# Decagonal-truncated tetrahedral duoprism

Decagonal-truncated tetrahedral duoprism
Rank5
TypeUniform
Notation
Bowers style acronymDatut
Coxeter diagramx10o x3x3o ()
Elements
Tera4 triangular-decagonal duoprisms, 10 truncated tetrahedral prisms, 4 hexagonal-decagonal duoprisms
Cells40 triangular prisms, 40 hexagonal prisms, 10 truncated tetrahedra, 6+12 decagonal prisms
Faces40 triangles, 60+120 squares, 40 hexagons, 12 decagons
Edges60+120+120
Vertices120
Vertex figureDigonal disphenoidal pyramid, edge lengths 1, 3, 3 (base triangle), (5+5)/2 (top), 2 (side edges)
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {\frac {23+4{\sqrt {5}}}{8}}}\approx 1.99826}$
Hypervolume${\displaystyle {\frac {115{\sqrt {10+4{\sqrt {5}}}}}{24}}\approx 20.85574}$
Diteral anglesTuttip–tut–tuttip: 144°
Tradedip-dip-hadedip: ${\displaystyle \arccos \left(-{\frac {1}{3}}\right)\approx 109.47122^{\circ }}$
Hadedip–dip–hadedip: ${\displaystyle \arccos \left({\frac {1}{3}}\right)\approx 70.52877^{\circ }}$
Central density1
Number of external pieces18
Level of complexity30
Related polytopes
ArmyDatut
RegimentDatut
DualDecagonal-triakis tetrahedral duotegum
ConjugateDecagrammic-truncated tetrahedral duoprism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryA3×I2(10), order 480
ConvexYes
NatureTame

The decagonal-truncated tetrahedral duoprism or datut is a convex uniform duoprism that consists of 10 truncated tetrahedral prisms, 4 hexagonal-decagonal duoprisms, and 4 triangular-decagonal duoprisms. Each vertex joins 2 truncated tetrahedral prisms, 1 triangular-decagonal duoprism, and 2 hexagonal-decagonal duoprisms.

## Vertex coordinates

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

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

## Representations

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

• x10o x3x3o (full symmetry)
• x5x x3x3o (decagons as dipentagons)