Brand-New Class of Shapes Discovered for the First Time in 400 Years

For the first time in 400 years, a new class of shapes has been described.

Meet the Goldberg polyhedra.

Two neurologists recently published a paper that describes a new class of shapes. (Image source: Schein et al. via Proceedings of the National Academy of Sciences)
Two neurologists recently published a paper that describes a new class of shapes. (Image source: Schein et al. via Proceedings of the National Academy of Sciences)

Now, Goldberg polyhedrons were actually first described in the 1930s by mathematician Michael Goldberg. They are symmetrical shapes are made up of pentagons and hexagons. But neurologists at the University of California at Los Angeles, described as “polyhedron lovers” by Science News believe they’ve discovered a new class of convex, equilateral polyhedra and named it after Goldberg, though they seem to disagree with Goldberg’s original shapes being considered 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,” study co-author Stan Schein told the Conversation.

“This is the first new class of convex, equilateral polyhedra with icosahedral symmetry in 400 years,” Schein, who discovered the shape with his colleague James Gayed, said, according to Science News.

The neurologists’ findings were published in the journal Proceedings of the National Academy of Sciences last week. In the paper’s abstract, the authors described this new class of convex equilateral polyhedrons.

The Goldberg polyhedra is “nearly spherical.” The faces of hexagons making up the shape have equal sides but unequal angles, while the “3gons, 4gons or 5gons” are regular, meaning they have equal side lengths and angles.

Here’s a bit more of a technical explanation about the shape from the study’s abstract:

We begin by decorating each of the triangular facets of a tetrahedron, an octahedron, or an icosahedron with the T vertices and connecting edges of a “Goldberg triangle.” We obtain the unique set of internal angles in each planar face of each polyhedron by solving a system of n equations and n variables, where the equations set the dihedral angle discrepancy about different types of edge to zero, and the variables are a subset of the internal angles in 6gons. Like the faces in Kepler’s rhombic polyhedra, the 6gon faces in Goldberg polyhedra are equilateral and planar but not equiangular. We show that there is just a single tetrahedral Goldberg polyhedron, a single octahedral one, and a systematic, countable infinity of icosahedral ones, one for each Goldberg triangle. Unlike carbon fullerenes and faceted viruses, the icosahedral Goldberg polyhedra are nearly spherical. The reasoning and techniques presented here will enable discovery of still more classes of convex equilateral polyhedra with polyhedral symmetry.

They discovered this new class of polyhedron while studying the human eye and the protein clathrin.

David Craven with the University of Birmingham explained to the Conversation that if one were to “take a cube and blow it up like a balloon,” its faces would bulge. To Schein and Gayed, this bulge would rule it out of being a polyhedron given that it wouldn’t have a planar — flat — face.

A Goldberg polyhedron described by Michael Goldberg. (Image source: Wikimedia)
A Goldberg polyhedron described by Michael Goldberg. (Image source: Wikimedia)

“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,” Craven told the Conversation.

To address this issue, which the researchers called the dihedral angle discrepancy, they found a way to flatten all the faces by making the dihedral angle discrepancy zero.

Branko Grunbaum, a mathematician at the University of Washington, told Science News the findings are “correct, and the result is new.”

As for how this discovery could be applied to other fields, the Conversation reported that it could impact research of virus structures.

(H/T: Daily Mail)