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After 400 years, mathematicians find a new class of solid shapes

The work of the Greek polymath Plato has kept millions of people busy for millennia. A few among them have been mathematicians…

Not so special anymore. fdecomite

The work of the Greek polymath Plato has kept millions of people busy for millennia. A few among them have been mathematicians who have obsessed about Platonic solids, a class of geometric forms that are highly regular and are commonly found in nature.

Since Plato’s work, two other classes of equilateral convex polyhedra, as the collective of these shapes are called, have been found: Archimedean solids (including truncated icosahedron) and Kepler solids (including rhombic polyhedra). Nearly 400 years after the last class was described, researchers claim that they may have now invented a new, fourth class, which they call Goldberg polyhedra. Also, they believe that their rules show that an infinite number of such classes could exist.

Platonic love for geometry

Equilateral convex polyhedra need to have certain characteristics. First, each of the sides of the polyhedra needs to be of the same length. Second, the shape must be completely solid: that is, it must have a well-defined inside and outside that is separated by the shape itself. Third, any point on a line that connects two points in a shape must never fall outside the shape.

Platonic solids, the first class of such shapes, are well known. They consist of five different shapes: tetrahedron, cube, octahedron, dodecahedron and icosahedron. They have four, six, eight, twelve and twenty faces, respectively.

Platonic solids in ascending order of number of faces. nasablueshift

These highly regular structures are commonly found in nature. For instance, the carbon atoms in a diamond are arranged in a tetrahedral shape. Common salt and fool’s gold (iron sulfide) form cubic crystals, and calcium fluoride forms octahedral crystals.

The new discovery comes from researchers who were inspired by finding such interesting polyhedra in their own work that involved the human eye. Stan Schein at the University of California in Los Angeles was studying the retina of the eye when he became interested in the structure of protein called clathrin. Clathrin is involved in moving resources inside and outside cells, and in that process it forms only a handful number of shapes. These shapes intrigued Schein, who ended up coming up with a mathematical explanation for the phenomenon.

Goldberg polyhedron.

During this work, Schein came across the work of 20th century mathematician Michael Goldberg who described a set of new shapes, which have been named after him, as Goldberg polyhedra. The easiest Goldberg polyhedron to imagine looks like a blown-up football, as the shape is made of many pentagons and hexagons connected to each other in a symmetrical manner (see image to the left).

However, Schein believes that Goldberg’s shapes – or cages, as geometers call them – are not polyhedra. “It may be confusing because Goldberg called them polyhedra, a perfectly sensible name to a graph theorist, but to a geometer, polyhedra require planar faces,” Schein said.

Instead, in a new paper in the Proceedings of the National Academy of Sciences, Schein and his colleague James Gayed have described that a fourth class of convex polyhedra, which given Goldberg’s influence they want to call Goldberg polyhedra, even at the cost of confusing others.

Blown up dodecahedron. stblaize

A crude way to describe Schein and Gayed’s work, according to David Craven at the University of Birmingham, “is to take a cube and blow it up like a balloon” – which would make its faces bulge (see image to the right). The point at which the new shapes breaks the third rule – which is, any point on a line that connects two points in that shape falls outside the shape – is what Schein and Gayed care about most.

Craven said, “There are two problems: the bulging of the faces, whether it creates a shape like a saddle, and how you turn those bulging faces into multi-faceted shapes. The first is relatively easy to solve. The second is the main problem. Here one can draw hexagons on the side of the bulge, but these hexagons won’t be flat. The question is whether you can push and pull all these hexagons around to make each and everyone of them flat.”

During the imagined bulging process, even one that involves replacing the bulge with multiple hexagons, as Craven points out, there will be formation of internal angles. These angles formed between lines of the same faces – referred to as dihedral angle discrepancies – means that, according to Schein and Gayed, the shape is no longer a polyhedron. Instead they claimed to have found a way of making those angles zero, which makes all the faces flat, and what is left is a true convex polyhedron (see image below).

Their rules, they claim, can be applied to develop other classes of convex polyhedra. These shapes will be with more and more faces, and in that sense there should be an infinite variety of them.

