# Dodecagonal duoprismatic prism

Dodecagonal duoprismatic prism
Rank5
TypeUniform
Notation
Bowers style acronymTwatwip
Coxeter diagramx x12o x12o ()
Elements
Tera24 square-dodecagonal duoprisms, 2 dodecagonal duoprisms
Cells144 cubes, 24+48 dodecagonal prisms
Faces288+288 squares, 48 dodecagons
Edges144+576
Vertices288
Vertex figureTetragonal disphenoidal pyramid, edge lengths 2+3 (disphenoid bases) and 2 (remaining edges)
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {17+8{\sqrt {3}}}}{2}}\approx 2.77743}$
Hypervolume${\displaystyle 9(7+4{\sqrt {3}})\approx 125.35383}$
Height1
Central density1
Number of external pieces26
Level of complexity15
Related polytopes
ArmyTwatwip
RegimentTwatwip
DualDodecagonal duotegmatic tegum
ConjugateDodecagrammic duoprismatic prism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryI2(12)≀S2×A1, order 2304
ConvexYes
NatureTame

The dodecagonal duoprismatic prism or twatwip, also known as the dodecagonal-dodecagonal prismatic duoprism, is a convex uniform duoprism that consists of 2 dodecagonal duoprisms and 24 square-dodecagonal duoprisms. Each vertex joins 4 square-dodecagonal duoprisms and 1 dodecagonal duoprism. Being a prism based on an orbiform polytope, it is also a convex segmentoteron.

This polyteron can be alternated into a hexagonal duoantiprismatic antiprism, although it cannot be made uniform. Half of the dodecagons can also be alternated into long ditrigons to create a hexagonal-hexagonal prismatic prismantiprismoid, which is also nonuniform.

## Vertex coordinates

The vertices of an dodecagonal duoprismatic prism of edge length 1 are given by:

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

## Representations

A dodecagonal duoprismatic prism has the following Coxeter diagrams:

• x x12o x12o () (full symmetry)
• x x6x x6x () (dodecagons as dihexagons)
• xx12oo xx12oo&#x (dodecagonal duoprism atop dodecagonal duoprism)
• xx6xx xx6xx&#x