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

February 18, 2014 by Akshat Rathi, The Conversation
Not so special anymore. Credit: 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.

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.

Platonic solids in ascending order of number of faces. nasablueshift

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.

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.

Goldberg polyhedron.

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.

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

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.

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Display comments: newest first

not rated yet Feb 18, 2014
I hadn't noticed before; the number of sides on the faces is: 3, 4, 3, 5, 3, 6.
Will the next two shapes be 3 and 7?
5 / 5 (3) Feb 18, 2014
"the next two"??? There. are. five. Platonic. Solids...Five.
There are 13 Archimedean solids. (excluding ∞ prisims and the (elongated) gryobicupola, and two left-hand reflections)
There are 4 Kepler-Poinsot solids.
There are 92 Johnson solids.
I don't know how many possible Goldberg polyhedra there are: the article implies the faces are non-planar, so I expect the number to be unrestricted.
Wikipedia has illustrations of many of these under the appropriate title.
At the least, you should know, as every school child does, the Platonic Solids.
not rated yet Feb 18, 2014
Sorry for my wording error.
Change "Will the next two shapes..." to "Would the next two shapes..."

I know that there will never be new Platonic solids, if for no other reason that Plato is dead.
I was suggesting a hypothetical, something that every school child should learn to recognize.
5 / 5 (3) Feb 18, 2014
Fun stuff. The "blowing up of a cube with subdivisions" looks exactly like the things I was trying out when implementing an automatic hexahedral mesher for finite element analysis some 12 years back. (As there keeping each element convex is also important).

Weird how the same geometries pop up all over the place.
5 / 5 (1) Feb 18, 2014
Fun stuff. The "blowing up of a cube with subdivisions" looks exactly like the things I was trying out when implementing an automatic hexahedral mesher for finite element analysis some 12 years back. (As there keeping each element convex is also important).

Weird how the same geometries pop up all over the place.

if you look at it in terms of the structural stability produced by the balance of forces or energetic equilibrium inherent is such shapes, then not at all surprising.

Whydening Gyre
5 / 5 (2) Feb 18, 2014
It's the way the universe counts. PHI (ve)
5 / 5 (6) Feb 19, 2014

No offense mate, but I don't think you quite understand the topology of Platonic solids. There will never be another Platonic solid, beyond the five Platonic solids...and it has nothing to do with the fact that Plato is dead. It has to do with the intrinsic geometry of 3-space (our physical universe). The defining characteristics of a Platonic solid is that:
-- All of it's faces must be congruent (identical), convex regular polygons
-- Each face only intersects at their edges.
-- And the most important constraint...the same number of polygons (faces) meet at each vertex (corner).

With these geometric criteria, there are only five possible Platonic solids, built up of either triangles (3 solids), squares (1 solid) and pentagons (1 solid). There is a very simple geometric proof of this.
3.7 / 5 (3) Feb 19, 2014
Thank you for the notion of "intrinsic geometry". Platonic solids are used as presuppositions for the creation of the ideal (Νους) and visible world (Σύμπαν):
-Tetrahedron for fire,
-cube for earth,
-octahedron for air,
-dodecahedron for ether
-and icosahedron for water,
(Timaios, 48 Β - 50 C).
The defining characteristics are also represent the notion of "justice", as the "golden bond", among the " numbered Elements" of the Universe.
Result of the ancient Mathematics:The reality of symmetry and the asymmetry, in nature. As long the Universe is countable, it is also Knowledgeable.
5 / 5 (2) Feb 19, 2014
I do understand that. The "Plato is dead" was an attempt at humor. I know that our geometry is bounded by dimensions.

My initial comment was simply that I hadn't noticed what appeared to be a progressive sequence (erroneously adding "six"). I don't know why I hadn't seen it before, and was (more or less) idly speculating on what it would signify if it were a sequence. The idea of 4 and 5D shapes and how they might relate back to familiar geometry was intriguing.
The only reason this has gone beyond mere idle, groundless comment of "hey, it looks like a pattern!" is... well, it doesn't seem like there is a reason, does it?
I appreciate your measured response, and your desire to fill what looks like a gap in my knowledge. Aristotle knows, there are plenty of them. I've also found that asking nonsense questions, such as "what if there were more than five platonic solids?", sometimes results in another gap being filled. Or, more frequently, more nonsense.
5 / 5 (1) Feb 23, 2014
We quilters were there before You: http://mathtouris...mes.html
I have seen websites that go further into this and one author speculated on the patters in Iranian rugs.
1 / 5 (1) Feb 23, 2014
@ SoylentGrin
Plato Solids 3, 4, 3, 5, 3
Plain Tessellations 3, 4, 6
3 x 4 x 5 x 6 = 360 ; '360' Degrees in circle
'14th' ASRP @ vertex 5-7-7 = 365 1/7 degrees (days)
'15th' ASRP @ vertex 3-5-4-5 = 366 degrees (leap days)
5 / 5 (2) Feb 24, 2014
This article falls short by failing to mention geodesic domes and the work of Buckminster Fuller and others. The sentence "dome-shaped buildings are never circular in shape" is just wrong. Geodesic domes can approximate spheres as accurately as you want. Triangles are more rigid than hexagons and pentagons, so if the interest is building things that are strong and efficient, it's wise to relax the preference that all edges be equal length, and use the topological duals, which are triangulated polyhedra.

Also note: in this sentence "any point on a line that connects two points in a shape must never fall outside the shape", the word "line" should be replaced with "line segment", to correctly define convexity.

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