>Hooke died in 1703, the same year that Newton became president of the Royal Society. There is no record for Royal Society ownership of this Beale painting. All Newton had to do was leave it behind when the Society moved official residence in 1710, thereby ridding himself (and history) of hard evidence of Hooke’s claim. </p>
<p>Where the painting has been during the intervening centuries is a matter of conjecture. When it first came to light at a Christie’s auction in the 1960s, it was ironically labeled as a portrait of Isaac Newton. Sotheby’s, the last public auctioneer of the work in 2006, has not revealed the buyer’s identity. I hope the current owner comes forward and sells the portrait to the Royal Society. That’s where it belongs, at long last. I would love to see the original.</p>
<p><em>This article has been updated to clarify Hooke’s work on elliptical orbits.</em></p><img src="https://counter.theconversation.com/content/141681/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Larry Griffing does not work for, consult, own shares in or receive funding from any company or organization that would benefit from this article, and has disclosed no relevant affiliations beyond their academic appointment.</span></em></p>Online sleuthing and deductive reasoning identifies what appears to be the only existent portrait painted of the celebrated scientist during his lifetime.Larry Griffing, Associate Professor of Biology, Texas A&M UniversityLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/1324042020-02-25T14:43:03Z2020-02-25T14:43:03ZKatherine Johnson: NASA mathematician and much-needed role model<figure><img src="https://images.theconversation.com/files/317098/original/file-20200225-24680-8cth84.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">
</span> <span class="attribution"><span class="source">NASA/Bob Nye</span></span></figcaption></figure><p>Katherine Johnson, <a href="https://www.theguardian.com/science/2020/feb/24/katherine-johnson-obituary">who has died</a> at the age of 101, was an <a href="https://en.wikipedia.org/wiki/Katherine_Johnson">amazing woman</a>. But up until a few years ago, hardly anyone had heard of her or her achievements. She was a mathematician and she worked for NASA. But on paper neither of those facts would make her stand out from the crowd. Add a few more facts – she was a woman, she was black and working in the US in the 1950s to early 1960s – and the scale of her success becomes more apparent. </p>
<p>Johnson’s story and significant contributions to the US space programme, along with those of Dorothy Vaughan (a computer scientist) and Mary Jackson (an engineer), were brought to widespread public attention by the <a href="https://www.space.com/34486-hidden-figures-author-margot-shetterly-interview.html">2016 book Hidden Figures</a> by <a href="http://margotleeshetterly.com/">Margot Lee Shetterly</a> and <a href="https://theconversation.com/hidden-figures-takes-us-back-to-a-time-when-computers-were-people-women-and-black-72303">film of the same name</a>.</p>
<p>I have rarely watched a film that has moved me as much as Hidden Figures did when I first saw it. And I have seen it at least twice since when I have led discussions about the significance of the film, drawing on my own experience of working in the space industry. In telling Johnson and her colleagues’ stories, the film shed light not only on advances in technology but also the status of black people in society and the role of women in the workplace and in science. </p>
<p>Katherine Coleman was born in 1918 in West Virginia and showed very early on that she was no ordinary child. Her ability in mathematics was such that she continued her schooling beyond high school (very unusual for African-American children at that time) and had graduated from college by the time she was 18. Katherine became a wife, a mother and a teacher, and her story might have ended there, if it hadn’t been for her drive to continue with her mathematics. </p>
<h2>Human computer</h2>
<p>In the 1950s, the US government was continuing to develop its flight capabilities, for which it required computers. Not the super-fast electronic technology of today, or even lumbering mechanical valve-driven machinery, but people. Johnson became one of a group of human computers who calculated (using slide rule and log tables) the flight dynamics of aircraft to help improve their safety and operation. </p>
<p>In 1958, she joined the newly formed NASA, where she calculated the flight trajectory for the missions of first American in space Alan Shepard and first American to orbit the Earth, John Glenn. Glenn <a href="https://www.nasa.gov/content/katherine-johnson-biography">apparently personally requested</a> that Johnson verify the flight trajectory that had been worked out by one of the new electronic computers. She would later work on the Apollo moon missions, helping synch the lunar module with the orbiting command and service module, and then the space shuttle programme. </p>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/317104/original/file-20200225-24672-1z0nb44.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/317104/original/file-20200225-24672-1z0nb44.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=571&fit=crop&dpr=1 600w, https://images.theconversation.com/files/317104/original/file-20200225-24672-1z0nb44.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=571&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/317104/original/file-20200225-24672-1z0nb44.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=571&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/317104/original/file-20200225-24672-1z0nb44.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=718&fit=crop&dpr=1 754w, https://images.theconversation.com/files/317104/original/file-20200225-24672-1z0nb44.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=718&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/317104/original/file-20200225-24672-1z0nb44.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=718&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">Johnson calculated the trajectory for the first US manned orbital space flight.</span>
<span class="attribution"><a class="source" href="https://www.nasa.gov/image-feature/liftoff-of-john-glenns-friendship-7-feb-20-1962">NASA</a></span>
</figcaption>
</figure>
<p>But Johnson was also a “<a href="https://www.newscientist.com/article/2118526-when-computers-were-human-the-black-women-behind-nasas-success/">coloured computer</a>” at a time when laws still enforced racial segregation and there was still much opposition to integration and equal rights for non-white US citizens. As such, she had to use the separate restrooms and eating facilities set aside for non-white staff. </p>
<p>She was also a woman working in a man’s world, a world where most of the staff wore suit, shirt and tie and left for home each evening to find dinner cooked and waiting for them. Katherine had to juggle home and work, like so many women today. But her workplace was 1950s and 60s NASA, where women’s place was lowly. They didn’t speak at meetings or get their names acknowledged as authors of reports. As depicted in Hidden Figures, Johnson demanded to be recognised for her work. And she was. Eventually.</p>
<p>In her final years, she received much acclaim, including the award of the <a href="https://www.nasa.gov/image-feature/langley/katherine-johnson-receives-presidential-medal-of-freedom">Presidential Medal of Freedom</a> by Barack Obama, as well, of course, as the publicity of Hidden Figures. But for a long time women didn’t have such visible role models working in science or space.</p>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/317100/original/file-20200225-24694-w0ovwu.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/317100/original/file-20200225-24694-w0ovwu.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=415&fit=crop&dpr=1 600w, https://images.theconversation.com/files/317100/original/file-20200225-24694-w0ovwu.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=415&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/317100/original/file-20200225-24694-w0ovwu.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=415&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/317100/original/file-20200225-24694-w0ovwu.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=521&fit=crop&dpr=1 754w, https://images.theconversation.com/files/317100/original/file-20200225-24694-w0ovwu.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=521&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/317100/original/file-20200225-24694-w0ovwu.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=521&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">Katherine Johnson received the Presidential Medal of Freedom.</span>
<span class="attribution"><a class="source" href="https://www.nasa.gov/image-feature/langley/katherine-johnson-receives-presidential-medal-of-freedom">NASA/Bill Ingalis</a></span>
</figcaption>
</figure>
<p>I should hesitate to mention my career in an article lauding the achievements of Johnson. I have not had the same barriers to progress that she had and I have been fortunate that the people I have worked with have never patronised or ignored me in the way that Johnson was treated. </p>
<p>And the space industry has come a long way in the half century that it has existed. We have rules about equality and discrimination and dozens of schemes set up to <a href="https://www.gov.uk/government/news/uk-space-agency-joins-women-in-aerospace-europe-organisation">encourage diversity</a> in the workplace. </p>
<h2>Making your voice heard</h2>
<p>But yesterday I received an invitation to a meeting of UK senior space scientists and engineers. There were 20 names on the list, only three of which were women. It will be a gathering of grey suits. I shall wear pink or bright yellow. Because it is still necessary to stand out to make your voice heard. And I am a confident and successful scientist.</p>
<p>It is stories like Johnson’s that need to be told. Where are our role models today? Where are the women who will inspire our students to become scientists and engineers? As an example, in 2019 the BBC published a list of <a href="https://www.bbc.co.uk/news/world-50042279">100 trailblazing women</a>, of which only four were scientists and just one an engineer.</p>
<p>Johnson has left an amazing legacy: as a mathematician, she helped NASA to put humans into space. But as an African-American woman, her legacy is perhaps even greater. She has given us a role model, showing that if we have the determination, our skills and talents can take us as high as we wish to fly.</p><img src="https://counter.theconversation.com/content/132404/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Monica Grady works for the Open University. She is a Senior Research Fellow at the Natural History Museum in London and Chancellor of Liverpool Hope University. She receives funding from the UK Space Agency, STFC and the European Space Agency</span></em></p>The pioneering African-American “computer” has died aged 101.Monica Grady, Professor of Planetary and Space Sciences, The Open UniversityLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/1228452019-12-07T17:22:31Z2019-12-07T17:22:31ZNicolas Bourbaki: The greatest mathematician who never was<figure><img src="https://images.theconversation.com/files/305022/original/file-20191203-67017-b4wfus.png?ixlib=rb-1.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">Bourbaki Congress of 1938.</span> <span class="attribution"><a class="source" href="https://commons.wikimedia.org/wiki/File:Bourbaki_congress1938.png">Wikimedia</a></span></figcaption></figure><p>By many measures, Nicolas Bourbaki ranks among the greatest mathematicians of the 20th century. </p>
<p>Largely unknown today, Bourbaki is likely the last mathematician to master nearly all aspects of the field. A consummate collaborator, he made fundamental contributions to important mathematical fields such as set theory and functional analysis. He also revolutionized mathematics by emphasizing rigor in place of conjecture. </p>
<p>There’s just one problem: Nicolas Bourbaki never existed.</p>
<h2>Never existed?</h2>
<p>While it is now widely accepted that there never was a Nicolas Bourbaki, there is evidence to the contrary. </p>
<p>For example, there are <a href="http://www.neverendingbooks.org/when-was-the-bourbaki-wedding">wedding announcements</a> for his daughter Betty, a baptismal certificate in his name and <a href="https://doi.org/10.1007/BF03028596">an impressive family lineage</a> extending back to an ancestor Napoleon raised as his own son.</p>
<figure class="align-left zoomable">
<a href="https://images.theconversation.com/files/305024/original/file-20191203-67028-1g5a38v.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/305024/original/file-20191203-67028-1g5a38v.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=237&fit=clip" srcset="https://images.theconversation.com/files/305024/original/file-20191203-67028-1g5a38v.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=830&fit=crop&dpr=1 600w, https://images.theconversation.com/files/305024/original/file-20191203-67028-1g5a38v.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=830&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/305024/original/file-20191203-67028-1g5a38v.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=830&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/305024/original/file-20191203-67028-1g5a38v.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=1044&fit=crop&dpr=1 754w, https://images.theconversation.com/files/305024/original/file-20191203-67028-1g5a38v.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=1044&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/305024/original/file-20191203-67028-1g5a38v.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=1044&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">The cover of the first volume in Bourbaki’s textbook.</span>
<span class="attribution"><a class="source" href="https://commons.wikimedia.org/wiki/File:Bourbaki,_Theorie_des_ensembles_maitrier.jpg">Maitrier/Wikimedia</a>, <a class="license" href="http://creativecommons.org/licenses/by-sa/4.0/">CC BY-SA</a></span>
</figcaption>
</figure>
<p><a href="https://www.maa.org/press/maa-reviews/mathematical-apocrypha">Even the professional mathematics community</a> was misled for a time. When Ralph Boas, an editor of the journal Mathematical Reviews, wrote that Bourbaki was a pseudonym, he was promptly refuted by none other than Bourbaki himself. Bourbaki countered with a letter stating that B.O.A.S. actually just was an acronym of the last names of the editors of the Reviews. </p>
<p>These cases of confused identity were not all fun and games. For example, <a href="https://enacademic.com/dic.nsf/enwiki/912">it is alleged</a> that, while visiting Finland at the outset of World War II, French mathematician André Weil was investigated for spying. The authorities found suspicious papers in his possession: a fake identity, a set of business cards and even invitations from the Russian Academy of Science – all in Bourbaki’s name. Supposedly, Weil was freed only after an officer recognized him as a preeminent mathematician. </p>
<h2>Who was Bourbaki?</h2>
<p>If Bourbaki never existed, who – or what – was he?</p>
<p>The name <a href="http://www.mcs.csueastbay.edu/%7Emalek/Mathlinks/Bourbaki.html">Nicolas Bourbaki</a> first appeared in a place rocked by turmoil at a volatile time in history: Paris in 1934.</p>
<p>World War I had wiped out a generation of French intellectuals. As a result, the standard university-level calculus textbook had been written more than two and half decades before and was out of date.</p>
<p><a href="https://www.ams.org/notices/199803/borel.pdf">Newly minted professors André Weil and Henri Cartan</a> wanted a rigorous method to teach Stokes’ theorem, a key result of calculus. <a href="https://doi.org/10.1007/BF03025255">After realizing that others had similar concerns</a>, Weil organized a meeting. It took place December 10, 1934 at a Parisian café called Capoulade.</p>
<figure class="align-left zoomable">
<a href="https://images.theconversation.com/files/305690/original/file-20191206-90588-1ffppop.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/305690/original/file-20191206-90588-1ffppop.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=237&fit=clip" srcset="https://images.theconversation.com/files/305690/original/file-20191206-90588-1ffppop.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=402&fit=crop&dpr=1 600w, https://images.theconversation.com/files/305690/original/file-20191206-90588-1ffppop.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=402&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/305690/original/file-20191206-90588-1ffppop.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=402&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/305690/original/file-20191206-90588-1ffppop.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=505&fit=crop&dpr=1 754w, https://images.theconversation.com/files/305690/original/file-20191206-90588-1ffppop.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=505&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/305690/original/file-20191206-90588-1ffppop.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=505&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">Cafe Capoulade in 1943.</span>
<span class="attribution"><a class="source" href="https://en.wikipedia.org/wiki/File:Bundesarchiv_Bild_101I-247-0775-09,_Deutsche_Soldaten_in_einem_Pariser_Stra%C3%9Fencaf%C3%A9.jpg">Langhaus, German Federal Archive/Wikimedia</a>, <a class="license" href="http://creativecommons.org/licenses/by-sa/4.0/">CC BY-SA</a></span>
</figcaption>
</figure>
<p>The nine mathematicians in attendance <a href="https://books.google.com/books?id=IieQDwAAQBAJ&ppis=_e&dq=%22to+define+for+25+years+the+syllabus+for+the+certificate+in+differential+and+integral+calculus+by+writing,+collectively,+a+treatise+on+analysis,%E2%80%9D&source=gbs_navlinks_s">agreed to write a textbook</a> “to define for 25 years the syllabus for the certificate in differential and integral calculus by writing, collectively, a treatise on analysis,” which they hoped to complete in just six months. </p>
