Generating Closed Shapes from Regular Tilings

Abstract

Closed geometrical shapes may be obtained from a hexagonal tiling by substituting 12 pentagons for hexagons and fusing the appropriate edges. Such geometrical shapes describe the fullerenes, demonstrating that this mathematical rule has utility in structural chemistry. Closure from a hexagonal tiling may also be obtained by substituting six squares; or four triangles; or combinations of pentagons (one point each), squares (two points each), and triangles (three points each) so that the total is 12 points. Similar recipes exist for obtaining closed shapes from tetragonal or trigonal tilings. Structures thus obtained may evolve into additional structures having the same symmetry in a manner similar to the evolution of the Archimedean solids from the Platonic solids. Important examples include the deltahedra, a group of eight convex polyhedra consisting of all triangles and belonging to high-symmetry point groups. These polyhedra evolve into additional high symmetry shapes, many of which have already found u...