# Decagonal-truncated dodecahedral duoprism

Decagonal-truncated dodecahedral duoprism
Rank5
TypeUniform
Notation
Bowers style acronymDatid
Coxeter diagramx10o x5x3o ()
Elements
Tera20 triangular-decagonal duoprisms, 12 decagonal duoprisms
Cells200 triangular prisms, 30+60+120 decagonal prisms, 10 truncated dodecahedra
Faces200 triangles, 300+600 squares, 60+120 decagons
Edges300+600+600
Vertices600
Vertex figureDigonal disphenoidal pyramid, edge lengths 1, (5+5)/2, (5+5)/2 (base triangle), (5+5)/2 (top), 2 (side edges)
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {\frac {49+19{\sqrt {5}}}{4}}}\approx 3.38167}$
Hypervolume${\displaystyle 25{\frac {\sqrt {197290+88222{\sqrt {5}}}}{24}}\approx 654.31294}$
Diteral anglesTiddip–tid–tiddip: 144°
Tradedip–dip–dedip: ${\displaystyle \arccos \left(-{\sqrt {\frac {5+2{\sqrt {5}}}{15}}}\right)\approx 142.62263^{\circ }}$
Dedip–dip–dedip: ${\displaystyle \arccos \left(-{\frac {\sqrt {5}}{5}}\right)\approx 116.56505^{\circ }}$
Dedip–dip–tiddip: 90°
Central density1
Number of external pieces42
Level of complexity30
Related polytopes
ArmyDatid
RegimentDatid
DualDecagonal-triakis icosahedral duotegum
ConjugateDecagrammic-quasitruncated great stellated dodecahedral duoprism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryH3×I2(10), order 2400
ConvexYes
NatureTame

The decagonal-truncated dodecahedral duoprism or datid is a convex uniform duoprism that consists of 10 truncated dodecahedral prisms, 12 decagonal duoprisms, and 20 triangular-decagonal duoprisms. Each vertex joins 2 truncated dodecahedral prisms, 1 triangular-decagonal duoprism, and 2 decagonal duoprisms.

## Vertex coordinates

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

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

## Representations

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

• x10o x5x3o (full symmetry)
• x5x x5x3o (decagons as dipentagons)