# Icosahedral prism

(Redirected from Ipe)
Icosahedral prism
Rank4
TypeUniform
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${\displaystyle {\sqrt {\frac {7+{\sqrt {5}}}{8}}}\approx 1.07448}$
Hypervolume${\displaystyle 5{\frac {3+{\sqrt {5}}}{12}}\approx 2.18169}$
Dichoral anglesTrip–4–trip: ${\displaystyle \arccos \left(-{\frac {\sqrt {5}}{3}}\right)\approx 138.18969^{\circ }}$
Ike–3–trip: 90°
Height1
Central density1
Number of external pieces22
Level of complexity4
Related polytopes
ArmyIpe
RegimentIpe
DualDodecahedral tegum
ConjugateGreat icosahedral prism
Abstract & topological properties
Flag count960
Euler characteristic0
OrientableYes
Properties
SymmetryH3×A1, order 240
ConvexYes
NatureTame

The icosahedral prism 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:

• ${\displaystyle \left(0,\,\pm {\frac {1}{2}},\,\pm {\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {1}{2}}\right).}$

## 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.