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.