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New class of polyhedra discovered

Construction of cages with polyhedral symmetry from Goldberg triangles.

Construction of cages with polyhedral symmetry from Goldberg triangles.

Geometric forms have fascinated mathematicians since ancient times. Now researchers in California say they have discovered a new class of polyhedron that may already exist in nature and could help lead to novel buildings and other kinds of artificial structures. The findings are detailed in the Proceedings of the National Academy of Sciences.

The Greeks discovered the five Platonic polyhedra — the four-sided tetrahedron, the six-sided cube or hexahedron, the eight-sided octahedron, the 12-sided dodecahedron, and the 20-sided icosahedron — as well as the 13 Archimedean polyhedra, which include the truncated icosahedron that resembles soccer balls and the famous carbon molecule known as buckminsterfullerene. Astronomer and mathematician Johannes Kepler added two rhombic polyhedra. All in all, these were the only known classes of polyhedron that were equilateral, that is, with edges of equal length; convex, meaning they did not have concave dents on their surfaces; and with the high symmetry displayed by the Platonic polyhedra.

Now researchers Stan Schein and James Maurice Gayed at the University of California, Los Angeles, have discovered what they said is a fourth class of convex equilateral polyhedra with polyhedral symmetry. Natural structures with such geometries might exist among viruses.

To create these new polyhedra, one starts with a tetrahedron, an octahedron, or an icosahedron. The facets of these polyhedra are equilateral triangles.

One then draws a triangle on a mesh or tiling of hexagons, creating what are called Goldberg triangles. Next, one places such triangles – including vertices and edges from the hexagonal tiling – on each of the aforementioned
polyhedron’s facets. Finally, one adds edges to join the vertices of the Goldberg triangles (see Figure).

The resulting cages of vertices and edges are often called Goldberg polyhedra, first discovered in 1937 by Michael Goldberg. However, Schein noted “the word ‘polyhedron’ means different things to different groups of mathematicians.” To geometers, the facets in a polyhedron must be flat planes. The cages often called Goldberg polyhedra generally do not have flat planar facets.

By tinkering with the angles in the hexagons in Goldberg cages, Schein and Gayed found it was possible to construct shapes that were geometrically polyhedral. Moreover, these were convex, equilateral and had polyhedral symmetry. They call their discovery Goldberg polyhedra. (Although the hexagons in these polyhedra are equilateral, most are not equiangular, and so look squished in comparison to the regular polygon.)

They found there is only a single Goldberg polyhedron based on the tetrahedron, only one based on the octahedron, but an infinite number based on the icosahedron. Intriguingly, Goldberg polyhedra based on icosahedra are nearly spherical.

“We hope the shapes will be useful for objects that need to approach a spherical shape,” Schein says.

Spheres offer the most volume for surface area of any shape. Spheres are also the strongest shapes if one measures average strength at all points. Nearly spherical Goldberg polyhedra could be used in nature and engineering for structures that hold as much as possible or that can withstand internal or external pressures. For instance, the researchers noted that it is well known that some viruses are symmetrical and nearly spherical in shape, just like Goldberg polyhedra.

Categories: Biophysics and Computational Biology | Mathematics
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