CONVEX EQUILATERAL POLYHEDRA WITH POLYHEDRAL SYMMETRY

摘要

A new class of polyhedron is constructed by decorating each of the triangular facets of an icosahedron with the T vertices and connecting edges of a “Goldberg triangle.” A unique set of internal angles in each planar face of each new polyhedron is then obtained, for example 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, where the independent variables are a subset of the internal angles in 6 gons. Alternatively, an iterative method that solves for angles within each hexagonal ring may be solved for that nulls dihedral angle discrepancy throughout the polyhedron. The 6 gon faces in the resulting “Goldberg polyhedra” are equilateral and planar, but not equiangular, and nearly spherical.

主权项

1. A method for designing a convex equilateral cage structure comprising:
selecting a Goldberg triangle comprising an equilateral triangle having three vertices that are each positioned on a center of a hexagon in a hexagonal tiling such that the equilateral triangle overlies a plurality of vertices from the hexagonal tiling, wherein the Goldberg triangle further comprises the plurality of vertices and each line segment connecting any two of the plurality of vertices; transferring the Goldberg triangle to each of the twenty faces of an icosahedron; adding connecting line segments that connect corresponding vertices across adjacent Goldberg triangles such that the Goldberg triangle line segments and the connecting line segments define a non-polyhedral cage, wherein the non-polyhedral cage comprises only trivalent vertices; and transforming the non-polyhedral cage such that the transformed cage comprises a plurality of hexagons and a plurality of pentagons, and the transformed cage is equilateral and convex.