# Icosahedral prism

Icosahedral prism
Rank4
TypeUniform
SpaceSpherical
Notation
Bowers style acronymIpe
Coxeter diagramx o5o3x ()
Elements
Cells20 triangular prisms, 2 icosahedra
Faces40 triangles, 30 squares
Edges12+60
Vertices24
Vertex figurePentagonal pyramid, edge lengths 1 (base), 2 (legs)
Measures (edge length 1)
Circumradius${\displaystyle \sqrt{\frac{7+\sqrt5}{8}} ≈ 1.07448}$
Hypervolume${\displaystyle 5\frac{3+\sqrt5}{12} ≈ 2.18169}$
Dichoral anglesTrip–4–trip: ${\displaystyle \arccos\left(-\frac{\sqrt5}{3}\right) ≈ 138.18969°}$
Ike–3–trip: 90°
Height1
Central density1
Number of pieces22
Level of complexity4
Related polytopes
ArmyIpe
RegimentIpe
DualDodecahedral tegum
ConjugateGreat icosahedral prism
Abstract properties
Flag count960
Euler characteristic0
Topological properties
OrientableYes
Properties
SymmetryH3×A1, order 240
ConvexYes
NatureTame

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. 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:

• ${\displaystyle \left(0,\,±\frac12,\,±\frac{1+\sqrt5}{4},\,±\frac12\right).}$

## Representations

An icosahedral prism has the following Coxeter diagrams:

• x o5o3x (full symmetry)
• x2s3s4o () (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.