# Hexagonal-icosahedral duoprism

Hexagonal-icosahedral duoprism
Rank5
TypeUniform
Notation
Bowers style acronymHike
Coxeter diagramx6o o5o3x
Elements
Tera20 triangular-hexagonal duoprisms, 6 icosahedral prisms
Cells120 triangular prisms, 30 hexagonal prisms, 6 icosahedra
Faces120 triangles, 180 squares, 12 hexagons
Edges72+180
Vertices72
Vertex figurePentagonal scalene, edge lengths 1 (base pentagon), 3 (top), 2 (sides)
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {\frac {13+{\sqrt {5}}}{8}}}\approx 1.38004}$
Hypervolume${\displaystyle {\frac {15{\sqrt {3}}+5{\sqrt {15}}}{8}}\approx 5.66821}$
Diteral anglesThiddip–hip–thiddip: ${\displaystyle \arccos \left(-{\frac {\sqrt {5}}{3}}\right)\approx 138.18969^{\circ }}$
Ipe–ike–ipe: 120°
Thiddip–trip–ipe: 90°
Central density1
Number of external pieces26
Level of complexity10
Related polytopes
ArmyHike
RegimentHike
DualHexagonal-dodecahedral duotegum
ConjugateHexagonal-great icosahedral duoprism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryH3×G2, order 1440
ConvexYes
NatureTame

The hexagonal-icosahedral duoprism or hike is a convex uniform duoprism that consists of 6 icosahedral prisms and 20 triangular-hexagonal duoprisms. Each vertex joins 2 icosahedral prisms and 5 triangular-hexagonal duoprisms.

## Vertex coordinates

The vertices of a triangular-icosahedral duoprism of edge length 1 are given by all even permutations of the last three coordinates of:

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

## Representations

A hexagonal-icosahedral duoprism has the following Coxeter diagrams:

• x6o o5o3x (full symmetry)
• x3x o5o3x (hexagons as ditrigons)