# Hexagonal duoprismatic prism

Hexagonal duoprismatic prism
Rank5
TypeUniform
Notation
Bowers style acronymHahip
Coxeter diagramx x6o x6o ()
Elements
Tera12 square-hexagonal duoprisms, 2 hexagonal duoprisms
Cells36 cubes, 12+24 hexagonal prisms
Faces72+72 squares, 24 hexagons
Edges36+144
Vertices72
Vertex figureTetragonal disphenoidal pyramid, edge lengths 3 (disphenoid bases) and 2 (remaining edges)
Measures (edge length 1)
Circumradius${\displaystyle {\frac {3}{2}}=1.5}$
Hypervolume${\displaystyle {\frac {27}{4}}=6.75}$
Diteral anglesShiddip–hip–shiddip: 120°
Shiddip–cube–shiddip: 90°
Hiddip–hip–shiddip: 90°
Height1
Central density1
Number of external pieces14
Level of complexity15
Related polytopes
ArmyHahip
RegimentHahip
DualHexagonal duotegmatic tegum
ConjugateHexagonal duoprismatic prism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryG2≀S2×A1, order 576
ConvexYes
NatureTame

The hexagonal duoprismatic prism or hahip, also known as the hexagonal-hexagonal prismatic duoprism, is a convex uniform duoprism that consists of 2 hexagonal duoprisms and 12 square-hexagonal duoprisms. Each vertex joins 4 square-hexagonal duoprisms and 1 hexagonal duoprism. Being a prism based on an orbiform polytope, it is also a convex segmentoteron.

This polyteron can be alternated into a triangular duoantiprismatic antiprism, although it cannot be made uniform.

## Vertex coordinates

The vertices of a hexagonal duoprismatic prism of edge length 1 are given by:

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

## Representations

A hexagonal duoprismatic prism has the following Coxeter diagrams:

• x x6o x6o (full symmetry)
• x x3x x3x () (hexagons as ditrigons)
• xx6oo xx6oo&#x (hexagonal duoprism atop hexagonal duoprism)
• xx3xx xx3xx&#x