# Truncated icosahedral prism

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Truncated icosahedral prism
Rank4
TypeUniform
SpaceSpherical
Notation
Bowers style acronymTipe
Coxeter diagramx o5x3x ()
Elements
Cells12 pentagonal prisms, 20 hexagonal prisms, 2 truncated icosahedra
Faces30+60 squares, 24 pentagons, 40 hexagons
Edges60+60+120
Vertices120
Vertex figureSphenoid, edge lengths (1+5)/2, 3, 3 (base), 2 (legs)
Measures (edge length 1)
Circumradius${\displaystyle \sqrt{\frac{31+9\sqrt5}{8}} ≈ 2.52796}$
Hypervolume${\displaystyle \frac{125+43\sqrt5}{4} ≈ 55.28773}$
Dichoral anglesPip–4–hip: ${\displaystyle \arccos\left(-\sqrt{\frac{5+2\sqrt5}{15}}\right) ≈ 142.62263°}$
Hip–4–hip: ${\displaystyle \arccos\left(-\frac{\sqrt5}{3}\right) ≈ 138.18968°}$
Ti–5–pip: 90°
Ti–6–hip: 90°
Height1
Central density1
Number of pieces34
Level of complexity12
Related polytopes
ArmyTipe
RegimentTipe
DualPentakis dodecahedral tegum
ConjugateTruncated great icosahedral prism
Abstract properties
Flag count2880
Euler characteristic0
Topological properties
OrientableYes
Properties
SymmetryH3×A1, order 240
ConvexYes
NatureTame

The truncated icosahedral prism or tipe is a prismatic uniform polychoron that consists of 2 truncated icosahedra, 12 pentagonal prisms, and 20 hexagonal prisms. Each vertex joins 1 truncated icosahedron, 1 pentagonal prism, and 2 hexagonal prisms. It is a prism based on the truncated icosahedron. As such it is also a CRF segmentochoron (designated K-4.127 on Richard Klitzing's list).

## Vertex coordinates

Coordinates for the vertices of a truncated icosahedral prism of edge length 1 are given by all even permutations of the first three coordinates of:

• ${\displaystyle \left(0,\,±\frac12,\,±3\frac{1+\sqrt5}{4},\,±\frac12\right),}$
• ${\displaystyle \left(±\frac12,\,±\frac{5+\sqrt5}{4},\,±\frac{1+\sqrt5}{2},\,±\frac12\right),}$
• ${\displaystyle \left(±\frac{1+\sqrt5}{4},\,±1,\,±\frac{2+\sqrt5}{2},\,±\frac12\right).}$

## Representations

A truncated icosahedral prism has the following Coxeter diagrams:

• x o5x3x (full symmetry)
• oo5xx3xx&#x (bases considered separately)