<p>As a joke, they named themselves after an <a href="https://www.bourbakipanorama.ch/en/museum/history/">old French general</a> who had been duped in the Franco-Prussian war.</p>
<p>As they proceeded, their original goal of elucidating Stokes’ theorem expanded to laying out the foundations of all mathematics. Eventually, they began to hold regular Bourbaki “conferences” three times a year to discuss new chapters for the treatise. </p>
<p>Individual members were encouraged to engage with all aspects of the effort, to ensure that the treatise would be accessible to nonspecialists. <a href="https://www.maa.org/sites/default/files/pdf/upload_library/22/Ford/Dieudonne(2).pdf">According to one of the founders</a>, spectators invariably came away with the impression that they were witnessing “a gathering of madmen.” They could not imagine how people, shouting – “sometimes three or four at the same time” – could ever come up with something “intelligent.”</p>
<p>Top mathematicians from across Europe, intrigued by the group’s work and style, joined to augment the group’s ranks. Over time, the name Bourbaki became a collective pseudonym for dozens of influential mathematicians spanning generations, including Weil, Dieudonne, Schwartz, Borel, Grothendieck and many others. </p>
<p>Since then, the group which has added new members over time, has proved to have a profound impact on mathematics, certainly rivaling any of its individual contributors.</p>
<h2>Profound impact</h2>
<p>Mathematicians have made a plethora of important contributions under Bourbaki’s name. </p>
<p>To name a few, the group introduced the null set symbol; the ubiquitous terms injective, surjective, bijective; and generalizations of many important theorems, including the Bourbaki-Witt theorem, the <a href="https://doi.org/10.1016/0315-0860(78)90136-2">Jacobson-Bourbaki</a> theorem and the Bourbaki-Banach-Alaoglu theorem. </p>
<p>Their text, “Elements of Mathematics,” has swelled to more than 6,000 pages. It provides a “<a href="https://doi.org/10.1007/s00283-017-9763-5">solid foundation for the whole body of modern mathematics</a>,” according to mathematician Barbara Pieronkiewicz.</p>
<figure class="align-right zoomable">
<a href="https://images.theconversation.com/files/305589/original/file-20191206-90574-9zerab.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/305589/original/file-20191206-90574-9zerab.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=237&fit=clip" srcset="https://images.theconversation.com/files/305589/original/file-20191206-90574-9zerab.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=906&fit=crop&dpr=1 600w, https://images.theconversation.com/files/305589/original/file-20191206-90574-9zerab.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=906&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/305589/original/file-20191206-90574-9zerab.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=906&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/305589/original/file-20191206-90574-9zerab.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=1138&fit=crop&dpr=1 754w, https://images.theconversation.com/files/305589/original/file-20191206-90574-9zerab.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=1138&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/305589/original/file-20191206-90574-9zerab.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=1138&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
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<span class="caption">The Henri Poincaré Institute, where Bourbaki seminars are regularly held.</span>
<span class="attribution"><a class="source" href="https://commons.wikimedia.org/wiki/File:Institut_Henri-Poincar%C3%A9.jpg">Antoine Taveneaux/Wikimedia</a>, <a class="license" href="http://creativecommons.org/licenses/by-sa/4.0/">CC BY-SA</a></span>
</figcaption>
</figure>
<p>Bourbaki’s influence is still alive and well. Now in “his” 80th year of research, in 2016 “he” published the 11th volume of the “Elements of Mathematics.” The Bourbaki group, with its ever-changing cast of members, still holds <a href="http://www.bourbaki.ens.fr/seminaires/2020/index.html#seminaire">regular seminars</a> at the University of Paris.</p>
<p>Partly thanks to the breadth and significance of “his” mathematical contributions, and also because – ageless, unchanging and operating in multiple places at once – “he” seems to defy the very laws of physics, Bourbaki’s mathematical prowess will likely never be equaled.</p>
<p>[ <em>Get the best of The Conversation, every weekend.</em> <a href="https://theconversation.com/us/newsletters/weekly-highlights-61?utm_source=TCUS&utm_medium=inline-link&utm_campaign=newsletter-text&utm_content=weeklybest">Sign up for our weekly newsletter</a>. ]</p><img src="https://counter.theconversation.com/content/122845/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>David Gunderman does not work for, consult, own shares in or receive funding from any company or organization that would benefit from this article, and has disclosed no relevant affiliations beyond their academic appointment.</span></em></p>Largely unknown today, Bourbaki was the last mathematician to master nearly all aspects of the field. There’s just one problem: Bourbaki never existed.David Gunderman, Ph.D. Candidate in Applied Mathematics, University of Colorado BoulderLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/1143452019-03-29T10:43:32Z2019-03-29T10:43:32ZWant to fix gerrymandering? Then the Supreme Court needs to listen to mathematicians<figure><img src="https://images.theconversation.com/files/266364/original/file-20190328-139364-3lxfco.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">Activists at the Supreme Court opposed to partisan gerrymandering hold up representations of congressional districts from North Carolina, left, and Maryland, right.</span> <span class="attribution"><a class="source" href="http://www.apimages.com/metadata/Index/Supreme-Court-Redistricting/c09bf9952d4b4906b03efdcacdf935de/6/0">AP Photo/Carolyn Kaster</a></span></figcaption></figure><p>“Are we in Maryland’s third congressional district?” Karen asked on a recent visit to the UMBC campus. Despite zooming into the district’s map on Wikipedia, neither of us could tell. With good reason – “<a href="https://www.washingtonpost.com/news/wonk/wp/2014/05/15/americas-most-gerrymandered-congressional-districts/">the praying mantis</a>,” as the third has been called, has one of the most flagrantly gerrymandered boundaries in the country. (The university sits just outside, as we later found.)</p>
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<span class="caption">Maryland’s third congressional district.</span>
<span class="attribution"><a class="source" href="https://en.wikipedia.org/wiki/Maryland%27s_3rd_congressional_district#/media/File:Maryland_US_Congressional_District_3_(since_2013).tif">Wikimedia</a></span>
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<p>Welcome to Democrat-controlled Maryland. The state, along with Republican-controlled North Carolina, defended its congressional districting against the charge of unlawful partisan gerrymandering in hearings <a href="https://www.nytimes.com/2019/03/26/us/politics/gerrymandering-supreme-court.html">at the U.S. Supreme Court on March 26</a>. </p>
<p>One might think that a map that confounds two mathematicians must be in clear violation of the law. Indeed, political scientists and mathematicians have worked together to propose several <a href="https://www.ams.org/journals/notices/201801/rnoti-p37.pdf">geometrical criteria</a> for drawing voting districts of logical contiguous shapes, which are now in use in various U.S. states.</p>
<p>But here’s the rub: Gerrymandering in itself is not unconstitutional. For the Supreme Court to rule against a particular map, plaintiffs need to establish that the map infringes on some constitutional right, such as their right to equal protection or free expression. This creates a problem. Geometrical criteria don’t detect partisanship. Other traditional criteria, like ensuring each district has the same population, can also be easily satisfied in an otherwise unfairly designed state map.</p>
<p>How then to define a standard to identify partisan gerrymandering that is egregious enough to be illegal? Mathematical scientists have already come up with promising solutions, but we are concerned that the Supreme Court may not take their advice when it issues its decision in June.</p>
<h2>Searching for answers</h2>
<p>The Supreme Court has grappled with the question of manageable standards at least since 1986 – long enough for Justice Antonin Scalia to <a href="https://scholar.google.com/scholar_case?case=16656282825028631654&hl=en&as_sdt=6&as_vis=1&oi=scholarr">declare in a 2004 ruling</a> that since one hadn’t emerged yet, the issue of partisan gerrymandering was not legally decidable, and therefore, no further appeals should be considered. </p>
<p>It was only Justice Anthony Kennedy’s separate concurrence that kept the door open. He cautioned against abandoning the search for a standard too soon, saying that “technology is both a threat and a promise.” In other words, technological advances would probably exacerbate the gerrymandering problem, but they could also provide a solution.</p>
<p>The problem has worsened, just as Kennedy predicted. Computer programs can now generate a profusion of redistricted maps, all of which satisfy traditional constraints such as contiguity and equal population across districts. Then, the majority party can just pick the map most favorable to it.</p>
<p>This was demonstrated in <a href="https://www.nytimes.com/2017/10/06/opinion/sunday/computers-gerrymandering-wisconsin.html">Wisconsin’s 2018 elections</a>. Computer-boosted gerrymandered maps supersized the Republicans’ 13-seat edge to a 25-seat majority, even though <a href="https://www.jsonline.com/story/news/blogs/wisconsin-voter/2018/12/06/wisconsin-gerrymandering-data-shows-stark-impact-redistricting/2219092002/">Democrats won 53 percent</a> of the total statewide vote. </p>
<p>We expect new congressional districts drawn countrywide after the 2020 census will be subject to even more ferocious computer-driven gerrymandering.</p>
<h2>Math to the rescue</h2>
<p>But the second part of Kennedy’s prediction has also come true. The same tools that produce drastically gerrymandered maps can be used to draw fair maps. </p>
<p>The first step is to generate – without partisan intent – a vast number of maps that adhere to traditional redistricting criteria. This creates a database against which any proposed map can be compared, by using a suitable mathematical formula that measures partisanship. Through this process, maps with extreme bias will appear as clear outliers, much like data points near the outer ends of a bell curve. </p>
<p>The <a href="https://www.policymap.com/2017/08/a-deeper-look-at-gerrymandering/">“efficiency gap”</a> is one such mathematical formula. It measures how efficiently one party’s votes get used and how much the other party’s votes get wasted. For example, a map might pack voters together to minimize their influence in other districts, or spread them out so they don’t form an effective bloc. </p>
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<p>Alternative formulas exist as well. In fact, we recommend using a collection of formulas, rather than just one, to compensate for the limitations of each. </p>
<p><a href="https://www.samsi.info/programs-and-activities/research-workshops/quantitative-redistricting/">Recent conferences</a> on redistricting have seen the mathematics and statistics communities coalesce around this “outlier approach.” </p>
<h2>Overcoming skepticism</h2>
<p>Getting the Supreme Court to accept this approach, however, will require overcoming the skepticism some conservative justices have expressed toward the use of mathematics and statistics in setting legal standards. </p>
<p>During <a href="https://www.supremecourt.gov/oral_arguments/argument_transcripts/2017/16-1161_mjn0.pdf">October 2017 oral arguments</a> for a challenge to the Wisconsin maps, for instance, Chief Justice John Roberts characterized the efficiency gap as “sociological gobbledygook,” while Justice Neil Gorsuch said that the idea of using multiple formulas for measuring gerrymandering was like adding “a pinch of this, a pinch of that” to his steak rub. Roberts also fretted that the country would dismiss statistical formulas as “a bunch of baloney” and suspect the court of political favoritism in adopting them.</p>
<p>At the <a href="https://www.supremecourt.gov/oral_arguments/argument_transcripts/2018/18-422_5hd5.pdf">March 26 hearings</a> for the North Carolina challenge, conservative justices were more measured and mathematically savvy in expressing their reservations. This time, the “outlier approach” took center stage. Affirmed in the <a href="https://www.brennancenter.org/sites/default/files/legal-work/CC_LWV_v_Rucho_MemorandumOpinion_01.09.18.pdf">lower court decision</a> and explained in an <a href="https://www.brennancenter.org/sites/default/files/legal-work/2019-02-12-Grofman%20Amicus%20Brief.pdf">amicus brief</a>, it was also endorsed in oral arguments by Justices Elena Kagan and Sonia Sotomayor. The key doubts came from Justices Samuel Alito, Gorsuch and Brett Kavanaugh, who questioned the feasibility of defining an “outlier” in practice – in particular, setting a range of numerical parameters that would demarcate permissible maps from nonpermissible ones. </p>
<p>The answer to such objections, adeptly addressed in an <a href="https://www.supremecourt.gov/DocketPDF/18/18-422/91120/20190307163214118_18-422%20Brief%20for%20Amicus%20Curiae%20Eric%20S.%20Lander.pdf?">amicus brief by MIT’s Eric Lander</a>, is twofold. First, the maps being challenged are so biased that they are extreme outliers. They would show up as anomalies under any test for partisanship. So there is no need for the Supreme Court to set a numerical cutoff level at this stage – though a threshold may, indeed, evolve in the future. Secondly, such an extreme outlier approach is already an indispensable tool in several areas of national importance. For example, it is used to <a href="https://mcnp.lanl.gov/pdf_files/la-ur-09-3136.pdf">test nuclear safety</a>, <a href="https://www.wsj.com/articles/as-forecasts-go-you-can-bet-on-monte-carlo-1470994203">predict hurricanes</a> and <a href="https://www.occ.treas.gov/news-issuances/news-releases/2019/nr-occ-2019-13.html">assess the health of financial institutions</a>.</p>
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<span class="caption">Partisan gerrymandering has also been a hot topic in Pennsylvania.</span>
<span class="attribution"><a class="source" href="http://www.apimages.com/metadata/Index/Election-2018-Redistricting/0aa1769955524602abaa931cb00a61f4/23/0">AP Photo/Keith Srakocic</a></span>
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<p>Moreover, this approach has already been shown to work smoothly in gerrymandering cases as well, such as in <a href="https://www.axios.com/pennsylvania-gerrymandering-tom-wolf-congressional-map-e4a1a2d0-65eb-4282-b048-811c2a3865b9.html">one from Pennsylvania</a>. Moon Duchin, a Tufts University math professor, used it to analyze – in a report requested by Governor Tom Wolf – newly proposed maps for fairness. A map drawn by the GOP state legislature clearly stood out as an extreme outlier among over a billion maps generated, both when evaluated using the efficiency gap and under another measure of partisanship called the mean-median score. Based on <a href="https://www.governor.pa.gov/wp-content/uploads/2018/02/md-report.pdf">Duchin’s report</a>, the governor rejected the map proposed by the GOP. </p>
<p>We expect that, pushed by citizen groups, an increasing number of states will incorporate math into redistricting procedures. Last year, for instance, Missouri approved <a href="https://www.kmov.com/news/missouri-amendment-explained/article_b4cace6a-d3de-11e8-9470-4b2875ebf73c.html">Amendment 1</a>, prescribing <a href="https://www.sos.mo.gov/CMSImages/Elections/Petitions/2018-048.pdf">detailed mathematical rules</a> that must be followed to ensure the fairness of redrawn districts. Although the rules rely heavily on just the efficiency gap – and lawmakers may try to <a href="https://www.columbiamissourian.com/news/state_news/voters-approved-clean-missouri-but-lawmakers-want-them-to-reconsider/article_4a4739e4-404d-11e9-b735-bfff863b5ed4.html">altogether annul them</a> – the fact that ordinary citizens <a href="https://www.columbiamissourian.com/news/elections/amendment-voters-strongly-support-clean-missouri-redistricting-plan-ethics-reform/article_6d2a9728-e155-11e8-b58e-43f1d7945f4e.html">voted overwhelmingly</a> (62 percent to 38 percent) in favor of such a math-incorporating measure is truly precedent-setting.</p>
<p>Such developments were noted in the March 26 oral arguments, when some justices wondered whether, in light of state initiatives, the Supreme Court really had to step in. As the citizens’ attorneys pointed out, however, there are very few states east of the Mississippi where such citizen initiatives are allowed. (North Carolina is not one of them.) It behooves the court to take the lead nationally. </p>
<p>Enhanced by computer power, partisan gerrymandering poses a burgeoning threat to the American way of democracy. Workable standards based on sound mathematical principles may be the only tools to counter this threat. We urge the Supreme Court to be receptive to such standards, thereby enabling citizens to protect their right to fair representation.</p><img src="https://counter.theconversation.com/content/114345/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>In the past I have volunteered for League of Women Voters, and serve as a redistricting resource for Common Cause MN. I also served on the Minnesota Citizens Redistricting Commission in 2010-11. I served as a AAAS Congressional Fellow in 2013-14 with Senator Al Franken (worked on higher education policy).
