# Dodecagonal-cubic duoprism

Dodecagonal-cubic duoprism
Rank5
TypeUniform
Notation
Bowers style acronymTwacube
Coxeter diagramx12o x4o3o ()
Elements
Tera12 tesseracts, 6 square-dodecagonal duoprisms
Cells12+72 cubes, 12 dodecagonal prisms
Faces72+144 squares, 8 dodecagons
Edges96+144
Vertices96
Vertex figureTriangular scalene, edge lengths 2+3 (top), 2 (base triangle and sides)
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {11+4{\sqrt {3}}}}{2}}\approx 2.11709}$
Hypervolume${\displaystyle 3(2+{\sqrt {3}})\approx 11.19615}$
Diteral anglesTes–cube–tes: 150°
Height1
Central density1
Number of external pieces18
Level of complexity10
Related polytopes
ArmyTwacube
RegimentTwacube
DualDodecagonal-octahedral duotegum
ConjugateDodecagrammic-cubic duoprism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryB3×I2(12), order 1152
ConvexYes
NatureTame

The dodecagonal-cubic duoprism or twacube, also known as the square-dodecagonal duoprismatic prism, is a convex uniform duoprism that consists of 12 tesseracts and 6 square-dodecagonal duoprisms. Each vertex joins 2 tesseracts and 3 square-dodecagonal duoprisms. It is a duoprism based on a square and a dodecagonal prism, which makes it a convex segmentoteron.

This polyteron can be alternated into a hexagonal-tetrahedral duoantiprism, although it cannot be made uniform. The dodecagons can also be alternated into long ditrigons to create a tetrahedral-hexagonal prismantiprismoid, which is also nonuniform.

## Vertex coordinates

The vertices of a dodecagonal-cubic duoprism of edge length 1 are given by:

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

## Representations

A dodecagonal-cubic duoprism has the following Coxeter diagrams:

• x12o x4o3o () (full symmetry)
• x x4o x12o () (square-dodecagonal duoprismatic prism)
• x x x x12o () (dodecagonal prismatic prismatic prism)