# Hexagonal-dodecagonal duoprismatic prism

Hexagonal-dodecagonal duoprismatic prism
Rank5
TypeUniform
Notation
Bowers style acronymHatwip
Coxeter diagramx x6o x12o
Elements
Tera12 square-hexagonal duoprisms, 6 square-dodecagonal duoprisms, 2 hexagonal-dodecagonal duoprisms
Cells72 cubes, 6+12 dodecagonal prisms, 12+24 hexagonal prisms
Faces72+72+144 squares, 24 hexagons, 12 dodecagons
Edges72+144+144
Vertices144
Vertex figureDigonal disphenoidal pyramid, edge lengths 3 (disphenoid base 1), 2+3 (disphenoid base 2), 2 (remaining edges)
Measures (edge length 1)
Circumradius${\displaystyle {\frac {1+2{\sqrt {3}}}{2}}\approx 2.23205}$
Hypervolume${\displaystyle 9{\frac {3+2{\sqrt {3}}}{2}}\approx 29.08846}$
Diteral anglesShiddip–hip–shiddip: 150°
Height1
Central density1
Number of external pieces20
Level of complexity30
Related polytopes
ArmyHatwip
RegimentHatwip
DualHexagonal-dodecagonal duotegmatic tegum
ConjugateHexagonal-dodecagrammic duoprismatic prism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryG2×I2(12)×A1, order 576
ConvexYes
NatureTame

The hexagonal-dodecagonal duoprismatic prism or hatwip, also known as the hexagonal-dodecagonal prismatic duoprism, is a convex uniform duoprism that consists of 2 hexagonal-dodecagonal duoprisms, 6 square-dodecagonal duoprisms, and 12 square-hexagonal duoprisms. Each vertex joins 2 square-hexagonal duoprisms, 2 square-dodecagonal duoprisms, and 1 hexagonal-dodecagonal duoprism. Being a prism based on an orbiform polytope, it is also a convex segmentoteron.

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

## Vertex coordinates

The vertices of a hexagonal-dodecagonal duoprismatic prism of edge length 1 are given by all permutations of the third and fourth coordinates of:

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

## Representations

A hexagonal-dodecagonal duoprismatic prism has the following Coxeter diagrams:

• x x6o x12o (full symmetry)
• x x3x x12o (hexagons as ditrigons)
• x x6o x6x (dodecagons as dihexagons)
• x x3x x6x
• xx6oo xx12oo&#x (hexagonal-dodecagonal duoprism atop hexagonal-dodecagonal duoprism)
• xx3xx xx12oo&#x
• xx6oo xx6xx&#x
• xx3xx xx6xx&#x