# Octagonal-icosahedral duoprism

Octagonal-icosahedral duoprism
Rank5
TypeUniform
Notation
Bowers style acronymOike
Coxeter diagramx8o o5o3x
Elements
Tera20 triangular-octagonal duoprisms, 8 icosahedral prisms
Cells160 triangular prisms, 30 octagonal prisms, 8 icosahedra
Faces160 triangles, 240 squares, 12 octagons
Edges96+240
Vertices96
Vertex figurePentagonal scalene, edge lengths 1 (base pentagon), 2+2 (top), 2 (sides)
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {\frac {13+4{\sqrt {2}}+{\sqrt {5}}}{8}}}\approx 1.61605}$
Hypervolume${\displaystyle 5{\frac {3+3{\sqrt {2}}+{\sqrt {5}}+{\sqrt {10}}}{6}}\approx 10.53416}$
Diteral anglesTodip–op–todip: ${\displaystyle \arccos \left(-{\frac {\sqrt {5}}{3}}\right)\approx 138.18969^{\circ }}$
Ipe–ike–ipe: 135°
Todip–trip–ipe: 90°
Central density1
Number of external pieces28
Level of complexity20
Related polytopes
ArmyOike
RegimentOike
DualOctagonal-dodecahedral duotegum
ConjugatesOctagrammic-icosahedral duoprism, Octagonal-great icosahedral duoprism, Octagrammic-great icosahedral duoprism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryH3×I2(8), order 1920
ConvexYes
NatureTame

The octagonal-icosahedral duoprism or oike is a convex uniform duoprism that consists of 8 icosahedral prisms and 20 triangular-octagonal duoprisms. Each vertex joins 2 icosahedral prisms and 5 triangular-octagonal 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:

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

## Representations

An octagonal-icosahedral duoprism has the following Coxeter diagrams:

• x8o o5o3x (full symmetry)
• x4x o5o3x (octagons as ditetragons)