I currently work for the American Mathematical Society, a non-profit, which has put out a statement on redistricting. We are not registered lobbyists.
</span></em></p><p class="fine-print"><em><span>Manil Suri does not work for, consult, own shares in or receive funding from any company or organization that would benefit from this article, and has disclosed no relevant affiliations beyond their academic appointment.</span></em></p>Supreme Court justices have previously called statistical methods of measuring partisan gerrymandering ‘sociological gobbledygook’ and ‘a bunch of baloney.’Manil Suri, Professor of Mathematics and Statistics, University of Maryland, Baltimore CountyKaren Saxe, Professor of Mathematics, Emerita, Macalester CollegeLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/1008872018-08-01T20:19:15Z2018-08-01T20:19:15ZAn Australian takes top honours in the prestigious Fields Medal in mathematics<figure><img src="https://images.theconversation.com/files/230155/original/file-20180801-136655-a4dm6y.jpg?ixlib=rb-1.1.0&rect=4%2C109%2C2912%2C2072&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">Australian mathematician Akshay Venkatesh honoured in this year's Fields Medals.</span> </figcaption></figure><p>Australian mathematician Akshay Venkatesh is one of four winners of the Fields Medal, <a href="https://www.mathunion.org/imu-awards/fields-medal/fields-medals-2018">announced overnight</a> at the International Congress of Mathematics, in Brazil.</p>
<p>The 36-year-old now becomes only the second Australian to win a Fields Medal, described as the Nobel Prize for mathematics. The previous Australian winner was <a href="http://www.math.ucla.edu/%7Etao/">Terence Tao</a> in 2006.</p>
<p>The <a href="https://www.mathunion.org/imu-awards/fields-medal">medal</a> is awarded every four years to researchers under 40 years old. It recognises their outstanding mathematical achievement for existing work and for the promise of future achievement. </p>
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Read more:
<a href="https://theconversation.com/written-in-stone-the-worlds-first-trigonometry-revealed-in-an-ancient-babylonian-tablet-81472">Written in stone: the world's first trigonometry revealed in an ancient Babylonian tablet</a>
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<p>Akshay’s medal is in <a href="https://www.mathunion.org/fileadmin/IMU/Prizes/Fields/2018/Venkatesh-Citation.pdf">recognition</a> for “his synthesis of analytic number theory, homogeneous dynamics, topology and representation theory”.</p>
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<h2>Showed early promise</h2>
<p>Akshay Venkatesh is currently a <a href="https://mathematics.stanford.edu/people/name/akshay-venkatesh/">professor of mathematics at Stanford University</a>, in the United States, and will soon be moving to join the faculty at the Institute for Advanced Study at Princeton.</p>
<p>But he started out as a student at the University of Western Australia, studying physics and mathematics.</p>
<p>I was doing my undergraduate studies at the same time and he skipped first year mathematics and jumped straight into my second year classes. He was only 13 at the time, while the the rest of the students were 18 and 19 years old. It was clear that he was a very talented individual with exceptional abilities in mathematics.</p>
<p>He topped all his classes and I remember one assignment in third year where he managed to show that something that we had been asked to prove was, in fact, false. The lecturer had missed out an important condition that was required to make it work, but Akshay was the only one to spot it. </p>
<p>Another example of Akshay’s early interest in mathematics comes from one of his mentors at UWA, my colleague Professor Cheryl Praeger, an Australian Academy of Science Fellow. She said:</p>
<blockquote>
<p>At our first meeting I was speaking with Akshay’s mother Svetha, while Akshay was sitting at a table in my office reading my blackboard which contained fragments from a supervision of one of my PhD students, just completed.</p>
<p>At Akshay’s request I explained what the problem was. He coped with quite a lot of detail and I found that he could easily grasp the essence of the research.</p>
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<p>Initially Praeger was worried about the 16-year-old heading to Princeton in 1998, but he soon settled in and it allowed him to return to his love of number theory.</p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/230161/original/file-20180801-136655-giuqc2.png?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/230161/original/file-20180801-136655-giuqc2.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/230161/original/file-20180801-136655-giuqc2.png?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=338&fit=crop&dpr=1 600w, https://images.theconversation.com/files/230161/original/file-20180801-136655-giuqc2.png?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=338&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/230161/original/file-20180801-136655-giuqc2.png?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=338&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/230161/original/file-20180801-136655-giuqc2.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=424&fit=crop&dpr=1 754w, https://images.theconversation.com/files/230161/original/file-20180801-136655-giuqc2.png?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=424&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/230161/original/file-20180801-136655-giuqc2.png?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=424&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">Akshay graduated at an early age from UWA before heading to Princeton to study for a PhD.</span>
<span class="attribution"><span class="source">Australian Academy of Science</span></span>
</figcaption>
</figure>
<h2>A love of number theory</h2>
<p><a href="https://www.britannica.com/science/number-theory">Number theory</a>, Akshay’s main area of research, is one of the central areas of mathematics. It is notorious for having problems that are very easy to state but which turn out to be extremely difficult to solve.</p>
<p>One of Akshay’s major contributions has been to use ideas from many other areas of mathematics (such as representation theory, ergodic theory and topology) to help solve some of these very difficult problems.</p>
<p>One of the fundamental questions of number theory is to understand the distribution of the prime numbers – a <a href="https://www.britannica.com/science/prime-number">prime number</a> is one that is divisible only by 1 and itself, such as 2, 3, 5, 7, 11, 13 and so on. Mathematicians have been trying to understand these numbers since the ancient Greeks.</p>
<p>One of the main tools used in the study of prime numbers is what are called L-functions – the most famous of which is the <a href="https://www.britannica.com/science/Riemann-zeta-function">Riemann zeta function</a> introduced in the 1800s.</p>
<p>One of Akshay’s major results was joint work with the French mathematician Philippe Michel that solved the <a href="https://link.springer.com/article/10.1007/s10240-010-0025-8">“subconvexity problem”</a> for a large family of L-functions. </p>
<p>This problem had been identified in 1999 as one of the four most important problems in the area, and involved providing good upper bounds on the values that these functions can take. He was then able to apply its solution to the study of homogeneous dynamics and the theory of quadratic forms.</p>
<h2>The Fields Medal</h2>
<figure class="align-right zoomable">
<a href="https://images.theconversation.com/files/230162/original/file-20180801-136649-pa58cm.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/230162/original/file-20180801-136649-pa58cm.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=237&fit=clip" srcset="https://images.theconversation.com/files/230162/original/file-20180801-136649-pa58cm.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=577&fit=crop&dpr=1 600w, https://images.theconversation.com/files/230162/original/file-20180801-136649-pa58cm.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=577&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/230162/original/file-20180801-136649-pa58cm.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=577&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/230162/original/file-20180801-136649-pa58cm.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=725&fit=crop&dpr=1 754w, https://images.theconversation.com/files/230162/original/file-20180801-136649-pa58cm.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=725&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/230162/original/file-20180801-136649-pa58cm.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=725&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">The Fields Medal.</span>
<span class="attribution"><a class="source" href="https://commons.wikimedia.org/wiki/File:FieldsMedalFront.jpg">Stefan Zachow/International Mathematical Union</a></span>
</figcaption>
</figure>
<p>The <a href="https://www.mathunion.org/imu-awards/fields-medal">Fields Medal</a>, regarded as one of the most important honours in mathematics, is named after the Canadian mathematician John Charles Fields (1863–1932). He conceived the award to celebrate the great achievements in the area.</p>
<p>In addition to a gold medal, a winner receives CA$15,000 (A$15,500).</p>
<p>It is an incredible honour to receive a Fields Medal, which is made even more difficult by the fact that you need to be under 40 at the start of the year to receive it. </p>
<p>The award is great recognition for the significant work that Akshay has done and hopefully it inspires other students to study mathematics.</p>
<hr>
<p>
<em>
<strong>
Read more:
<a href="https://theconversation.com/universities-can-help-recruit-more-science-and-maths-teachers-but-they-cant-do-it-alone-99998">Universities can help recruit more science and maths teachers, but they can't do it alone</a>
</strong>
</em>
</p>
<hr>
<p>The Fields Medal is usually awarded to two to four mathematics. This year’s other winners are:</p>
<ul>
<li><p>Alessio Figalli, 34, professor at ETH in Zurich (Switzerland) who works on differential equations.</p></li>
<li><p>Caucher Birkar, 40, professor at Cambridge University (UK), who works in algebraic geometry.</p></li>
<li><p>Peter Scholze, 30, professor at Bonn University (Germany), who also works in algebraic geometry.</p></li>
</ul><img src="https://counter.theconversation.com/content/100887/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Michael Giudici receives funding from the Australian Research Council. </span></em></p>I was in second year at the University of Western Australia when Akshay Venkatesh skipped first year maths and jumped straight into my classes. He was 13 at the time. Now he’s won a prestigious award.Michael Giudici, Professor, The University of Western AustraliaLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/898782018-01-12T12:11:59Z2018-01-12T12:11:59ZWhy do we need to know about prime numbers with millions of digits?<figure><img src="https://images.theconversation.com/files/201796/original/file-20180112-101489-x4wvpo.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">
</span> <span class="attribution"><span class="source">SeventyFour via Shutterstock</span></span></figcaption></figure><p>Prime numbers are <a href="https://theconversation.com/your-numbers-up-a-case-for-the-usefulness-of-useless-maths-11799">more than just numbers</a> that can only be divided by themselves and one. They are a mathematical mystery, the secrets of which mathematicians have been trying to uncover ever since Euclid <a href="https://www.math.utah.edu/%7Epa/math/q2.html">proved</a> that they have no end.</p>
<p>An ongoing project – the <a href="https://www.mersenne.org">Great Internet Mersenne Prime Search</a> – which aims to discover more and more primes of a particularly rare kind, has recently resulted in the discovery of the largest prime number known to date. Stretching to 23,249,425 digits, it is so large that it would easily fill 9,000 book pages. By comparison, the number of atoms in the entire observable universe is estimated to have no more than 100 digits. </p>
<p>The number, simply written as 2⁷⁷²³²⁹¹⁷-1 (two to the power of 77,232,917, minus one) was <a href="https://theconversation.com/largest-known-prime-number-discovered-why-it-matters-89743">found by a volunteer</a> who had dedicated <a href="https://www.mersenne.org/primes/press/M77232917.html">14 years of computing time</a> to the endeavour. </p>
<p>You may be wondering, if the number stretches to more than 23m digits, why we need to know about it? Surely the most important numbers are the ones that we can use to quantify our world? That’s not the case. We need to know about the properties of different numbers so that we can not only keep developing the technology we rely on, but also keep it secure.</p>
<h2>Secrecy with prime numbers</h2>
<p>One of the most widely used applications of prime numbers in computing is <a href="https://theconversation.com/the-rsa-algorithm-or-how-to-send-private-love-letters-13191">the RSA encryption system</a>. In 1978, Ron Rivest, Adi Shamir and Leonard Adleman combined some simple, known facts about numbers to create RSA. The system they developed allows for the secure transmission of information – such as credit card numbers – online.</p>
<p>The first ingredient required for the algorithm are two large prime numbers. The larger the numbers, the safer the encryption. The counting numbers one, two, three, four, and so on – also called the natural numbers – are, obviously, extremely useful here. But the prime numbers are the building blocks of all natural numbers and so even more important. </p>
<p>Take the number 70 for example. Division shows that it is the product of two and 35. Further, 35 is the product of five and seven. So 70 is the product of three smaller numbers: two, five, and seven. This is the end of the road for 70, since none of these can be further broken down. We have found the primal components that make up 70, giving its prime factorisation. </p>
<p>Multiplying two numbers, even if very large, is perhaps tedious but a straightforward task. Finding prime factorisation, on the other hand, is extremely hard, and that is precisely what the RSA system takes advantage of. </p>
<figure>
<iframe width="440" height="260" src="https://www.youtube.com/embed/rLUpFYH80i0?wmode=transparent&start=0" frameborder="0" allowfullscreen=""></iframe>
</figure>
<p>Suppose that Alice and Bob wish to communicate secretly over the internet. They require an encryption system. If they first meet in person, they can devise a method for encryption and decryption that only they will know, but if the initial communication is online, they need to first openly communicate the encryption system itself – a risky business. </p>
<p>However, if Alice chooses two large prime numbers, computes their product, and communicates this openly, finding out what her original prime numbers were will be a very difficult task, as only she knows the factors. </p>
<p>So Alice communicates her product to Bob, keeping her factors secret. Bob uses the product to encrypt his message to Alice, which can only be decrypted using the factors that she knows. If Eve is eavesdropping, she cannot decipher Bob’s message unless she acquires Alice’s factors, which were never communicated. If Eve tries to break the product down into its prime factors – even using the fastest supercomputer – no known algorithm exists that can accomplish that before the sun will explode. </p>
<h2>The primal quest</h2>
