Some Archimedean Solids in the Icosahedral Lattice
Requires a Wolfram Notebook System
Interact on desktop, mobile and cloud with the free Wolfram Player or other Wolfram Language products.
This Demonstration considers a set of points of integral linear combinations , where the are six vertices of a regular icosahedron and the coefficients to are integers between and or and , for a total of or points. The coordinates of the are the six permutations of , , and (the golden ratio). The convex hull of is a triacontahedron. Certain choices of linear combinations give the vertices of a dodecahedron, an icosidodecahedron, a truncated dodecahedron, and a truncated icosahedon. Given one vertex on a solid, all the other vertices are points in that are at the same distance as the given one.[more]
To get a rhombicosidodecahedron or a great rhombicosidodecahedron, coefficients of absolute value 3 or 5 are needed.[less]
Contributed by: Izidor Hafner (April 2013)
Open content licensed under CC BY-NC-SA
The set of all triplets with integer coefficients is called phi space.