# Octagonal-tetrahedral duoprism

Octagonal-tetrahedral duoprism
Rank5
TypeUniform
Notation
Bowers style acronymOtet
Coxeter diagramx8o x3o3o ()
Elements
Tera8 tetrahedral prisms, 4 triangular-octagonal duoprisms
Cells8 tetrahedra, 32 triangular prisms, 6 octagonal prisms
Faces32 triangles, 48 squares, 4 octagons
Edges32+48
Vertices32
Vertex figureTriangular scalene, edge lengths 1 (base triangle), 2+2 (top), 2 (sides)
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {\frac {11+4{\sqrt {2}}}{8}}}\approx 1.44295}$
Hypervolume${\displaystyle {\frac {2+{\sqrt {2}}}{6}}\approx 0.56904}$
Diteral anglesTepe–tet–tepe: 135°
Tepe–trip–todip: 90°
Todip–op–todip: ${\displaystyle \arccos {\left({\frac {1}{3}}\right)}\approx 70.52878^{\circ }}$
HeightsOc atop todip: ${\displaystyle {\frac {\sqrt {6}}{3}}\approx 0.81650}$
Op atop perp op: ${\displaystyle {\frac {\sqrt {2}}{2}}\approx 0.70711}$
Central density1
Number of external pieces12
Level of complexity10
Related polytopes
ArmyOtet
RegimentOtet
DualOctagonal-tetrahedral duotegum
ConjugateOctagrammic-tetrahedral duoprism
Abstract & topological properties
Flag count3840
Euler characteristic2
OrientableYes
Properties
SymmetryA3×I2(8), order 384
ConvexYes
NatureTame

The octagonal-tetrahedral duoprism or otet is a convex uniform duoprism that consists of 8 tetrahedral prisms and 4 triangular-octagonal duoprisms. Each vertex joins 2 tetrahedral prisms and 3 triangular-octagonal duoprisms.

## Vertex coordinates

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

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

## Representations

An octagonal-tetrahedral duoprism has the following Coxeter diagrams:

• x8o x3o3o (full symmetry)
• x4x x3o3o (octagons as ditetragons)
• ox3oo xx8oo&#x (octagon atop triangular-octagonal duoprism)
• ox xo xx8oo&#x (octagonal prism atop orthogonal octagonal prism)