Icosahedral prism

The icosahedral prism or ipe is a prismatic uniform polychoron that consists of 2 icosahedra and 20 triangular prisms. Each vertex joins 1 icosahedron and 5 triangular prisms. As the name suggests, it is a prism based on the icosahedron. As such it is also a convex segmentochoron (designated K-4.36 in Richard Klitzing's list).

Vertex coordinates
The vertices of an icosahedral prism of edge length 1 are given by all even permutations and all sign changes of the first three coordinates of:
 * $$\left(0,\,±\frac12,\,±\frac{1+\sqrt5}{4},\,±\frac12\right).$$

Representations
An icosahedral prism has the following Coxeter diagrams:


 * x o5o3x (full symmetry)
 * x2s3ss4o (bases as pyritohedral symmetry)
 * x2s3s3s (as snub tetrahedral prism)
 * oo5oo3xx&#x (bases seen separately)
 * xxxx oxoo5ooxo&#xt (H2×A1 axial, edge-first)

Related polychora
An icosahedral prism can be cut into a central pentagonal antiprismatic prism augmented with 2 pentagonal pyramidal prisms.

The regiment of the icosahedral prism also contains the great dodecahedral prism.