<p>Large prime numbers are used prominently in other cryptosystems too. The faster computers get, the larger the numbers they can crack. For modern applications, prime numbers measuring hundreds of digits suffice. These numbers are minuscule in comparison to the giant recently discovered. In fact, the new prime is so large that – at present – no conceivable technological advancement in computing speed could lead to a need to use it for cryptographic safety. It is even likely that the risks posed by the looming quantum computers wouldn’t need such monster numbers to be made safe. </p>
<p>It is neither safer cryptosystems nor improving computers that drove the latest Mersenne discovery, however. It is mathematicians’ need to uncover the jewels inside the chest labelled “prime numbers” that fuels the ongoing quest. This is a primal desire that starts with counting one, two, three, and drives us to the frontiers of research. The fact that online commerce has been revolutionised is almost an accident.</p>
<p>The celebrated British mathematician <a href="https://theconversation.com/the-man-who-taught-infinity-how-gh-hardy-tamed-srinivasa-ramanujans-genius-57585">Godfrey Harold Hardy</a> said: “Pure mathematics is on the whole distinctly more useful than applied. For what is useful above all is technique, and mathematical technique is taught mainly through pure mathematics”. Whether or not huge prime numbers, such as the 50th known Mersenne prime with its millions of digits, will ever be found useful is, at least to Hardy, an irrelevant question. The merit of knowing these numbers lies in quenching the human race’s intellectual thirst that started with Euclid’s proof of the infinitude of primes and still goes on today.</p><img src="https://counter.theconversation.com/content/89878/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Ittay Weiss does not work for, consult, own shares in or receive funding from any company or organisation that would benefit from this article, and has disclosed no relevant affiliations beyond their academic appointment.</span></em></p>Prime numbers are a mathematical mystery.Ittay Weiss, Teaching Fellow, Department of Mathematics, University of PortsmouthLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/843322017-09-21T11:02:15Z2017-09-21T11:02:15ZFive ways ancient India changed the world – with maths<figure><img src="https://images.theconversation.com/files/186896/original/file-20170920-16437-hxdak9.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">Bakhshali manuscript.</span> <span class="attribution"><span class="source">Bodleian Libraries, University of Oxford</span></span></figcaption></figure><p>It should come as no surprise that the first recorded use of the number zero, <a href="https://www.theguardian.com/science/2017/sep/14/much-ado-about-nothing-ancient-indian-text-contains-earliest-zero-symbol">recently discovered</a> to be made as early as the 3rd or 4th century, happened in India. Mathematics on the Indian subcontinent has a rich history <a href="http://www.tifr.res.in/%7Earchaeo/FOP/FoP%20papers/ancmathsources_Dani.pdf">going back over 3,000 years</a> and thrived for centuries before similar advances were made in Europe, with its influence meanwhile spreading to China and the Middle East.</p>
<p>As well as giving us the concept of zero, Indian mathematicians made seminal contributions to the study of <a href="http://www-history.mcs.st-andrews.ac.uk/HistTopics/Indian_mathematics.html">trigonometry, algebra, arithmetic and negative numbers among other areas</a>. Perhaps most significantly, the decimal system that we still employ worldwide today was first seen in India.</p>
<h2>The number system</h2>
<p>As far back as 1200 BC, mathematical knowledge was being written down as part of a large body of knowledge known as <a href="https://www.ancient.eu/The_Vedas/">the Vedas</a>. In these texts, numbers were commonly expressed as <a href="http://www.thehindu.com/sci-tech/science/understanding-ancient-indian-mathematics/article2747006.ece">combinations of powers of ten</a>. For example, 365 might be expressed as three hundreds (3x10²), six tens (6x10¹) and five units (5x10⁰), though each power of ten was represented with a name rather than a set of symbols. It is <a href="http://www.tifr.res.in/%7Earchaeo/FOP/FoP%20papers/ancmathsources_Dani.pdf">reasonable to believe</a> that this representation using powers of ten played a crucial role in the development of the decimal-place value system in India.</p>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/186842/original/file-20170920-16414-6jy46n.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/186842/original/file-20170920-16414-6jy46n.png?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=126&fit=crop&dpr=1 600w, https://images.theconversation.com/files/186842/original/file-20170920-16414-6jy46n.png?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=126&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/186842/original/file-20170920-16414-6jy46n.png?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=126&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/186842/original/file-20170920-16414-6jy46n.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=158&fit=crop&dpr=1 754w, https://images.theconversation.com/files/186842/original/file-20170920-16414-6jy46n.png?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=158&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/186842/original/file-20170920-16414-6jy46n.png?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=158&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">Brahmi numerals.</span>
<span class="attribution"><a class="source" href="https://en.wikipedia.org/wiki/Brahmi_numerals#/media/File:Indian_numerals_100AD.svg">Wikimedia</a></span>
</figcaption>
</figure>
<p>From the <a href="http://www-history.mcs.st-andrews.ac.uk/HistTopics/Indian_mathematics.html">third century BC</a>, we also have written evidence of the <a href="http://www-history.mcs.st-andrews.ac.uk/HistTopics/Indian_numerals.html">Brahmi numerals</a>, the precursors to the modern, Indian or Hindu-Arabic numeral system that most of the world uses today. Once zero was introduced, almost all of the mathematical mechanics would be in place to enable ancient Indians to study higher mathematics. </p>
<h2>The concept of zero</h2>
<p>Zero itself has a much longer history. The <a href="http://www.bodleian.ox.ac.uk/news/2017/sep-14">recently dated first recorded zeros</a>, in what is known as the Bakhshali manuscript, were simple placeholders – a tool to distinguish 100 from 10. Similar marks had already been seen in the <a href="https://www.scientificamerican.com/article/history-of-zero/">Babylonian and Mayan cultures in the early centuries AD</a> and arguably in <a href="https://www.scientificamerican.com/article/history-of-zero/">Sumerian mathematics as early as 3000-2000 BC</a>.</p>
<p>But only in India did the placeholder symbol for nothing progress to become a <a href="https://theconversation.com/nothing-matters-how-the-invention-of-zero-helped-create-modern-mathematics-84232">number in its own right</a>. The advent of the concept of zero allowed numbers to be written efficiently and reliably. In turn, this allowed for effective record-keeping that meant important financial calculations could be checked retroactively, ensuring the honest actions of all involved. Zero was a significant step on the route to the <a href="https://blog.sciencemuseum.org.uk/illuminating-india-starring-oldest-recorded-origins-zero-bakhshali-manuscript/">democratisation of mathematics</a>.</p>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/187001/original/file-20170921-8179-14x8av.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/187001/original/file-20170921-8179-14x8av.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=400&fit=crop&dpr=1 600w, https://images.theconversation.com/files/187001/original/file-20170921-8179-14x8av.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=400&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/187001/original/file-20170921-8179-14x8av.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=400&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/187001/original/file-20170921-8179-14x8av.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=503&fit=crop&dpr=1 754w, https://images.theconversation.com/files/187001/original/file-20170921-8179-14x8av.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=503&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/187001/original/file-20170921-8179-14x8av.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=503&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">No abacus needed.</span>
<span class="attribution"><span class="source">Shutterstock</span></span>
</figcaption>
</figure>
<p>These accessible mechanical tools for working with mathematical concepts, in combination with a strong and open scholastic and scientific culture, meant that, by around 600AD, all the ingredients were in place for an explosion of mathematical discoveries in India. In comparison, these sorts of tools were not popularised in the West until the early 13th century, though <a href="http://www.springer.com/gb/book/9780387407371">Fibonnacci’s book liber abaci</a>. </p>
<h2>Solutions of quadratic equations</h2>
<p>In the seventh century, the first written evidence of the rules for working with zero were formalised in the <a href="https://archive.org/details/Brahmasphutasiddhanta_Vol_1">Brahmasputha Siddhanta</a>. In his seminal text, the astronomer <a href="http://www.storyofmathematics.com/indian_brahmagupta.html">Brahmagupta</a> introduced rules for solving quadratic equations (so beloved of secondary school mathematics students) and for computing square roots.</p>
<h2>Rules for negative numbers</h2>
<p>Brahmagupta also demonstrated rules for working with negative numbers. He referred to <a href="https://nrich.maths.org/5961">positive numbers as fortunes and negative numbers as debts</a>. He wrote down rules that have been interpreted by translators as: “A fortune subtracted from zero is a debt,” and “a debt subtracted from zero is a fortune”.</p>
<p>This latter statement is the same as the rule we learn in school, that if you subtract a negative number, it is the same as adding a positive number. Brahmagupta also knew that “The product of a debt and a fortune is a debt” – a positive number multiplied by a negative is a negative. </p>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/187008/original/file-20170921-8211-139ku7u.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/187008/original/file-20170921-8211-139ku7u.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=400&fit=crop&dpr=1 600w, https://images.theconversation.com/files/187008/original/file-20170921-8211-139ku7u.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=400&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/187008/original/file-20170921-8211-139ku7u.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=400&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/187008/original/file-20170921-8211-139ku7u.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=503&fit=crop&dpr=1 754w, https://images.theconversation.com/files/187008/original/file-20170921-8211-139ku7u.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=503&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/187008/original/file-20170921-8211-139ku7u.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=503&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">Negative cows.</span>
<span class="attribution"><span class="source">Shutterstock</span></span>
</figcaption>
</figure>
<p>For the large part, European mathematicians were reluctant to accept negative numbers as meaningful. Many took the view that <a href="https://books.google.co.uk/books?id=STKX4qadFTkC&pg=PA56&redir_esc=y#v=onepage&q&f=false">negative numbers were absurd</a>. They reasoned that numbers were developed for counting and questioned what you could count with negative numbers. Indian and Chinese mathematicians recognised early on that one answer to this question was debts.</p>
<p>For example, in a primitive farming context, if one farmer owes another farmer 7 cows, then effectively the first farmer has -7 cows. If the first farmer goes out to buy some animals to repay his debt, he has to buy 7 cows and give them to the second farmer in order to bring his cow tally back to 0. From then on, every cow he buys goes to his positive total. </p>
<h2>Basis for calculus</h2>
<p>This reluctance to adopt negative numbers, and indeed zero, held European mathematics back for many years. Gottfried Wilhelm Leibniz was one of the first Europeans to use zero and the negatives in a systematic way in his <a href="https://books.google.co.uk/books?id=CXG6CgAAQBAJ&pg=PA165&lpg=PA165&dq=Leibniz+zero+negatives+calculus&source=bl&ots=NsKOzdZL7Y&sig=dE2KJvCXPFovF4uyFdgHMJOAQr8&hl=en&sa=X&ved=0ahUKEwjdxKv8_LPWAhXhAcAKHR0XBcUQ6AEIMjAC#v=onepage&q=Leibniz%20zero%20negatives%20calculus&f=false">development of calculus</a> in the late 17th century. Calculus is used to measure rates of changes and is important in almost every branch of science, notably underpinning many key discoveries in modern physics.</p>
<figure class="align-right ">
<img alt="" src="https://images.theconversation.com/files/187017/original/file-20170921-8194-1ypsmdq.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=237&fit=clip" srcset="https://images.theconversation.com/files/187017/original/file-20170921-8194-1ypsmdq.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=759&fit=crop&dpr=1 600w, https://images.theconversation.com/files/187017/original/file-20170921-8194-1ypsmdq.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=759&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/187017/original/file-20170921-8194-1ypsmdq.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=759&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/187017/original/file-20170921-8194-1ypsmdq.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=954&fit=crop&dpr=1 754w, https://images.theconversation.com/files/187017/original/file-20170921-8194-1ypsmdq.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=954&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/187017/original/file-20170921-8194-1ypsmdq.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=954&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">Leibniz: Beaten to it by 500 years.</span>
</figcaption>
</figure>
<p>But <a href="http://www-groups.dcs.st-and.ac.uk/history/Biographies/Bhaskara_II.html">Indian mathematician Bhāskara</a> had already discovered many of Leibniz’s ideas <a href="https://ijrier.com/published-papers/volume-1/issue-8/origin-of-concept-of-calculus-in-india.pdf">over 500 years earlier</a>. Bhāskara, also made major contributions to algebra, arithmetic, geometry and trigonometry. He provided many results, for example on the solutions of certain “Doiphantine” equations, that <a href="https://www.amazon.co.uk/Mathematical-Achievements-Pre-modern-Mathematicians-Elsevier/dp/0123979137#reader_0123979137">would not be rediscovered in Europe for centuries</a>.</p>
<p><a href="https://link.springer.com/referenceworkentry/10.1007%2F978-1-4020-4425-0_8683">The Kerala school of astronomy and mathematics</a>, founded by <a href="https://en.wikipedia.org/wiki/Madhava_of_Sangamagrama">Madhava of Sangamagrama</a> in the 1300s, was responsible for many firsts in mathematics, including the use of mathematical induction and some early calculus-related results. Although no systematic rules for calculus were developed by the Kerala school, its proponents first conceived of many of the results that would <a href="http://www.jstor.org/stable/1558972?origin=crossref&seq=1#page_scan_tab_contents">later be repeated in Europe</a> including Taylor series expansions, infinitessimals and differentiation. </p>