Playing with shapes

Such mathematical discoveries don’t have immediate applications, but often many are found. For example, dome-shaped buildings are never circular in shape. Instead they are built like half-cut Goldberg polyhedra, consisting of many regular shapes that give more strength to the structure than using round-shaped construction material.

Only the one in the right bottom corner is a convex polyhedra. Stan Schein/PNAS

However, there may be some immediate applications. The new rules create polyhedra that have structures similar to viruses or fullerenes, a carbon allotrope. The fact that there has been no “cure” against influenza, or common flu, shows that stopping viruses is hard. But if we are able to describe the structure of a virus accurately, we get a step closer to finding a way of fighting them.

If nothing else, Schein’s work will invoke mathematicians to find other interesting geometric shapes, now that equilateral convex polyhedra may have been done with.

Update: The post was corrected to clarify that it refers to equilateral convex polyhedra.

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13 Comments sorted by

  1. James Hill

    Industrial Designer

    Richard Buckminster Fuller claimed convex polyhedra in his 1956 US patent on Geodesic Domes, and those patented buildings are certainly circular in shape, in contradiction to the description in the article.
    Further there is no structural reason to claim that round structural components are less "strong" than straight ones.
    It is more complicated that that.
    Anyway, those difficulties could be excluded from the article without harming its message.

    1. Andrew Mole

      Structural Engineer

      In reply to James Hill

      Trusses are constructed from straight members because they have more resistance to buckling. Thus the statement is indeed correct if you understand it as referring to structural domes built in a polyhedral form. Incidentally, I don't think geodesic domes qualify as polyhedra as not all of the sides are of the same length.

      "Each of the sides of the polyhedra needs to be of the same length"

    2. James Hill

      Industrial Designer

      In reply to Andrew Mole

      Perhaps Andrew, but you may find that thin shell domes have been analysed using membrane theory, and that the distribution of stresses in a thin shell dome follow those in multi strut geodesic domes composed of straight elements.
      I suggest to you that the forces which lead to buckling are so large that straight elements would not confer any greater advantage than curved elements, in the thin shell structures to which the author refers
      Then there is the problem for "Regular " polyhedra which you…

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    3. Andrew Mole

      Structural Engineer

      In reply to James Hill

      I think that we have some confusion due to the inaccuracies in the article. The author has now clarified that he was referring to "equilateral convex polyhedra". I followed the author in his definition. In this case, the geodesic "domes" from Buckminster Fuller do not qualify.

      In your latest post you are referring to shells. That is a little off-topic to either geodesic domes or what the author was referring to, which are lattice shells, except insofar as they can be seen as approximating to…

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    4. Andrew Mole

      Structural Engineer

      In reply to Andrew Mole

      Incidentally, if we take the case of a snap-through buckling case (which can be seen as being similar to thin shell buckling), the forces could indeed be large, and in a lattice truss with pin joints these will be axial forces. At that point, if they are passing through a curved member, that member will need to be made stronger to cope with the curvature. The curved member will also be less stiff axially, which will also have some effect.

      If the joints are rigid and the lattice members are working in flexure, then this changes things, but I thinkmthe same principle still applies - straight is better than curved between the nodes. Which were you referring to?

  2. Grace Jenkins


    Question: how can better describing the shape of a virus help you stop a virus?

  3. Peter Walford

    Practical Theoretician

    Very interesting article: thanks for including mathematics in the Conversation.

  4. Aaron Muir Hamilton

    logged in via Facebook

    Interesting, but this looks like a π = 4 proof on the surface, I guess I'll have to read /their/ report.

  5. Dan Winter

    logged in via Facebook

    platonic symmetry is useful primarily as an intro to the wave mechanics of the natural structures- pent symmetry structures (based on golden ratio) like water clathrates, and fullerenes- are electrically useful primarily because the pent symmetry array (characteristic of all life) - causes an electrical field which is conjugate / centripetal / and NEGENTROPIC - (why fullerenes are such a hot topic in chemistry- the CENTER has a containment field)-
    hence the origin of biologic structures is also negentropic / centripetal / and conjugate - (the reason biofields appear self organizing - originally discovering in physics of phase conjugation);.. animations- equations on this: