# Dodecagonal-hexagonal antiprismatic duoprism

Dodecagonal-hexagonal antiprismatic duoprism
Rank5
TypeUniform
Notation
Bowers style acronymTwahap
Coxeter diagramx12o s2s12o ()
Elements
Tera12 hexagonal antiprismatic prisms, 12 triangular-dodecagonal duoprisms, 2 hexagonal-dodecagonal duoprisms
Cells144 triangular prisms, 24 hexagonal prisms, 12 hexagonal antiprisms, 12+12 dodecagonal prisms
Faces144 triangles, 24 hexagons, 144+144 squares, 12 dodecagons
Edges144+144+144
Vertices144
Vertex figureIsosceles-trapezoidal scalene, edge lengths 1, 1, 1, 3 (base trapezoid), 2+3 (top), 2 (side edges)
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {11+5{\sqrt {3}}}}{2}}\approx 2.21699}$
Hypervolume${\displaystyle 3{\sqrt {38+22{\sqrt {3}}}}\approx 26.17147}$
Diteral anglesHappip–hap–happip: 150°
Titwadip–twip–titwadip: = ${\displaystyle \arccos \left({\frac {1-2{\sqrt {3}}}{3}}\right)\approx 145.22189^{\circ }}$
Titwadip–twip–hitwadip: = ${\displaystyle \arccos \left({\frac {3-2{\sqrt {3}}}{3}}\right)\approx 98.89943^{\circ }}$
Height${\displaystyle {\sqrt {{\sqrt {3}}-1}}\approx 0.85560}$
Central density1
Number of external pieces26
Level of complexity40
Related polytopes
ArmyTwahap
RegimentTwahap
DualDodecagonal-hexagonal antitegmatic duotegum
ConjugateDodecagrammic-hexagonal antiprismatic duoprism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryI2(12)×I2(12)×A1+, order 576
ConvexYes
NatureTame

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

## Vertex coordinates

The vertices of a dodecagonal-hexagonal antiprismatic duoprism of edge length 1 are given by all permutations of the first two coordinates of:

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

## Representations

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

• x12o s2s12o () (full symmetry; hexagonal antiprisms as alternated dodecagonal prisms)
• x12o s2s6s () (hexagonal antiprisms as alternated dihexagonal prisms)
• x6x s2s12o () (dodecagons as dihexagons)
• x6x s2s6s ()