# Decagonal-octahedral duoprism

Decagonal-octahedral duoprism
Rank5
TypeUniform
Notation
Bowers style acronymDoct
Coxeter diagramx10o o4o3x ()
Elements
Tera10 octahedral prisms, 8 triangular-decagonal duoprisms
Cells80 triangular prisms, 10 octahedra, 12 decagonal prisms
Faces80 triangles, 120 squares, 6 decagons
Edges60+120
Vertices60
Vertex figureSquare scalene, edge lengths 1 (base square), (5+5)/2 (top), 2 (sides)
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {\frac {4+{\sqrt {5}}}{2}}}\approx 1.76580}$
Hypervolume${\displaystyle {\frac {5{\sqrt {10+4{\sqrt {5}})}}}{6}}\approx 3.62708}$
Diteral anglesOpe–oct–ope: 144°
Tradedip–dip–tradedip: ${\displaystyle \arccos \left(-{\frac {1}{3}}\right)\approx 109.47122^{\circ }}$
Height${\displaystyle {\frac {\sqrt {6}}{3}}\approx 0.81650}$
Central density1
Number of external pieces=18
Level of complexity10
Related polytopes
ArmyDoct
RegimentDoct
DualDecagonal-cubic duotegum
ConjugateDecagrammic-octahedral duoprism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryB3×I2(10), order 960
ConvexYes
NatureTame

The decagonal-octahedral duoprism or doct is a convex uniform duoprism that consists of 10 octahedral prisms and 8 triangular-decagonal duoprisms. Each vertex joins 2 octahedral prisms and 4 triangular-decagonal duoprisms.

## Vertex coordinates

The vertices of a decagonal-octahedral duoprism of edge length 1 are given by all permutations and sign changes of the last three coordinates of:

• ${\displaystyle \left(0,\,\pm {\frac {1+{\sqrt {5}}}{2}},\,0,\,0,\,{\frac {\sqrt {2}}{2}}\right),}$
• ${\displaystyle \left(\pm {\sqrt {\frac {5+{\sqrt {5}}}{8}}},\,\pm {\frac {3+{\sqrt {5}}}{4}},\,0,\,0,\,{\frac {\sqrt {2}}{2}}\right),}$
• ${\displaystyle \left(\pm {\frac {\sqrt {5+2{\sqrt {5}}}}{2}},\,\pm {\frac {1}{2}},\,0,\,0,\,{\frac {\sqrt {2}}{2}}\right).}$

## Representations

A triangular-octahedral duoprism has the following Coxeter diagrams:

• x10o o4o3x (full symmetry)
• x5x o4o3x (decagons as dipentagons)
• x10o o3x3o (octahedra as tetratetrahedra)
• x5x o3x3o (decagons as dipentagons and octahedra as tetratetrahedra)