Twenty equilateral triangles on thirty edges, five meeting at each vertex. Plato gives it to water, the most mobile and least bounded of the elements, and the icosahedron is the roundest of the five, with the most faces and the largest volume for its surface. Because five triangles meet at each vertex it carries the pentagonal symmetry, and the golden ratio enters with it.
Sacred Ratio
phi (the extreme and mean ratio sits in its coordinates and in its relation to the dodecahedron)
Construction
Twelve vertices arranged as three mutually perpendicular golden rectangles; thirty equal edges follow. Euclid constructs it inscribed in a sphere at Elements XIII.16 and proves the extreme and mean ratio in the book. It is the dual of the dodecahedron.