<p>The leap, made in India, that transformed zero from a simple placeholder to a number in its own right indicates the mathematically enlightened culture that was flourishing on the subcontinent at a time when Europe was stuck in the dark ages. Although its reputation <a href="http://www.cbc.ca/news/technology/calculus-created-in-india-250-years-before-newton-study-1.632433">suffers from the Eurocentric bias</a>, the subcontinent has a strong mathematical heritage, which it continues into the 21st century by <a href="http://m.ranker.com/list/famous-mathematicians-from-india/reference?page=1">providing key players at the forefront of every branch of mathematics</a>.</p><img src="https://counter.theconversation.com/content/84332/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Christian Yates does not work for, consult, own shares in or receive funding from any company or organisation that would benefit from this article, and has disclosed no relevant affiliations beyond their academic appointment.</span></em></p>High school students can blame ancient India for quadratic equations and calculus.Christian Yates, Senior Lecturer in Mathematical Biology, University of BathLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/811932017-07-18T11:44:42Z2017-07-18T11:44:42ZMaryam Mirzakhani’s success showed us the challenges women in maths still face<figure><img src="https://images.theconversation.com/files/178626/original/file-20170718-10341-ztw0sb.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">
</span> <span class="attribution"><span class="source">Shutterstock</span></span></figcaption></figure><p>When Marin Alsop conducted the Last Night of the Proms, <a href="http://www.bbc.co.uk/news/entertainment-arts-24008355">she said</a> that she was “quite shocked that it can be 2013 and there can still be firsts for women”. The following year, <a href="https://theconversation.com/maryam-mirzakhani-was-a-role-model-for-more-than-just-her-mathematics-81143">Maryam Mirzakhani</a> became the first woman to win the Fields Medal, awarded to mathematicians under 40 for their contribution to the subject. Now, after Mirzakhani’s <a href="http://www.newyorker.com/tech/elements/maryam-mirzakhanis-pioneering-mathematical-legacy">sad death</a> from breast cancer at the age of 40, I am struck by the stark reality that in the 80 years since the award was first given, there has still been only one female winner so far.</p>
<p>It is important to note that Mirzakhani was a pioneer in other ways too. She was the first Iranian Fields medallist. Her <a href="https://www.theguardian.com/science/alexs-adventures-in-numberland/2014/aug/13/fields-medals-2014-maths-avila-bhargava-hairer-mirzakhani">mathematical research</a> on the geometry of complex surfaces was groundbreaking, and has opened up new horizons for others to explore. When she was awarded the Fields Medal, Mirzakhani immediately became an inspiration for many young mathematicians. I was delighted that, finally, a woman had won this prize, the highest accolade that the mathematical community awards. But I was also dismayed that it had taken so long. </p>
<p>For centuries, it was socially unacceptable, or even impossible, for women to study mathematics at the highest levels. As a young mathematician, I was inspired by the story of <a href="http://www-groups.dcs.st-and.ac.uk/history/Biographies/Germain.html">Sophie Germain</a>, who taught herself maths under the bedclothes in the face of opposition from her parents. She went on to make substantial contributions to mathematicians’ understanding of one of the most famous problems in mathematics: Fermat’s Last Theorem.</p>
<p>In 1890, <a href="http://www.diverse.cam.ac.uk/stories/fawcett/">Philippa Fawcett</a> was the highest performing maths student at the University of Cambridge. Yet women were not included in the main ranked list so the honour of Senior Wrangler (top student) went to a man, even though Fawcett scored higher. In the US in the 1940s, the mathematician <a href="https://www.awm-math.org/noetherbrochure/Robinson82.html">Julia Robinson</a> was not allowed to teach at the University of California at Berkeley, because her husband worked there and “nepotism rules” prevented them both working in the same department. She went on to play a major role in solving the tenth of Hilbert’s famous list of 23 problems.</p>
<p>Happily, things have moved on. These days, my own department and many others are actively seeking ways to improve gender diversity and inclusivity (as well as other forms of diversity). The <a href="http://www.ecu.ac.uk/equality-charters/athena-swan/">Athena SWAN</a> scheme provides recognition to universities and departments who are making serious attempts in this area.</p>
<p>It requires effort and active engagement to change culture. Progress is being made, albeit slowly. The “<a href="https://theconversation.com/stopping-the-brain-drain-of-women-scientists-22802">leaky pipeline</a>” of academia (which sees women drop out at every level) means that even though gender diversity is improving amongst mathematics undergraduates, the balance is not great among postdocs and worse still among professors. <a href="https://www.lms.ac.uk/sites/lms.ac.uk/files/Benchmarking%20Data%20Updated%20for%202011-2015%20April%202016_0.pdf">Recent data</a> from the London Mathematical Society showed that from 2014 to 2015 around 40% of UK mathematics undergraduates were female, but only 9% of UK mathematics professors were female.</p>
<p>Of course the same phenomenon occurs in many walks of life, not just academia. There is lots being done to try to understand why this is the case in mathematics. Recruitment practices are being improved, and academics are being trained in unconscious bias. Perhaps a problem that is distinctive to mathematics (and closely related subjects) is cultural. There is sometimes an unhelpful, and in my opinion incorrect, perception that one has to be <a href="https://www.theguardian.com/education/2016/mar/26/reckon-you-were-born-without-a-brain-for-maths-highly-unlikely">some sort of genius</a> to succeed in mathematics, and this can be off-putting.</p>
<h2>Signs of progress</h2>
<p>Anecdotally, my impression is that there is more awareness now than say 15 years ago of the need to increase diversity within mathematics at all levels of seniority. At the same time, there is a danger that this might make the gender imbalance clearer to school students who might in turn feel inhibited in their desire to study mathematics. My personal hope is for all young people to experience the joys and frustrations, the creativity and the practicality of mathematics, so that those who wish to can take their mathematical studies further, regardless of gender or any other factor. </p>
<p>There are signs of progress in improving diversity in mathematics (of all forms, not just gender), but it’s taking a long time, and there are more firsts to come. We are still waiting for the first female winner of the <a href="http://www.abelprize.no/c53673/seksjon/vis.html?tid=53719">Abel Prize</a>, another major accolade in mathematics. Tragically, Mirzakhani died much too soon, but her mathematical contributions will live on, both in theorems and ideas that others can build on and also in inspiring future generations.</p>
<p>I look forward to the day when women winning the Fields Medal receive acclaim for their outstanding mathematical achievement, without reference to their gender. As Robinson wrote:</p>
<blockquote>
<p>What I really am is a mathematician. Rather than being remembered as the first woman this or that, I would prefer to be remembered, as a mathematician should, simply for the theorems I have proved and the problems I have solved.</p>
</blockquote><img src="https://counter.theconversation.com/content/81193/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Vicky Neale does not work for, consult, own shares in or receive funding from any company or organisation that would benefit from this article, and has disclosed no relevant affiliations beyond their academic appointment.</span></em></p>It took 80 years for a woman to be awarded the highest prize in mathematics, the Fields Medal.Vicky Neale, Whitehead Lecturer at the Mathematical Institute and Supernumerary Fellow at Balliol College, University of OxfordLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/659632016-10-13T18:06:42Z2016-10-13T18:06:42ZYes, mathematics can be decolonised. Here’s how to begin<figure><img src="https://images.theconversation.com/files/139426/original/image-20160927-14593-1rf92dt.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">
</span> <span class="attribution"><span class="source">Shutterstock</span></span></figcaption></figure><p>At a time when decolonisation, part of which involves changing the content of what’s taught, is dominating debate at many universities, the discipline of mathematics presents an interesting case. </p>
<p>But it’s not obvious how mathematics can be decolonised at the level of content. This means that those within the discipline must consider other aspects: curriculum processes, such as critical thinking and problem solving; pedagogy – how the subject is taught and, as a number of people have <a href="https://www.routledge.com/Mathematical-Relationships-in-Education-Identities-and-Participation/Black-Mendick-Solomon/p/book/9780415996846">argued</a>, addressing the issue of identity. </p>
<p>Students’ mathematical identities – how they see themselves as learners of mathematics and the extent to which mathematics is meaningful to them – are important when thinking about teaching and learning in mathematics.</p>
<p>In his book <em><a href="https://www.amazon.com/Leading-Change-leadership-university-Educational/dp/113889026X">Leading for change</a></em>, South African educationist Jonathan Jansen suggests that transforming university campuses into deracialised spaces requires attention to both the academic and the human project. I take the human project to mean how students see themselves. What might this mean for mathematics?</p>
<h2>So what is mathematics?</h2>
<p>For starters, it’s important to explore what mathematics actually is.</p>
<p>Mathematician and academic Jo Boaler <a href="http://eu.wiley.com/WileyCDA/WileyTitle/productCd-0470894520.html">points out</a> that mathematics is the only subject where students and mathematicians give very different answers to this question.</p>
<p>Mathematicians view the subject as an exciting, creative endeavour in which problem solving, curiosity, excitement, intuition and perseverance play important roles – albeit in relation to abstract objects of study. </p>
<p>For school and even undergraduate mathematics students, these aspects of mathematics are often not experienced and remain opaque. Students tend to believe that mathematics is a set of procedures to be followed. They think only particularly gifted people can do and understand these procedures. This suggests that the way mathematics is usually taught doesn’t provide opportunities for accessing mathematical knowledge. It doesn’t allow students to identify with mathematics, nor make them aspire to become mathematicians.</p>
<p>As a result, mathematics has a problem with diversity. All over the world, black and women mathematicians remain rare. They <a href="https://www.youcubed.org/think-it-up/ideas-of-giftedness-hurt-students/">simply don’t</a> take mathematics at higher academic levels as much as their white and male peers.</p>
<p>One reason for this is given by <a href="https://www.youcubed.org/think-it-up/ideas-of-giftedness-hurt-students/">a study</a> in the US, which showed that the more a field attributes success to giftedness rather than effort, the fewer female and black academics are in that field. This is because the field perpetuates stereotypes about who belongs in the field. The same study found that mathematics professors hold the most fixed ideas about giftedness. </p>
<p>But this view of giftedness versus effort is not borne out by research. A number of scholars <a href="http://www.ams.org/notices/200102/rev-devlin.pdf">have argued</a> that all people are capable of learning mathematics, to high levels.</p>
<p>This suggests that a lot of the “bad press” around mathematics as a subject and discipline lies with how it is taught and learned.</p>
<h2>What is learning?</h2>
<p>When scholars theorise learning, the thinking always happens in two directions: to the past, and to the future. </p>
<p>Some see learning as building on current knowledge in a <a href="http://rer.sagepub.com/content/57/2/175.full.pdf">step-wise linear way</a>. Some see it as working <a href="http://infed.org/mobi/jerome-bruner-and-the-process-of-education/">in a spiral</a> –- coming back to old ideas in new ways. Still others view learning as <a href="https://books.google.co.za/books?id=x2XRiIm-3vAC&pg=RA4-PA1946&lpg=RA4-PA1946&dq=a+conception+of+knowledge+acquisition+and+its+implications+for+education&source=bl&ots=tO3N1Cedy1&sig=UYT9tzDyfKDubi0MxKIJK0eOS1A&hl=en&sa=X&redir_esc=y#v=onepage&q=a%20conception%20of%20knowledge%20acquisition%20and%20its%20implications%20for%20education&f=false">disrupting or transforming</a> current knowledge.</p>
<p>For teachers, working with current knowledge means finding ways to ascertain, predict, anticipate and think about students’ ideas – and finding ways to engage with these. An important part of students’ ideas about mathematics is how they see themselves in relation to mathematics. Research in schools <a href="http://eu.wiley.com/WileyCDA/WileyTitle/productCd-0470894520.html">has shown</a> that one of the key factors in students’ mathematics achievement is a teacher who believes that they can do mathematics.</p>
<p>The future is important because universities must produce future thinkers, leaders, professionals and citizens. These institutions are the bridge between the past and the future. </p>
<p>Educational theorist Etienne Wenger argues that learning is fundamentally about becoming <a href="https://www.amazon.com/Communities-Practice-Cognitive-Computational-Perspectives/dp/0521663636">a certain kind of person</a>. At universities, students are inducted into disciplines, fields and professions that require them to be certain kinds of people with certain orientations to the world, to knowledge, to other people and to practice. </p>
<p>Traditionally universities have focused on knowledge and hoped that identity will follow. This hasn’t been entirely unsuccessful. But to genuinely transform the academic project, universities must do explicit identity work with their students. Academics must engage in the human project, thinking about who their students are and what their previous experiences of mathematics and of learning mathematics have been.</p>
<h2>Towards genuine change</h2>
<p>There have been attempts to transform the content of school mathematics curricula. These include <a href="http://www.maa.org/publications/periodicals/maa-focus/ethnomathematics-shows-students-their-connections-math">ethnomathematics</a>, which excavates the mathematics in cultural objects, artefacts and practices; and critical mathematics, where mathematics is used to critique aspects of society and where students critique mathematics, for example, how algorithms structure our lives in ways which <a href="http://nymag.com/thecut/2016/09/cathy-oneils-weapons-of-math-destruction-math-is-biased.html">reproduce inequality</a>.</p>
<p>However, not all of mathematics can be accessed in these ways. For true epistemological access to mathematics, students need to study it systematically, as a body of knowledge in and of itself. This can be both empowering or disempowering.</p>
<p>Much, though certainly not all, of mathematics was created by <a href="https://theconversation.com/its-time-to-take-the-curriculum-back-from-dead-white-men-40268">dead white men</a>. But maths should and does belong to everybody. Everybody deserves access to its beauty and its power – and everybody should be able to push back when the discipline is used to destroy and oppress.</p>
