# Dodecagonal-square antiprismatic duoprism

Dodecagonal-square antiprismatic duoprism
Rank5
TypeUniform
Notation
Bowers style acronymTwasquap
Coxeter diagramx12o s2s8o ()
Elements
Tera12 square antiprismatic prisms, 8 triangular-dodecagonal duoprisms, 2 square-dodecagonal duoprisms
Cells96 triangular prisms, 24 cubes, 12 square antiprisms, 8+8 dodecagonal prisms
Faces96 triangles, 24+96+96 squares, 8 dodecagons
Edges96+96+96
Vertices96
Vertex figureIsosceles-trapezoidal scalene, edge lengths 1, 1, 1, 2 (base trapezoid), 2+3 (top), 2 (side edges)
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {\frac {20+{\sqrt {2}}+8{\sqrt {3}}}{8}}}\approx 2.09972}$
Hypervolume${\displaystyle {\sqrt {28+21{\sqrt {2}}+16{\sqrt {3}}+12{\sqrt {6}}}}\approx 10.71472}$
Diteral anglesSquappip–squap–squappip: 150°
Titwadip–twip–titwadip: = ${\displaystyle \arccos \left({\frac {1-2{\sqrt {2}}}{3}}\right)\approx 127.55160^{\circ }}$
Titwadip–twip–sitwadip: = ${\displaystyle \arccos \left({\frac {{\sqrt {3}}-{\sqrt {6}}}{3}}\right)\approx 103.83616^{\circ }}$
Height${\displaystyle {\frac {\sqrt[{4}]{8}}{2}}\approx 0.84090}$
Central density1
Number of external pieces22
Level of complexity40
Related polytopes
ArmyTwasquap
RegimentTwasquap
DualDodecagonal-square antitegmatic duotegum
ConjugateDodecagrammic-square antiprismatic duoprism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryI2(12)×I2(8)×A1+, order 384
ConvexYes
NatureTame

The dodecagonal-square antiprismatic duoprism or twasquap is a convex uniform duoprism that consists of 12 square antiprismatic prisms, 2 square-dodecagonal duoprisms, and 8 triangular-dodecagonal duoprisms. Each vertex joins 2 square antiprismatic prisms, 3 triangular-dodecagonal duoprisms, and 1 square-dodecagonal duoprism.

## Vertex coordinates

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

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

## Representations

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

• x12o s2s8o () (full symmetry; square antiprisms as alternated octagonal prisms)
• x12o s2s4s () (square antiprisms as alternated ditetragonal prisms)
• x6x s2s8o () (dodecagons as dihexagons)
• x6x s2s4s ()