# Decagonal-hexagonal antiprismatic duoprism

Decagonal-hexagonal antiprismatic duoprism
Rank5
TypeUniform
Notation
Bowers style acronymDahap
Coxeter diagramx10o s2s12o ()
Elements
Tera10 hexagonal antiprismatic prisms, 12 triangular-decagonal duoprisms, 2 hexagonal-decagonal duoprisms
Cells120 triangular prisms, 20 hexagonal prisms, 10 hexagonal antiprisms, 12+12 decagonal prisms
Faces120 triangles, 120+120 squares, 20 hexagons, 12 decagons
Edges120+120+120
Vertices120
Vertex figureIsosceles-trapezoidal scalene, edge lengths 1, 1, 1, 3 (base trapezoid), (5+5)/2 (top), 2 (side edges)
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {9+{\sqrt {3}}+2{\sqrt {5}}}}{2}}\approx 1.94963}$
Hypervolume${\displaystyle {\frac {5{\sqrt {2\left(5+5{\sqrt {3}}+2{\sqrt {5}}+2{\sqrt {15}}\right)}}}{2}}\approx 17.98553}$
Diteral anglesTradedip–dip–tradedip: = ${\displaystyle \arccos \left({\frac {1-2{\sqrt {3}}}{3}}\right)\approx 145.22189^{\circ }}$
Happip–hap–happip: 144°
Tradedip–dip–hadedip: = ${\displaystyle \arccos \left({\frac {3-2{\sqrt {3}}}{3}}\right)\approx 98.89943^{\circ }}$
Height${\displaystyle {\sqrt {{\sqrt {3}}-1}}\approx 0.85560}$
Central density1
Number of external pieces24
Level of complexity40
Related polytopes
ArmyDahap
RegimentDahap
DualDecagonal-hexagonal antitegmatic duotegum
ConjugateDecagrammic-hexagonal antiprismatic duoprism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryI2(10)×I2(12)×A1+, order 480
ConvexYes
NatureTame

The decagonal-hexagonal antiprismatic duoprism or dahap is a convex uniform duoprism that consists of 10 hexagonal antiprismatic prisms, 2 hexagonal-decagonal duoprisms, and 12 triangular-decagonal duoprisms. Each vertex joins 2 hexagonal antiprismatic prisms, 3 triangular-decagonal duoprisms, and 1 hexagonal-decagonal duoprism.

## Vertex coordinates

The vertices of a decagonal-hexagonal antiprismatic duoprism of edge length 1 are given by:

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

## Representations

A decagonal-hexagonal antiprismatic duoprism has the following Coxeter diagrams:

• x10o s2s12o () (full symmetry; hexagonal antiprisms as alternated dodecagonal prisms)
• x10o s2s6s () (hexagonal antiprisms as alternated dihexagonal prisms)
• x5x s2s12o () (decagons as dipentagons)
• x5x s2s6s ()