<p>To transform mathematics teaching and learning in ways that empower students, universities need to give students the theoretical grounding they need to access the subject and support them to identify with it –- to want to learn it, to become the mathematicians of the future, to enjoy and critique mathematics and its applications. </p>
<p>This means that as teachers, my colleagues and I need to believe – to know – that all students can do mathematics. This knowledge must be transmitted to them. They must be shown that mathematics is a human enterprise: it belongs to all, and it can be taken forward to transform society.</p><img src="https://counter.theconversation.com/content/65963/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Karin Brodie receives funding from the NRF for a research project on mathematical identities of high school learners. </span></em></p>Some have suggested that deracialising the academy requires all researchers, teachers and students to link knowledge and identity. What might this mean for mathematics?Karin Brodie, Professor of Education and Mathematics Education, University of the WitwatersrandLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/601682016-06-01T20:16:40Z2016-06-01T20:16:40ZWill computers replace humans in mathematics?<figure><img src="https://images.theconversation.com/files/124546/original/image-20160531-13810-16flhes.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">Computers are coming up with proofs in mathematics that are almost impossible for a human to check.</span> <span class="attribution"><span class="source">Shutterstock/Fernando Batista</span></span></figcaption></figure><p>Computers can be valuable tools for helping mathematicians solve problems but they can also play their own part in the discovery and proof of mathematical theorems.</p>
<p>Perhaps the first major result by a computer came 40 years ago, with proof for the <a>four-color theorem</a> – the assertion that any map (with certain reasonable conditions) can be coloured with just four distinct colours.</p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/124539/original/image-20160531-13773-1iv50nc.png?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/124539/original/image-20160531-13773-1iv50nc.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/124539/original/image-20160531-13773-1iv50nc.png?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=450&fit=crop&dpr=1 600w, https://images.theconversation.com/files/124539/original/image-20160531-13773-1iv50nc.png?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=450&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/124539/original/image-20160531-13773-1iv50nc.png?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=450&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/124539/original/image-20160531-13773-1iv50nc.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=566&fit=crop&dpr=1 754w, https://images.theconversation.com/files/124539/original/image-20160531-13773-1iv50nc.png?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=566&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/124539/original/image-20160531-13773-1iv50nc.png?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=566&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">No more that four colours are needed in this picture to make sure that no two touching shapes share the same colour.</span>
<span class="attribution"><a class="source" href="https://commons.wikimedia.org/wiki/File:Four_Colour_Map_Example.svg">Wikimedia/Inductiveload</a>, <a class="license" href="http://creativecommons.org/licenses/by-sa/4.0/">CC BY-SA</a></span>
</figcaption>
</figure>
<p>This was first proved by computer in 1976, although flaws were later found, and a <a href="http://www.ams.org/notices/200811/tx081101382p.pdf">corrected proof</a> was not completed until 1995.</p>
<p>In 2003, Thomas Hales, of the University of Pittsburgh, published a computer-based proof of <a href="http://experimentalmath.info/blog/2014/08/formal-proof-completed-for-keplers-conjecture-on-sphere-packing/">Kepler’s conjecture</a> that the familiar method of stacking oranges in the supermarket is the most space-efficient way of arranging equal-diameter spheres.</p>
<p>Although Hales published a proof in 2003, many mathematicians were not satisfied because the proof was accompanied by two gigabytes of computer output (a large amount at the time), and some of the computations could not be certified.</p>
<p>In response, Hales produced a <a href="http://experimentalmath.info/blog/2014/08/formal-proof-completed-for-keplers-conjecture-on-sphere-packing/">computer-verified formal proof</a> in 2014.</p>
<h2>The new kid on the block</h2>
<p>The latest development along this line is the <a href="http://www.nature.com/news/two-hundred-terabyte-maths-proof-is-largest-ever-1.19990">announcement this month in Nature</a> of a computer proof for what is known as the Boolean Pythagorean triples problem. </p>
<p>The assertion here is that the integers from one to 7,824 can be coloured either red or blue with the property that no set of three integers a, b and c that satisfy a<sup>2</sup> + b<sup>2</sup> = c<sup>2</sup> (Pythagoras’s Theorem where a, b and c form the sides of a right triangle) are all the same colour. For the integers from one to 7,825, this cannot be done.</p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/124463/original/image-20160530-7678-sa7jwl.png?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/124463/original/image-20160530-7678-sa7jwl.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/124463/original/image-20160530-7678-sa7jwl.png?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=360&fit=crop&dpr=1 600w, https://images.theconversation.com/files/124463/original/image-20160530-7678-sa7jwl.png?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=360&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/124463/original/image-20160530-7678-sa7jwl.png?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=360&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/124463/original/image-20160530-7678-sa7jwl.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=452&fit=crop&dpr=1 754w, https://images.theconversation.com/files/124463/original/image-20160530-7678-sa7jwl.png?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=452&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/124463/original/image-20160530-7678-sa7jwl.png?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=452&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">Pythagoras’s theorem for a right-angled triangle.</span>
<span class="attribution"><span class="source">The Conversation</span>, <a class="license" href="http://creativecommons.org/licenses/by/4.0/">CC BY</a></span>
</figcaption>
</figure>
<p>Even for small integers, it is hard to find a non-monochrome colouring. For instance, if five is red then one of 12 or 13 must be blue, since 5<sup>2</sup> + 12<sup>2</sup> = 13<sup>2</sup>; and one of three or four must also be blue, since 3<sup>2</sup> + 4<sup>2</sup> = 5<sup>2</sup>. Each choice has many constraints.</p>
<p>As it turns out, the number of possible ways to colour the integers from one to 7,825 is gigantic – more than 10<sup>2,300</sup> (a one followed by 2,300 zeroes). This number is far, far greater than the number of fundamental particles in the visible universe, which is a mere <a>10<sup>85</sup></a>. </p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/124693/original/image-20160601-1964-12ufmt4.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/124693/original/image-20160601-1964-12ufmt4.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/124693/original/image-20160601-1964-12ufmt4.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=524&fit=crop&dpr=1 600w, https://images.theconversation.com/files/124693/original/image-20160601-1964-12ufmt4.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=524&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/124693/original/image-20160601-1964-12ufmt4.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=524&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/124693/original/image-20160601-1964-12ufmt4.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=658&fit=crop&dpr=1 754w, https://images.theconversation.com/files/124693/original/image-20160601-1964-12ufmt4.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=658&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/124693/original/image-20160601-1964-12ufmt4.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=658&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">The numbers one to 7,824 can be coloured either red or blue so that no trio a, b and c that satisfies Pythagoras’s theorem are all the same colour. A white square can be either red or blue.</span>
<span class="attribution"><span class="source">Marijn Heule</span></span>
</figcaption>
</figure>
<p>But the researchers were able to sharply reduce this number by taking advantage of various symmetries and number theory properties, to “only” one trillion. The computer run to examine each of these one trillion cases required two days on 800 processors of the University of Texas’ <a>Stampede supercomputer</a>.</p>
<p>While direct applications of this result are unlikely, the ability to solve such difficult colouring problems is bound to have implications for coding and for security.</p>
<p>The Texas computation, which we estimate performed roughly 10<sup>19</sup> arithmetic operations, is still not the largest mathematical computation. A 2013 <a href="http://www.ams.org/notices/201307/rnoti-p844.pdf">computation</a> of digits of pi<sup>2</sup> by us and two IBM researchers did twice this many operations. </p>
<p>The Great Internet Mersenne Prime Search (<a href="http://www.mersenne.org">GIMPS</a>), a global network of computers search for the largest known prime numbers, routinely performs a total of <a href="https://www.sciencedaily.com/releases/2016/01/160120084917.htm">450 trillion calculations per second</a>, which every six hours exceeds the number of operations performed by the Texas calculation. </p>
<p>In computer output, though, the Texas calculation takes the cake for a mathematical computation – a staggering 200 terabytes, namely 2✕10<sup>14</sup> bytes, or 30,000 bytes for every human being on Earth.</p>
<p>How can one check such a sizeable output? Fortunately, the Boolean Pythagorean triple program produced a solution (shown in the image, above) that can be checked by a much smaller program.</p>
<p>This is akin to factoring a very large number c into two smaller factors a and b by computer, so that c = a ✕ b. It is often quite difficult to find the two factors a and b, but once found, it is a trivial task to multiply them together and verify that they work.</p>
<h2>Are mathematicians obsolete?</h2>
<p>So what do these developments mean? Are research mathematicians soon to join the ranks of <a href="http://www.nytimes.com/1997/05/12/nyregion/swift-and-slashing-computer-topples-kasparov.html">chess grandmasters</a>, <a href="http://www.nytimes.com/2011/02/17/science/17jeopardy-watson.html">Jeopardy champions</a>, <a href="http://www.geekwire.com/2016/more-layoffs-at-nordstrom/">retail clerks</a>, <a href="https://www.theguardian.com/technology/2016/feb/10/black-cab-drivers-uber-protest-london-traffic-standstill">taxi drivers</a>, <a href="http://www.cnet.com/news/driverless-truck-convoy-platoons-across-europe/">truck drivers</a>, <a href="http://www.huffingtonpost.com/entry/ibm-watson-radiology_us_55cbccf9e4b0898c48867c56">radiologists</a> and other professions threatened with obsolescence due to rapidly advancing technology?</p>
<p>Not quite. Mathematicians, like many other professionals, have for the large part embraced computation as a new mode of mathematical research, a development known as experimental mathematics, which has far-reaching implications.</p>
<p>So what exactly is experimental mathematics? It is best defined as a mode of research that employs computers as a “laboratory,” in the same sense that a physicist, chemist, biologist or engineer performs an experiment to, for example, gain insight and intuition, test and falsify conjecture, and confirm results proved by conventional means.</p>
<p>We have written on this topic at some length elsewhere – see our <a href="http://www.experimentalmath.info/books/">books</a> and <a href="https://www.carma.newcastle.edu.au/jon/papers.html#PAPERS">papers</a> for full technical details.</p>
<p>In one sense, there there is nothing fundamentally new in the experimental methodology of mathematical research. In the third century BCE, the great Greek mathematician Archimedes <a href="https://books.google.com/books?id=Vvj_AwAAQBAJ&pg=PA314#v=onepage">wrote</a>:</p>
<blockquote>
<p>For it is easier to supply the proof when we have previously acquired, by the [experimental] method, some knowledge of the questions than it is to find it without any previous knowledge.</p>
</blockquote>
<p>Galileo once reputedly wrote:</p>
<blockquote>
<p>All truths are easy to understand once they are discovered; the point is to discover them.</p>
</blockquote>
<p>Carl Friederich Gauss, 19th century mathematician and physicist, frequently employed computations to motivate his remarkable discoveries. He once wrote:</p>
<blockquote>
<p>I have the result, but I do not yet know how to get [prove] it.</p>
</blockquote>
<p>Computer-based experimental mathematics certainly has technology on its side. With every passing year, computer hardware advances with <a href="http://www.intel.com/content/www/us/en/silicon-innovations/moores-law-technology.html">Moore’s Law</a>, and mathematical computing software packages such as Maple, Mathematica, Sage and others become ever more powerful.</p>
<p>Already these systems are powerful enough to solve virtually any equation, derivative, integral or other task in undergraduate mathematics.</p>
<p>So while ordinary human-based proofs are still essential, the computer leads the way in assisting mathematicians to identify new theorems and chart a route to formal proof.</p>
<p>What’s more, one can argue that in many cases computations are more compelling than human-based proofs. Human proofs, after all, are subject to mistakes, oversights, and reliance on earlier results by others that may be unsound. </p>
<p><a href="http://www.intel.com/content/www/us/en/silicon-innovations/moores-law-technology.html">Andrew Wiles’</a> initial proof of <a href="http://simonsingh.net/books/fermats-last-theorem/the-whole-story/">Fermat’s Last Theorem</a> was later found to be flawed. This was fixed later.</p>
<p>Along this line, recently Alexander Yee and Shigeru Kondo computed <a href="http://www.numberworld.org/misc_runs/pi-12t/">12.1 trillion digits of pi</a>. To do this, they first computed somewhat more than 10 trillion base-16 digits, then they checked their computation by computing a section of base-16 digits near the end by a completely different algorithm, and compared the results. They matched perfectly.</p>
<p>So which is more reliable, a human-proved theorem hundreds of pages long, which only a handful of other mathematicians have read and verified in detail, or the Yee-Kondo result? Let’s face it, computation is arguably more reliable than proof in many cases.</p>
<h2>What does the future hold?</h2>
<p>There is every indication that research mathematicians will continue to work in respectful symbiosis with computers for the foreseeable future. Indeed, as this relationship and computer technology mature, mathematicians will become more comfortable leaving certain parts of a proof to computers. </p>
