# Truncated icosahedral prism

(Redirected from Tipe)
Truncated icosahedral prism
Rank4
TypeUniform
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${\displaystyle {\sqrt {\frac {31+9{\sqrt {5}}}{8}}}\approx 2.52796}$
Hypervolume${\displaystyle {\frac {125+43{\sqrt {5}}}{4}}\approx 55.28773}$
Dichoral anglesPip–4–hip: ${\displaystyle \arccos \left(-{\sqrt {\frac {5+2{\sqrt {5}}}{15}}}\right)\approx 142.62263^{\circ }}$
Hip–4–hip: ${\displaystyle \arccos \left(-{\frac {\sqrt {5}}{3}}\right)\approx 138.18968^{\circ }}$
Ti–5–pip: 90°
Ti–6–hip: 90°
Height1
Central density1
Number of external pieces34
Level of complexity12
Related polytopes
ArmyTipe
RegimentTipe
DualPentakis dodecahedral tegum
ConjugateTruncated great icosahedral prism
Abstract & topological properties
Flag count2880
Euler characteristic0
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:

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

## Representations

A truncated icosahedral prism has the following Coxeter diagrams:

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