# Decagonal-truncated octahedral duoprism

Decagonal-truncated octahedral duoprism
Rank5
TypeUniform
Notation
Bowers style acronymDatoe
Coxeter diagramx10o o4x3x ()
Elements
Tera6 square-decagonal duoprisms, 10 truncated octahedral prisms, 8 hexagonal-decagonal duoprisms
Cells60 cubes, 80 hexagonal prisms, 12+24 decagonal prisms, 10 truncated octahedra
Faces60+120+240 squares, 80 hexagons, 24 decagons
Edges120+240+240
Vertices240
Vertex figureDigonal disphenoidal pyramid, edge lengths 2, 3, 3 (base triangle), (5+5)/2 (top), 2 (side edges)
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {\frac {8+{\sqrt {5}}}{2}}}\approx 2.26231}$
Hypervolume${\displaystyle 20{\sqrt {10+4{\sqrt {5}}}}\approx 87.05004}$
Diteral anglesTope–toe–tope: 144°
Squadedip–dip–hadedip: ${\displaystyle \arccos \left(-{\frac {\sqrt {3}}{3}}\right)\approx 125.26439^{\circ }}$
Hadedip–dip–hadedip: ${\displaystyle \arccos \left(-{\frac {1}{3}}\right)\approx 109.47122^{\circ }}$
Central density1
Number of external pieces24
Level of complexity30
Related polytopes
ArmyDatoe
RegimentDatoe
DualDecagonal-tetrakis hexahedral duotegum
ConjugateDecagrammic-truncated octahedral duoprism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryB3×I2(10), order 960
ConvexYes
NatureTame

The decagonal-truncated octahedral duoprism or datoe is a convex uniform duoprism that consists of 10 truncated octahedral prisms, 8 hexagonal-decagonal duoprisms, and 6 square-decagonal duoprisms. Each vertex joins 2 truncated octahedral prisms, 1 square-decagonal duoprism, and 2 hexagonal-decagonal duoprisms.

This polyteron can be alternated into a pentagonal-pyritohedral icosahedral duoantiprism, although it cannot be made uniform.

## Vertex coordinates

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

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

## Representations

A decagonal-truncated octahedral duoprism has the following Coxeter diagrams:

• x10o o4x3x (full symmetry)
• x10o x3x3x
• x5x o4x3x (decagons as dipentagons)
• x5x x3x3x