<p>This very question was discussed in a June 2014 <a href="http://experimentalmath.info/blog/2014/11/breakthrough-prize-recipients-give-math-seminar-talks/">panel discussion</a> by the five inaugural <a href="https://breakthroughprize.org/?controller=Page&action=news&news_id=18">Breakthrough Prize in Mathematics</a> recipients for mathematics. The Australian-American mathematician Terence Tao expressed their consensus in these terms:</p>
<blockquote>
<p>Computers will certainly increase in power, but I expect that much of mathematics will continue to be done with humans working with computers.</p>
</blockquote>
<p>So don’t toss your algebra textbook quite yet. You will need it!</p><img src="https://counter.theconversation.com/content/60168/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Jonathan Borwein (Jon) receives funding from the Australian Research Council.</span></em></p><p class="fine-print"><em><span>David H. Bailey does not work for, consult, own shares in or receive funding from any company or organisation that would benefit from this article, and has disclosed no relevant affiliations beyond their academic appointment.</span></em></p>Computers are increasingly used to prove mathematical theorems. So does that mean human mathematicians will become obselete?Jonathan Borwein (Jon), Laureate Professor of Mathematics, University of NewcastleDavid H. Bailey, PhD; Lawrence Berkeley Laboratory (retired) and Research Fellow, University of California, DavisLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/557442016-03-11T17:22:29Z2016-03-11T17:22:29ZThe search for the value of pi<figure><img src="https://images.theconversation.com/files/114523/original/image-20160309-13712-1jxaozq.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">This "pi plate" shows some of the progress toward finding all the digits of pi.</span> <span class="attribution"><a class="source" href="https://upload.wikimedia.org/wikipedia/commons/d/d2/Pi_plate.jpg">Piledhigheranddeeper</a>, <a class="license" href="http://creativecommons.org/licenses/by-sa/4.0/">CC BY-SA</a></span></figcaption></figure><p>The number represented by pi (π) is used in calculations whenever something round (or nearly so) is involved, such as for circles, spheres, cylinders, cones and ellipses. Its value is necessary to compute many important quantities about these shapes, such as understanding the relationship between a circle’s radius and its circumference and area (circumference=2πr; area=πr<sup>2</sup>).</p>
<p>Pi also appears in the calculations to determine the area of an ellipse and in finding the radius, surface area and volume of a sphere.</p>
<p>Our world contains many round and near-round objects; finding the exact value of pi helps us build, manufacture and work with them more accurately.</p>
<p>Historically, people had only very coarse estimations of pi (such as 3, or 3.12, or 3.16), and while they knew these were estimates, they had no idea how far off they might be. </p>
<p>The search for the accurate value of pi led not only to more accuracy, but also to the development of new concepts and techniques, such as limits and iterative algorithms, which then became fundamental to new areas of mathematics.</p>
<h2>Finding the actual value of pi</h2>
<figure class="align-left zoomable">
<a href="https://images.theconversation.com/files/114531/original/image-20160309-13689-10j7dgu.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/114531/original/image-20160309-13689-10j7dgu.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=237&fit=clip" srcset="https://images.theconversation.com/files/114531/original/image-20160309-13689-10j7dgu.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=737&fit=crop&dpr=1 600w, https://images.theconversation.com/files/114531/original/image-20160309-13689-10j7dgu.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=737&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/114531/original/image-20160309-13689-10j7dgu.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=737&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/114531/original/image-20160309-13689-10j7dgu.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=926&fit=crop&dpr=1 754w, https://images.theconversation.com/files/114531/original/image-20160309-13689-10j7dgu.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=926&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/114531/original/image-20160309-13689-10j7dgu.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=926&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">Archimedes.</span>
<span class="attribution"><a class="source" href="https://commons.wikimedia.org/wiki/File:Archimedes1.jpg">André Thévet (1584)</a></span>
</figcaption>
</figure>
<p>Between 3,000 and 4,000 years ago, people used trial-and-error approximations of pi, without doing any math or considering potential errors. The earliest written approximations of pi are <a href="http://www.exploratorium.edu/pi/history_of_pi/">3.125 in Babylon</a> (1900-1600 B.C.) and <a href="http://www.ualr.edu/lasmoller/pi.html">3.1605 in ancient Egypt</a> (1650 B.C.). Both approximations start with 3.1 – pretty close to the actual value, but still relatively far off.</p>
<figure class="align-left zoomable">
<a href="https://images.theconversation.com/files/114533/original/image-20160309-13704-lrnh10.png?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/114533/original/image-20160309-13704-lrnh10.png?ixlib=rb-1.1.0&q=45&auto=format&w=237&fit=clip" srcset="https://images.theconversation.com/files/114533/original/image-20160309-13704-lrnh10.png?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=200&fit=crop&dpr=1 600w, https://images.theconversation.com/files/114533/original/image-20160309-13704-lrnh10.png?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=200&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/114533/original/image-20160309-13704-lrnh10.png?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=200&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/114533/original/image-20160309-13704-lrnh10.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=252&fit=crop&dpr=1 754w, https://images.theconversation.com/files/114533/original/image-20160309-13704-lrnh10.png?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=252&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/114533/original/image-20160309-13704-lrnh10.png?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=252&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">Archimedes’ method of calculating pi involved polygons with more and more sides.</span>
<span class="attribution"><a class="source" href="https://commons.wikimedia.org/wiki/File:Archimedes_pi.svg">Leszek Krupinski</a>, <a class="license" href="http://creativecommons.org/licenses/by-sa/4.0/">CC BY-SA</a></span>
</figcaption>
</figure>
<p>The first rigorous approach to finding the true value of pi was based on geometrical approximations. Around 250 B.C., the Greek mathematician Archimedes drew polygons both around the outside and within the interior of circles. Measuring the perimeters of those gave upper and lower bounds of the range containing pi. He started with hexagons; by using polygons with more and more sides, he ultimately calculated three accurate digits of pi: 3.14. Around A.D. 150, Greek-Roman scientist Ptolemy used this method to calculate a value of 3.1416.</p>
<figure class="align-right zoomable">
<a href="https://images.theconversation.com/files/114534/original/image-20160309-13693-ipve24.png?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/114534/original/image-20160309-13693-ipve24.png?ixlib=rb-1.1.0&q=45&auto=format&w=237&fit=clip" srcset="https://images.theconversation.com/files/114534/original/image-20160309-13693-ipve24.png?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=600&fit=crop&dpr=1 600w, https://images.theconversation.com/files/114534/original/image-20160309-13693-ipve24.png?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=600&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/114534/original/image-20160309-13693-ipve24.png?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=600&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/114534/original/image-20160309-13693-ipve24.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=754&fit=crop&dpr=1 754w, https://images.theconversation.com/files/114534/original/image-20160309-13693-ipve24.png?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=754&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/114534/original/image-20160309-13693-ipve24.png?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=754&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">Liu Hui’s method of calculating pi also used polygons, but in a slightly different way.</span>
<span class="attribution"><a class="source" href="https://commons.wikimedia.org/wiki/File:Liuhui_Pi_Inequality.svg">Gisling and Pbroks13</a>, <a class="license" href="http://creativecommons.org/licenses/by-sa/4.0/">CC BY-SA</a></span>
</figcaption>
</figure>
<p>Independently, around A.D. 265, Chinese mathematician Liu Hui created another simple polygon-based iterative algorithm. He proposed a very fast and efficient approximation method, which gave four accurate digits. Later, around A.D. 480, Zu Chongzhi adopted Liu Hui’s method and achieved seven digits of accuracy. This record held for another 800 years. </p>
<p>In 1630, Austrian astronomer Christoph Grienberger arrived at 38 digits, which is the most accurate approximation manually achieved using polygonal algorithms.</p>
<h2>Moving beyond polygons</h2>
<p>The development of infinite series techniques in the 16th and 17th centuries greatly enhanced people’s ability to approximate pi more efficiently. An infinite series is the sum (or much less commonly, product) of the terms of an infinite sequence, such as ½, ¼, 1/8, 1/16, … 1/(2<sup>n</sup>). The first written description of an infinite series that could be used to compute pi was laid out in Sanskrit verse by Indian astronomer Nilakantha Somayaji around 1500 A.D., the proof of which was presented around 1530 A.D. </p>
<figure class="align-right zoomable">
<a href="https://images.theconversation.com/files/114535/original/image-20160309-13734-1wgyx0o.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/114535/original/image-20160309-13734-1wgyx0o.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=237&fit=clip" srcset="https://images.theconversation.com/files/114535/original/image-20160309-13734-1wgyx0o.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=745&fit=crop&dpr=1 600w, https://images.theconversation.com/files/114535/original/image-20160309-13734-1wgyx0o.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=745&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/114535/original/image-20160309-13734-1wgyx0o.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=745&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/114535/original/image-20160309-13734-1wgyx0o.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=936&fit=crop&dpr=1 754w, https://images.theconversation.com/files/114535/original/image-20160309-13734-1wgyx0o.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=936&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/114535/original/image-20160309-13734-1wgyx0o.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=936&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">Sir Isaac Newton.</span>
<span class="attribution"><a class="source" href="https://commons.wikimedia.org/wiki/File:Sir_Isaac_Newton_(1642-1727)._Oil_painting_by_a_follower_of_Wellcome_L0016625.jpg">Wellcome Trust</a>, <a class="license" href="http://creativecommons.org/licenses/by/4.0/">CC BY</a></span>
</figcaption>
</figure>
<p>In 1665, English mathematician and physicist Isaac Newton used infinite series to compute pi to 15 digits using calculus he and German mathematician Gottfried Wilhelm Leibniz discovered. After that, the record kept being broken. It reached 71 digits in 1699, 100 digits in 1706, and 620 digits in 1956 – the best approximation achieved without the aid of a calculator or computer.</p>
<figure class="align-right zoomable">
<a href="https://images.theconversation.com/files/114536/original/image-20160309-13737-1igjx6f.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/114536/original/image-20160309-13737-1igjx6f.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=237&fit=clip" srcset="https://images.theconversation.com/files/114536/original/image-20160309-13737-1igjx6f.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=729&fit=crop&dpr=1 600w, https://images.theconversation.com/files/114536/original/image-20160309-13737-1igjx6f.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=729&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/114536/original/image-20160309-13737-1igjx6f.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=729&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/114536/original/image-20160309-13737-1igjx6f.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=916&fit=crop&dpr=1 754w, https://images.theconversation.com/files/114536/original/image-20160309-13737-1igjx6f.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=916&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/114536/original/image-20160309-13737-1igjx6f.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=916&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">Carl Louis Ferdinand von Lindemann.</span>
</figcaption>
</figure>
<p>In tandem with these calculations, mathematicians were researching other characteristics of pi. Swiss mathematician Johann Heinrich Lambert (1728-1777) first proved that pi is an irrational number – it has an infinite number of digits that never enter a repeating pattern. In 1882, German mathematician Ferdinand von Lindemann proved that pi cannot be expressed in a rational algebraic equation (<a href="http://sprott.physics.wisc.edu/pickover/trans.html">such as pi²=10</a> or 9pi<sup>4</sup> - 240pi<sup>2</sup> + 1492 = 0).</p>
<h2>Toward even more digits of pi</h2>
<p>Bursts of calculations of even more digits of pi followed the adoption of iterative algorithms, which repeatedly build an updated value by using a calculation performed on the previous value. A simple example of an iterative algorithm allows you to approximate the square root of 2 as follows, using the formula (x+2/x)/2:</p>
<ul>
<li>(2+2/2)/2 = 1.5</li>
<li>(<strong>1.5</strong>+2/<strong>1.5</strong>)/2 = 1.4167</li>
<li>(<strong>1.4167</strong>+2/<strong>1.4167</strong>)/2 = 1.4142, which is a very close approximation already.</li>
</ul>
<p>Advances toward more digits of pi came with the use of a Machin-like algorithm (a generalization of English mathematician <a href="http://mathworld.wolfram.com/Machin-LikeFormulas.html">John Machin’s formula</a> developed in 1706) and the <a href="http://mathfaculty.fullerton.edu/mathews/n2003/GaussianQuadMod.html">Gauss-Legendre algorithm</a> (late 18th century) in electronic computers (invented mid-20th century). In 1946, ENIAC, the first electronic general-purpose computer, calculated 2,037 digits of pi in 70 hours. The most recent calculation found <a href="http://www.numberworld.org/y-cruncher/">more than 13 trillion digits of pi</a> in 208 days!</p>
<p>It has been widely accepted that for most numerical calculations involving pi, a dozen digits provides sufficient precision. According to mathematicians <a href="http://www.springer.com/us/book/9783642567353?wt_mc=GoogleBooks.GoogleBooks.3.EN&token=gbgen#otherversion=9783540665724">Jörg Arndt and Christoph Haenel</a>, 39 digits are sufficient to perform most cosmological calculations, because that’s the accuracy necessary to calculate the circumference of the observable universe to within one atom’s diameter. Thereafter, more digits of pi are not of practical use in calculations; rather, today’s pursuit of more digits of pi is about testing supercomputers and numerical analysis algorithms. </p>
<h2>Calculating pi by yourself</h2>
