# Octagonal-truncated icosahedral duoprism

Octagonal-truncated icosahedral duoprism
Rank5
TypeUniform
Notation
Bowers style acronymOti
Coxeter diagramx8o o5x3x
Elements
Tera12 pentagonal-octagonal duoprisms, 20 hexagonal-octagonal duoprisms
Cells96 pentagonal prisms, 160 hexagonal prisms, 30+60 octagonal prisms, 8 truncated icosahedral prisms
Faces240+480 squares, 96 pentagons, 160 hexagons, 60 octagons
Edges240+480+480
Vertices480
Vertex figureDigonal disphenoidal pyramid, edge lengths (1+5)/2, 3, 3 (base triangle), 2+2 (top), 2 (side edges)
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {\frac {37+4{\sqrt {2}}+9{\sqrt {5}}}{8}}}\approx 2.80137}$
Hypervolume${\displaystyle {\frac {125+125{\sqrt {2}}+43{\sqrt {5}}+43{\sqrt {10}}}{2}}\approx 266.95278}$
Diteral anglesPodip–op–hodip: ${\displaystyle \arccos \left(-{\sqrt {\frac {5+2{\sqrt {5}}}{15}}}\right)\approx 142.62263^{\circ }}$
Hodip–op–hodip: ${\displaystyle \arccos \left(-{\frac {\sqrt {5}}{3}}\right)\approx 138.18968^{\circ }}$
Tipe–ti–tipe: 135°
Podip–pip–tipe: 90°
Hodip–hip–tipe: 90°
Central density1
Number of external pieces40
Level of complexity30
Related polytopes
ArmyOti
RegimentOti
DualOctagonal-pentakis dodecahedral duotegum
ConjugatesOctagrammic-truncated icosahedral duoprism, Octagonal-truncated great icosahedral duoprism, Octagrammic-truncated great icosahedral duoprism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryH3×I2(8), order 1920
ConvexYes
NatureTame

The octagonal-truncated icosahedral duoprism or oti is a convex uniform duoprism that consists of 8 truncated icosahedral prisms, 20 hexagonal-octagonal duoprisms, and 12 pentagonal-octagonal duoprisms. Each vertex joins 2 truncated icosahedral prisms, 1 pentagonal-octagonal duoprism, and 2 hexagonal-octagonal duoprisms.

## Vertex coordinates

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

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

## Representations

An octagonal-truncated icosahedral duoprism has the following Coxeter diagrams:

• x8o o5x3x (full symmetry)
• x4x o5x3x (octagons as ditetragons)