# Decagonal-tetrahedral duoprism

Decagonal-tetrahedral duoprism
Rank5
TypeUniform
Notation
Bowers style acronymDatet
Coxeter diagramx10o x3o3o ()
Elements
Tera10 tetrahedral prisms, 4 triangular-decagonal duoprisms
Cells10 tetrahedra, 40 triangular prisms6 decagonal prisms
Faces40 triangles, 60 squares, 4 decagons
Edges40+60
Vertices40
Vertex figureTriangular scalene, edge lengths 1 (base triangle), (5+5)/2 (top), 2 (sides)
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {\frac {15+4{\sqrt {5}}}{8}}}\approx 1.73004}$
Hypervolume${\displaystyle {\frac {5{\sqrt {10+4{\sqrt {5}}}}}{24}}\approx 0.90677}$
Diteral anglesTepe–tet–tepe: 144°
Tradedip–dip–tradedip: ${\displaystyle \arccos {\left({\frac {1}{3}}\right)}\approx 70.52878^{\circ }}$
HeightsDec atop tradedip: ${\displaystyle {\frac {\sqrt {6}}{3}}\approx 0.81650}$
Dip atop perp dip: ${\displaystyle {\frac {\sqrt {2}}{2}}\approx 0.70711}$
Central density1
Number of external pieces14
Level of complexity10
Related polytopes
ArmyDatet
RegimentDatet
DualDecagonal-tetrahedral duotegum
ConjugateDecagrammic-tetrahedral duoprism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryA3×I2(10), order 480
ConvexYes
NatureTame

The decagonal-tetrahedral duoprism or datet is a convex uniform duoprism that consists of 10 tetrahedral prisms and 4 triangular-decagonal duoprisms. Each vertex joins 2 tetrahedral prisms and 3 triangular-decagonal duoprisms.

## Vertex coordinates

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

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

## Representations

A decagonal-tetrahedral duoprism has the following Coxeter diagrams:

• x10o x3o3o (full symmetry)
• x5x x3o3o (decagons as dipentagons)