“Fourth class of convex equilateral polyhedron with polyhedral symmetry related to fullerenes and viruses”
Schein and Gayed, PNAS 111: 2920-2925 (2014)
Goldberg polyhedra* are beautiful solids with higher symmetry. Vertices are 3-degree (3 edges meet at each vertex) and faces are triangles and hexagons (tetrahedral symmetry), squares and hexagons (octahedral symmetry), or pentagons and hexagons (icosahedral symmetry). Simply put, starting with a tetrahedron (triangles x4), cube (squares x6), or dodecahedron (pentagons x12), make a space between each face and then evenly fill the gap with hexagons (3-regularly and symmetrically). Because hexagons are variable, they are infinite. A series of these solids includes some of Archimedean solids (truncated tetrahedron, truncated octahedron, and truncated icosahedron), which have flat faces and no valleys (convex polyhedra). However, in most cases, when you try to construct Goldberg polyhedra with regular polygons, curvatures are formed on the surface. Therefore, irregular polygons with different edge lengths, different included angles, or both may be necessary. Goldberg polyhedra seemed to be difficult to jump on the bandwagon of Platonic solids, Archimedean solids, and Kepler’s rhombic polyhedra, which are convex equilateral polyhedra with higher symmetries. In the paper, authors figured out that, while maintaining equal edge lengths, convex polyhedra are attained just by adjusting the included angles of hexagons. A new class of the polyhedra is created. Shown below is a merely equilateral example (angles of hexagons are not adjusted), constructed with regular polygons. *Goldberg (1937), Tohoku Math J 43: 104-108. See also, Caspar and Klug (1962), Cold Spring Harb Symp Quant Biol 27: 1–24
Tetrahedral Goldberg polyhedra
3-regular polyhedra of triangular and hexagonal faces with tetrahedral symmetry. An infinite series begins with a tetrahedron. Triangles are 4 in total. Triangles are located at vertices of a tetrahedral cell, and hexagons fill the gaps evenly. Starting from one triangle, move m-faces in one direction, turn 60 degrees, and move n-faces to reach the closest triangle. A pair of numbers (m,n) is useful to classify the structures: m, n=0 (type I), m=n (type II), and m≠n (type III). Only type III is chiral, and looks twisted. Type II and III are particularly interesting. (m,n)=(1,0) is type I, a tetrahedron V4F4. (m,n)=(1,1) is type II, a truncated tetrahedron V12F8. Shown below is (m,n)=(1,2) V28F16, which is type III, and its mirror-image is (m,n)=(2, 1). It looks like a twisted truncated tetrahedron or a blazing fire when put it upside down (the figure is not exactly a polyhedron due to curved faces).