An IcosiDodecahedron is the intersection of an icosahedron and a dodecahedron if these two are scaled
so that their corresponding edges intersect at their midpoints. See the following animation.
The IcosiDodecahedron is obtained in the same way by midpoint truncation from the dodecahedron or from the icosahedron.
The Archimedean solid that is often called "truncated IcosiDodecahedron" is not quite a truncation,
because vertex truncation produces rectangles rather than squares. An additional slight shift
of the vertices is required. Still, the animation looks like a truncation of the IcosiDodecahedron.
The IcosiDodecahedron inside its dual, the rhombic triacontaeder (wireframe).
The IcosiDodecahedron vertices touch the midpoints of the rhombic triacontaeder faces.
If one constructs the IcosiDodecahedron as the intersection of a dodecahedron and an icosahedron whose
edge midpoints have the same distance from the body midpoint, then the dual polyhedron has
the vertices of the dodecahedron as 3-edged vertices and it has the vertices of the icosahedron
as 5-edged vertices. Its faces are rhombi whose diagonals have the same lengths as the edge lengths
of the icosahedron and the dodecahedron.
If one constructs icosahedron and dodecahedron from a cube of edge length 1, then they have the same
edge lengths, namely the golden ration (sqrt(5) - 1)/2. The distance of opposite edge midpoints of
icosahedron and dodecahedron is 1 and (sqrt(5) + 1)/2. We need to scale the icosahedron with the
factor (sqrt(5) + 1)/2, because we want their edge midpoints to coincide. The ratio of the rhombi
diagonals is therefore (sqrt(5) + 1)/2 = 2/(sqrt(5) - 1). Hence they are called golden rhombi.
With this scaling the icosahedron edge lengths are 1.
