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.