<p>There are also fun and simple methods for estimating the value of pi. One of the best-known is a method called “<a href="http://www.eveandersson.com/pi/monte-carlo-circle">Monte Carlo</a>.” </p>
<figure class="align-left zoomable">
<a href="https://images.theconversation.com/files/114488/original/image-20160309-13717-1c27vzw.png?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/114488/original/image-20160309-13717-1c27vzw.png?ixlib=rb-1.1.0&q=45&auto=format&w=237&fit=clip" srcset="https://images.theconversation.com/files/114488/original/image-20160309-13717-1c27vzw.png?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=600&fit=crop&dpr=1 600w, https://images.theconversation.com/files/114488/original/image-20160309-13717-1c27vzw.png?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=600&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/114488/original/image-20160309-13717-1c27vzw.png?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=600&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/114488/original/image-20160309-13717-1c27vzw.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=754&fit=crop&dpr=1 754w, https://images.theconversation.com/files/114488/original/image-20160309-13717-1c27vzw.png?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=754&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/114488/original/image-20160309-13717-1c27vzw.png?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=754&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">A square with inscribed circle.</span>
<span class="attribution"><a class="source" href="https://commons.wikimedia.org/wiki/File:Square-circle.svg">Deweirdifier</a></span>
</figcaption>
</figure>
<p>The method is fairly simple. To try it at home, draw a circle and a square around it (as at left) on a piece of paper. Imagine the square’s sides are of length 2, so its area is 4; the circle’s diameter is therefore 2, and its area is pi. The ratio between their areas is pi/4, or about 0.7854.</p>
<p>Now pick up a pen, close your eyes and put dots on the square at random. If you do this enough times, and your efforts are truly random, eventually the percentage of times your dot landed inside the circle will approach 78.54% – or 0.7854.</p>
<p>Now you’ve joined the ranks of mathematicians who have calculated pi through the ages.</p><img src="https://counter.theconversation.com/content/55744/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Xiaojing Ye does not work for, consult, own shares in or receive funding from any company or organization that would benefit from this article, and has disclosed no relevant affiliations beyond their academic appointment.</span></em></p>On the occasion of Pi Day, a look at the history of calculating the actual, and increasingly exact, value of pi (π).Xiaojing Ye, Assistant Professor of Mathematics and Statistics, Georgia State UniversityLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/507772015-11-17T19:05:34Z2015-11-17T19:05:34ZThe Man Who Knew Infinity: a mathematician’s life comes to the movies<p>The movie <a href="http://www.imdb.com/title/tt0787524/">The Man Who Knew Infinity</a> is about <a href="http://www-history.mcs.st-andrews.ac.uk/Biographies/Ramanujan.html">Srinivasa Ramanujan</a>, who is generally viewed by mathematicians as one of the two most romantic figures in our discipline. (I shall say more about the other romantic later.)</p>
<p>Ramanujan (1887–1920) was born and died, aged just 32, in Southern India. But in one of the most extraordinary events in mathematical history, he spent the period of World War I in Trinity College Cambridge at the invitation of the leading British mathematician <a href="http://www-history.mcs.st-and.ac.uk/Biographies/Hardy.html">Godfrey Harold (G. H.) Hardy</a> (1877–1947) and his great collaborator <a href="http://www-history.mcs.st-and.ac.uk/Mathematicians/Littlewood.html">John E. Littlewood</a>. </p>
<p>To avoid having to issue spoiler alerts, I will not tell much of Ramanujan’s story here.</p>
<figure class="align-right zoomable">
<a href="https://images.theconversation.com/files/102116/original/image-20151117-4970-ulole9.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/102116/original/image-20151117-4970-ulole9.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=237&fit=clip" srcset="https://images.theconversation.com/files/102116/original/image-20151117-4970-ulole9.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=822&fit=crop&dpr=1 600w, https://images.theconversation.com/files/102116/original/image-20151117-4970-ulole9.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=822&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/102116/original/image-20151117-4970-ulole9.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=822&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/102116/original/image-20151117-4970-ulole9.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=1033&fit=crop&dpr=1 754w, https://images.theconversation.com/files/102116/original/image-20151117-4970-ulole9.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=1033&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/102116/original/image-20151117-4970-ulole9.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=1033&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">Srinivasa Ramanujan.</span>
<span class="attribution"><a class="source" href="https://commons.wikimedia.org/wiki/File:Srinivasa_Ramanujan_-_OPC_-_1.jpg">Wikimedia</a></span>
</figcaption>
</figure>
<p>Suffice to say that as a boy he refused to learn anything but mathematics, he was almost entirely self taught and his pre-Cambridge work is contained in a series of <a href="http://www.math.uiuc.edu/%7Eberndt/articles/aachen.pdf">Notebooks</a>.</p>
<p>The work he did after returning to India in 1919 is contained in the misleadingly named <a href="http://www.math.uiuc.edu/%7Eberndt/lostnotebookhistory.pdf">Lost Notebook</a>. It was lost and later found in the Wren library of the leading college for mathematics of the leading University in England. While in England Ramanujan became the first Indian Fellow both of Trinity and of the Royal Society.</p>
<h2>A man of numbers</h2>
<p>Ramanujan had an extraordinary ability to see patterns. While he rarely proved his results he left a host of evaluations of sums and integrals. He was especially expert in a part of number theory called modular forms which is of even more interest today than when he died.</p>
<p>The lost notebook initiated the study of <a href="http://www.maa.org/news/puzzle-solved-ramanujans-mock-theta-conjectures-0">mock theta functions</a> which are only now being fully understood. Fleshing out his Notebooks has only recently been completed principally by American mathematicians <a href="http://www.math.uiuc.edu/%7Eberndt/">Prof Bruce Berndt</a> and <a href="http://www.personal.psu.edu/gea1/">Prof George Andrews</a>. It comprises thousands of printed pages.</p>
<p>An old Indian friend, Swami Swaminathan, oversaw the Ramanujan Library in Madras over half a century ago. He commented that had Ramanujan been born ten years early he would have been unable to receive the education and financial assistance that made his pre-Cambridge work possible. </p>
<p>Swaminathan went on to say that had Ramanujan been born ten years later, he would have probably received a more robust and more ordinary education. In either case our version of Ramanujan would not exist.</p>
<h2>Ramanujan and me</h2>
<p>Ramanujan has been part of my life for as long as I can remember. My father David was a student of one of Hardy’s students. In our house “the bible” referred to Hardy’s masterpiece Divergent Series.</p>
<p>In 1962 on the 75th anniversary of Ramanujan’s birth the envelope (below) arrived at my parents’ house. A kind stranger had put the franked stamps on the back.</p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/102107/original/image-20151117-4961-c10mrd.png?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/102107/original/image-20151117-4961-c10mrd.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/102107/original/image-20151117-4961-c10mrd.png?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=386&fit=crop&dpr=1 600w, https://images.theconversation.com/files/102107/original/image-20151117-4961-c10mrd.png?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=386&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/102107/original/image-20151117-4961-c10mrd.png?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=386&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/102107/original/image-20151117-4961-c10mrd.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=485&fit=crop&dpr=1 754w, https://images.theconversation.com/files/102107/original/image-20151117-4961-c10mrd.png?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=485&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/102107/original/image-20151117-4961-c10mrd.png?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=485&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">The anonymous letter.</span>
<span class="attribution"><span class="license">Author provided</span></span>
</figcaption>
</figure>
<p>In 1987 I was fortunate enough to speak with my brother at the major centennial conference on Ramanujan, held at the University of Illinois. We had become experts on and had extended Ramanujan’s <a href="http://wayback.cecm.sfu.ca/organics/papers/borwein/">work on Pi</a>.</p>
<p>Highlights at the conference included the Nobel prize winning astronomer <a href="http://www.nobelprize.org/nobel_prizes/physics/laureates/1983/chandrasekhar-bio.html">Subrahmanyan Chandrasekhar</a>, who described how important Ramanujan’s success in England had been to the self-confidence of himself and the founders of modern India including <a href="http://www.bbc.co.uk/history/historic_figures/nehru_jawaharlal.shtml">Jawaharlal Nehru</a>, who became the first prime minister of independent India in 1947.</p>
<p>In 2008 David Leavitt published a novelised version of Ramanujan’s life entitled the <a href="http://www.davidleavittwriter.com/books/indianclerk.html">Indian Clerk</a>. While Leavitt captures much beautifully, as a novelist, he takes some sizeable liberties. In particular, he dramatically embellishes Hardy’s (closeted?) homosexuality. I prefer my novels as fables and my biographies straight.</p>
<p>In 2012 on the 125th anniversary of Ramanujan’s birth the Notices of the American Mathematical Society <a href="http://www.ams.org/notices/201211/index.html">published</a> eight articles on his work. This suite forcibly showed how Ramanujan’s reputation and impact continue to grow. </p>
<h2>Gifted with numbers</h2>
<p>There is one famous anecdote about Ramanujan that even a non-mathematician can appreciate. In 1917 Ramanujan was hospitalised in London. He was said to have tuberculosis but it is more likely this was to cover a failed suicide attempt.</p>
<p>Hardy took a cab to visit him. Not being good at small talk all Hardy could think to say was that the number of his cab, 1,729, was uninteresting.</p>
<p>Ramanujan replied that quite to the contrary it was the smallest number expressible as a sum of two cubes in two distinct ways:</p>
<blockquote>
<p>1,729 = 12<sup>3</sup> + 1<sup>3</sup> = 10<sup>3</sup> + 9<sup>3</sup></p>
</blockquote>
<p>This is know known as Ramanujan’s taxi-cab number.</p>
<h2>Mathematicians in the movies</h2>
<p>There has been a recent spate of books, plays and movies, and TV series about mathematicians and theoretical physicists: <a href="http://www.imdb.com/title/tt0268978/">A Beautiful Mind</a> (2001), <a href="http://www.imdb.com/title/tt0340057/">Copenhagen</a> (2002), <a href="http://www.imdb.com/title/tt0377107/">Proof</a> (2005) and last year’s Oscar winning movies <a href="http://www.imdb.com/title/tt2084970/">The Imitation Game</a> about Alan Turing and <a href="http://www.imdb.com/title/tt2980516/">The Theory of Everything</a> on Stephen Hawking. </p>
<p>When I have read the book on someone’s life, I frequently avoid the movie. As writer Michael Crichton <a href="http://www.abc.net.au/science/slab/crichton/story.htm">put it</a>:</p>
<blockquote>
<p>All professions look bad in the movies […] why should scientists expect to be treated differently?</p>
</blockquote>
<p>Such movies – even biopics – have to compress a life of the mind into 90 to 120 minutes and give a flavour of genius to the rest of us. Even more than the books on which they are based, they have to make the character more exotic (Turing) or better redeemed (John Nash in A Beautiful Mind) than in the book let alone real life. </p>
<p>So I tend to avoid the movies and to be satisfied with my own knowledge and the corresponding book which can take 500 pages and more if it needs to.</p>
<h2>The Man Who Knew Infinity</h2>
<p>But I do intend to see the movie of The Man Who Knew Infinity. Ramanujan’s presence has been too much a part of my life (intellectually and personally) for me to miss it.</p>
<figure>
<iframe width="440" height="260" src="https://www.youtube.com/embed/DaOZHN3pCS0?wmode=transparent&start=0" frameborder="0" allowfullscreen=""></iframe>
</figure>
<p>In the movie Hardy is played by Jeremy Irons while Stephen Fry plays Sir <a href="https://en.wikipedia.org/wiki/Francis_Spring">Francis Spring</a> who was an early advocate of Ramanujan in India.</p>
<p>Twenty-five year old <a href="http://www.imdb.com/name/nm2353862/">Dev Patel</a>, who acted in <a href="http://www.imdb.com/title/tt1010048/">Slumdog Millionaire</a> (2008), is Ramanujan.</p>
<p>I reviewed very favourably Robert Kanigel’s book <a href="http://www.robertkanigel.com/_i__b_the_man_who_knew_infinity__b___a_life_of_the_genius_ramanujan__i__58016.htm">The Man Who Knew Infinity: A Life of the Genius</a>, on which the movie is based.</p>
<p>The current movie has had the brilliant Canadian-born Fields Medalist and Princeon professor of mathematics <a href="https://en.wikipedia.org/wiki/Manjul_Bhargava">Manjul Bhargava</a> as technical advisor. Bhargava is also an expert tabla player who works in fields well aligned with Ramanujan’s opus. This augurs well for the movie’s accuracy.</p>
<h2>The other romantic</h2>
<p>The other romantic mathematician I alluded to earlier was the even more short-lived French revolutionary <a href="http://www-history.mcs.st-andrews.ac.uk/Biographies/Galois.html">Évariste Galois</a>.</p>
<p>Galois (1811–1832) died, aged 20, in a duel related to the famous female mathematician <a href="http://www-history.mcs.st-andrews.ac.uk/Mathematicians/Germain.html">Sophie Germain</a>. As the story goes, there is a note in the margin of the manuscript that Galois wrote the night before the duel. It read:</p>
<blockquote>
<p>There is something to complete in this demonstration. I do not have the time. </p>
</blockquote>
<p>It is this note which has led to the legend that Galois spent his last night writing out all he knew about group (Galois) theory. This story appears to have grown with the telling but his life would also make for a very interesting movie.</p>
<hr>
<p><em>The Man Who Knew Infinity is screening, Wednesday November 18 2015, at selected cinemas in Adelaide, Brisbane, Byron Bay, Canberra, Melbourne, Perth and Sydney as the closing movie in the <a href="http://www.britishfilmfestival.com.au/films/the-man-who-knew-infinity">British Film Festival</a>.</em></p><img src="https://counter.theconversation.com/content/50777/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Jonathan Borwein (Jon) receives funding from ARC</span></em></p>Srinivasa Ramanujan was one of the most brilliant mathematicians of the 20th century. His story is told in the movie The Man Who Knew Infinity, screening tonight in selected cinemas in Australia.Jonathan Borwein (Jon), Laureate Professor of Mathematics, University of NewcastleLicensed as Creative Commons – attribution, no derivatives.