Octagonal-icosahedral duoprism

Octagonal-icosahedral duoprism
(OFF file)
Rank	5
Type	Uniform
Notation
Bowers style acronym	Oike
Coxeter diagram	x8o o5o3x
Elements
Tera	20 triangular-octagonal duoprisms, 8 icosahedral prisms
Cells	160 triangular prisms, 30 octagonal prisms, 8 icosahedra
Faces	160 triangles, 240 squares, 12 octagons
Edges	96+240
Vertices	96
Vertex figure	Pentagonal scalene, edge lengths 1 (base pentagon), √2+√2 (top), √2 (sides)
Measures (edge length 1)
Circumradius	${\sqrt {\frac {13+4{\sqrt {2}}+{\sqrt {5}}}{8}}}\approx 1.61605$
Hypervolume	$5{\frac {3+3{\sqrt {2}}+{\sqrt {5}}+{\sqrt {10}}}{6}}\approx 10.53416$
Diteral angles	Todip–op–todip: $\arccos \left(-{\frac {\sqrt {5}}{3}}\right)\approx 138.18969^{\circ }$
	Ipe–ike–ipe: 135°
	Todip–trip–ipe: 90°
Central density	1
Number of external pieces	28
Level of complexity	20
Related polytopes
Army	Oike
Regiment	Oike
Dual	Octagonal-dodecahedral duotegum
Conjugates	Octagrammic-icosahedral duoprism, Octagonal-great icosahedral duoprism, Octagrammic-great icosahedral duoprism
Abstract & topological properties
Euler characteristic	2
Orientable	Yes
Properties
Symmetry	H3×I2(8), order 1920
Convex	Yes
Nature	Tame

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:

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

Representations

An octagonal-icosahedral duoprism has the following Coxeter diagrams:

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

External links

• Klitzing, Richard. "oike".