Fractals are exquisite structures produced by nature, hiding in plain sight all around us.
They are tricky to define precisely, though most are linked by a set of four common fractal features: infinite intricacy, zoom symmetry, complexity from simplicity and fractional dimensions – all of which will be explained below.
The next fern you encounter will provide a great illustration of these features if you pause for a closer look. First, notice that the shape of the fern is intricately detailed. Remarkably, you can see that the leaves are shaped like little copies of the branches.
In fact, the entire fern is mostly built up from the same basic shape repeated over and over again at ever smaller scales. Most astonishing of all, fractal mathematics reveals that this humble fern leaf is neither a one- nor an two-dimensional shape, but hovers somewhere in-between.
Exactly what shape does this fern have?
The classical Euclidean geometry taught in high-school leaves us at a loss to answer this simple question. Though cylinders and rectangles may be great for modelling the shapes of technology, there are precious few regular shapes to be found in the natural world.
How can we describe a fern as a precise mathematical shape? How can we build a mathematical model of this wonderful object? Enter a completely new world of beautiful shapes: a branch of mathematics known as fractal geometry.
1. Infinite Intricacy
Many patterns of nature are so irregular and fragmented that, compared with Euclid … Nature exhibits not simply a higher degree but an altogether different level of complexity. – Benoît Mandelbröt, The Fractal Geometry of Nature
In 1861, the discovery of the world’s first fractal sent shockwaves through the mathematical community.
If you pick up a pen and doodle a zig-zag, you should end up with a number of sharp corners connected by smooth lines. To show it could be done, the German mathematician Karl Weierstrass constructed a zig-zag that was so jagged, it was nothing but corners - the ultimate mathematical staccato.
No matter how many times the shape was magnified, any glimmer of a smooth line would invariably dissolve into a never-ending cascade of corners, packed ever-more tightly together. Weierstrass’ shape had irregular details at every possible scale - the first key feature of a fractal shape.
Mathematicians labelled Weierstrass’ shape as “pathological,” as it stood in defiance of the tried-and-tested tools of calculus that had been so painstakingly assembled over the previous few hundred years. It remained just a tantalising glimpse of a completely new kind of shape until modern computing power gave mathematicians the keys to the promised land.
2. Zoom Symmetry
I found myself, in other words, constructing a geometry … of things which had no geometry. – Benoît Mandelbröt, 1924-2010
The blossoming of fractal geometry into a new branch of mathematics is largely thanks to the Polish-born mathematician Benoît Mandelbröt and his seminal 1977 essay The Fractal Geometry of Nature.
Mandelbröt worked for IBM New York in the 1960s. With the company’s immense computing power at his disposal, he was able to explore the strange new world of fractals for the first time.
Perhaps the most famous fractal today is the Mandelbröt set (as shown below), named after its discoverer. To draw it exactly is impossible, but it can be approximated by painstakingly colouring each point in the plane separately.
To choose the right colour for a specific point, we apply a simple movement rule to the point over and over again and watch how long it takes for the point to “escape” off the page. Though practically impossible to create by hand, modern interactive applets (such as this one, created by British designer Paul Neave) allow you to create and explore these sets in real time.
These computer programs allow you to spot a new kind of symmetry associated with fractals. To mathematicians, a symmetry is an action that when applied to a shape will leave it looking (more or less) the same.
For instance, we say that a square has rotational symmetry because there’s no way to tell if a square has been spun around by 90 degrees when you weren’t looking.
The infinite intricacy of fractals permits them a completely new type of symmetry that isn’t found in ordinary shapes. Incredibly, zooming in on a small region of a fractal leaves you looking at the same shape you started with. Tiny bits of the fractal can look exactly the same as the whole.
Far from being a mathematical curiosity, this zoom symmetry can be found everywhere in nature - once you know to look for it.
3. Complexity from simplicity
Bottomless wonders spring from simple rules which are repeated without end. – Benoît Mandelbröt, 1924-2010
As Mandelbröt was putting fractals under the microscope, the British mathematician Michael Barnsley (currently of the Australian National University) was approaching the same objects from a different angle.
Though the geometry of fractal shapes is infinitely complex, a third trait of fractals is that their complexity arises from very simple core definitions. The shape of a fractal can be completely captured by a small list of mathematical mappings that describe exactly how the smaller copies are arranged to form the whole fractal.
Barnsley’s influential 1988 book Fractals Everywhere contained an algorithm, known as the Chaos Game, that allowed computers to quickly generate any fractal shape from its known mappings.
The Chaos Game took a starting point in space and tracked its motion as it hopped around. Each hop was determined by selecting one of the mappings at random.
Remarkably, no matter the starting point and the order in which the mappings were traversed, the point would quickly be sucked onto a “strange attractor” - the fractal shape - and once there, it would dance around on it forever.
These fractal attractors lie at the heart of Chaos Theory. Since the behaviour of a chaotic system also dances around a fractal attractor, the infinite intricacy of fractal shapes means the slightest nudge to the system can move the point off the attractor entirely.
Crucially, Barnsley found a way to take any desired shape and calculate its list of fractal mappings. Since the complex shape could be completely reconstructed from the simple maps, Barnsley’s algorithms were instrumental in the new field of image compression - allowing the original edition of Microsoft Encarta to pack tens of thousands of images onto a single CD.
4. Fractional dimensions
Nature has played a joke on the mathematicians. The 19th-Century mathematicians may have been lacking in imagination, but Nature was not. – F J Dyson, as quoted by Benoît Mandelbröt, The Fractal Nature of Geometry
The last and most striking feature of fractals is that are not one-, two- or three-dimensional, but somewhere in-between. Nature seems perfectly happy to use fractional dimensions, so we should be too. In order to do so, we must first clarify what we mean by “dimension”.
The idea of “dimension” has many different (but consistent) mathematical definitions. Intuitively, we can think of a shape’s dimension as a measure of how rough the shape is, or a score that reflects how well the shape fills up its surrounding space.
These intuitive ideas can be made mathematically precise. To illustrate a fractional dimension, think about a piece of paper, which is (practically) 2-dimensional. A solid sphere is 3-dimensional and fills up more space than the piece of paper.
Now crumple the paper into a ball. You now have a fractal-like shape that fills up more space than the paper, but not as much space as the solid sphere. It scores approximately 2.5 for its dimension.
Similarly, your lungs are about 2.97 dimensional - their fractal geometry allows them to pack lots of surface area (a few tennis courts) into a small volume (a few tennis balls). Packing such a huge surface area into your body provides you with the ability to extract enough oxygen to keep you alive.
Fractals can be found everywhere in the world around you, from a humble fern to the structure of the universe on the largest of scales.
Even certain parts of your anatomy are fractal, including your brain. If you are mindful of fractals, you will be struck by the sheer variety of places you can find them as you go about your daily routine - from clouds, plants and the landscape to church windows and laboratories …
Fractal mathematics not only allows us to begin modelling the shapes of nature, it can also reawaken our childlike wonder at the world around us.
With thanks to Jon Borwein.