# Decagonal-truncated cubic duoprism

Decagonal-truncated cubic duoprism
Rank5
TypeUniform
Notation
Bowers style acronymDatic
Coxeter diagramx10o x4x3o ()
Elements
Tera8 triangular-decagonal duoprisms, 10 truncated cubic prisms, 6 octagonal-decagonal duoprisms
Cells80 triangular prisms, 60 octagonal prisms, 12+24 decagonal prisms, 10 truncated cubes
Faces80 triangles, 120+240 squares, 60 octagons, 24 decagons
Edges120+240+240
Vertices240
Vertex figureDigonal disphenoidal pyramid, edge lengths 1, 2+2, 2+2 (base triangle), (5+5)/2 (top), 2 (side edges)
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {13+4{\sqrt {2}}+2{\sqrt {5}}}}{2}}\approx 2.40463}$
Hypervolume${\displaystyle 35{\frac {\sqrt {85+60{\sqrt {2}}+34{\sqrt {5}}+24{\sqrt {10}}}}{6}}\approx 104.63865}$
Diteral anglesTiccup–tic–ticcup: 144°
Tradedip–dip–odedip: ${\displaystyle \arccos \left(-{\frac {\sqrt {3}}{3}}\right)\approx 125.26439^{\circ }}$
Odedip–op–ticcup: 90°
Odedip–dip–odedip: 90°
Central density1
Number of external pieces24
Level of complexity30
Related polytopes
ArmyDatic
RegimentDatic
DualDecagonal-triakis octahedral duotegum
ConjugatesDecagrammic-truncated cubic duoprism, Decagonal-quasitruncated hexahedral duoprism, Decagrammic-quasitruncated hexahedral duoprism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryB3×I2(10), order 960
ConvexYes
NatureTame

The decagonal-truncated cubic duoprism or datic is a convex uniform duoprism that consists of 10 truncated cubic prisms, 6 octagonal-decagonal duoprisms, and 8 triangular-decagonal duoprisms. Each vertex joins 2 truncated cubic prisms, 1 triangular-decagonal duoprism, and 2 octagonal-decagonal duoprisms.

## Vertex coordinates

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

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

## Representations

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

• x10o x4x3o (full symmetry)
• x5x x4x3o (decagons as dipentagons)