# Octagonal-truncated tetrahedral duoprism

Octagonal-truncated tetrahedral duoprism
Rank5
TypeUniform
Notation
Bowers style acronymOtut
Coxeter diagramx8o x3x3o
Elements
Tera4 triangular-octagonal duoprisms, 8 truncated tetrahedral prisms, 4 hexagonal-octagonal duoprisms
Cells32 triangular prisms, 32 hexagonal prisms, 8 truncated tetrahedra, 6+12 octagonal prisms
Faces32 triangles, 48+96 squares, 32 hexagons, 12 octagons
Edges48+96+96
Vertices96
Vertex figureDigonal disphenoidal pyramid, edge lengths 1, 3, 3 (base triangle), 2+2 (top), 2 (side edges)
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {\frac {19+4{\sqrt {2}}}{8}}}\approx 1.75559}$
Hypervolume${\displaystyle 23{\frac {2+{\sqrt {2}}}{6}}\approx 13.08782}$
Diteral anglesTuttip–tut–tuttip: 135°
Todip-op-hodip: ${\displaystyle \arccos \left(-{\frac {1}{3}}\right)\approx 109.47122^{\circ }}$
Todip–trip–tuttip: 90°
Hodip-hip-tuttip: 90°
Hodip–op–hodip: ${\displaystyle \arccos \left({\frac {1}{3}}\right)\approx 70.52877^{\circ }}$
Central density1
Number of external pieces16
Level of complexity30
Related polytopes
ArmyOtut
RegimentOtut
DualOctagonal-triakis tetrahedral duotegum
ConjugateOctagrammic-truncated tetrahedral duoprism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryA3×I2(8), order 384
ConvexYes
NatureTame

The octagonal-truncated tetrahedral duoprism or otut is a convex uniform duoprism that consists of 8 truncated tetrahedral prisms, 4 hexagonal-octagonal duoprisms, and 4 triangular-octagonal duoprisms. Each vertex joins 2 truncated tetrahedral prisms, 1 triangular-octagonal duoprism, and 2 hexagonal-octagonal duoprisms.

## Vertex coordinates

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

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

## Representations

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

• x8o x3x3o (full symmetry)
• x4x x3x3o (octagons as ditetragons)