Decoration of the Truncated Tetrahedron-An Archimedean Polyhedron-To Produce a New Class of Convex Equilateral Polyhedra with Tetrahedral Symmetry

作者：Schein Stan^*; Yeh Alexander J; Coolsaet Kris; Gayed James M

来源：Symmetry-Basel, 2016, 8(8): 82.

DOI：10.3390/sym8080082

• Goldberg polyhedra
• cages
• fullerenes
• tilings
• cutouts

The Goldberg construction of symmetric cages involves pasting a patch cut out of a regular tiling onto the faces of a Platonic host polyhedron, resulting in a cage with the same symmetry as the host. For example, cutting equilateral triangular patches from a 6.6.6 tiling of hexagons and pasting them onto the full triangular faces of an icosahedron produces icosahedral fullerene cages. Here we show that pasting cutouts from a 6.6.6 tiling onto the full hexagonal and triangular faces of an Archimedean host polyhedron, the truncated tetrahedron, produces two series of tetrahedral (T-d) fullerene cages. Cages in the first series have 28n(2) vertices (n >= 1). Cages in the second (leapfrog) series have 3 x 28n(2). We can transform all of the cages of the first series and the smallest cage of the second series into geometrically convex equilateral polyhedra. With tetrahedral (T-d) symmetry, these new polyhedra constitute a new class of "convex equilateral polyhedra with polyhedral symmetry". We also show that none of the other Archimedean polyhedra, six with octahedral symmetry and six with icosahedral, can host full-face cutouts from regular tilings to produce cages with the host's polyhedral symmetry.

• 出版日期2016-8