# Heptagonal-dodecagonal duoprismatic prism

Heptagonal-dodecagonal duoprismatic prism
Rank5
TypeUniform
Notation
Bowers style acronymHetwip
Coxeter diagramx x7o x12o
Elements
Tera12 square-heptagonal duoprisms, 7 square-dodecagonal duoprisms, 2 heptagonal-dodecagonal duoprisms
Cells84 cubes, 7+14 dodecagonal prisms, 12+24 heptagonal prisms, 84 cubes
Faces84+84+168 squares, 24 heptagons, 14 dodecagons
Edges84+168+168
Vertices168
Vertex figureDigonal disphenoidal pyramid, edge lengths 2cos(π/7) (disphenoid base 1), 2+3 (disphenoid base 2), 2 (remaining edges)
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {9+4{\sqrt {3}}+{\frac {1}{\sin ^{2}{\frac {\pi }{7}}}}}}{2}}\approx 2.30435}$
Hypervolume${\displaystyle 21{\frac {2+{\sqrt {3}}}{4\tan {\frac {\pi }{7}}}}\approx 40.68584}$
Diteral anglesSquahedip–hep–squahedip: 150°
Sitwadip–twip–sitwadip: ${\displaystyle {\frac {5\pi }{7}}\approx 128.57143^{\circ }}$
Height1
Central density1
Number of external pieces21
Level of complexity30
Related polytopes
ArmyHetwip
RegimentHetwip
DualHeptagonal-dodecagonal duotegmatic tegum
ConjugatesHeptagonal-dodecagrammic duoprismatic prism, Heptagrammic-dodecagonal duoprismatic prism, Heptagrammic-dodecagrammic duoprismatic prism, Great heptagrammic-dodecagonal duoprismatic prism, Great heptagrammic-dodecagrammic duoprismatic prism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryI2(7)×I2(12)×A1, order 672
ConvexYes
NatureTame

The heptagonal-dodecagonal duoprismatic prism or hetwip, also known as the heptagonal-dodecagonal prismatic duoprism, is a convex uniform duoprism that consists of 2 heptagonal-dodecagonal duoprisms, 7 square-dodecagonal duoprisms, and 12 square-heptagonal duoprisms. Each vertex joins 2 square-heptagonal duoprisms, 2 square-dodecagonal duoprisms, and 1 heptagonal-dodecagonal duoprism. Being a prism based on an orbiform polytope, it is also a convex segmentoteron.

## Vertex coordinates

The vertices of a heptagonal-dodecagonal duoprismatic prism of edge length 2sin(π/7) are given by all permutations of the third and fourth coordinates of:

• ${\displaystyle \left(1,\,0,\,\pm (1+{\sqrt {3}})\sin {\frac {\pi }{7}},\,\pm (1+{\sqrt {3}})\sin {\frac {\pi }{7}},\,\pm \sin {\frac {\pi }{7}}\right),}$
• ${\displaystyle \left(\cos \left({\frac {j\pi }{7}}\right),\,\pm \sin \left({\frac {j\pi }{7}}\right),\,\pm (1+{\sqrt {3}})\sin {\frac {\pi }{7}},\,\pm (1+{\sqrt {3}})\sin {\frac {\pi }{7}},\,\pm \sin {\frac {\pi }{7}}\right),}$
• ${\displaystyle \left(1,\,0,\,\pm \sin {\frac {\pi }{7}},\,\pm (2+{\sqrt {3}})\sin {\frac {\pi }{7}},\,\pm \sin {\frac {\pi }{7}}\right),}$
• ${\displaystyle \left(\cos \left({\frac {j\pi }{7}}\right),\,\pm \sin \left({\frac {j\pi }{7}}\right),\,\pm \sin {\frac {\pi }{7}},\,\pm (2+{\sqrt {3}})\sin {\frac {\pi }{7}},\,\pm \sin {\frac {\pi }{7}}\right),}$

where j = 2, 4, 6.

## Representations

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

• x x7o x12o (full symmetry)
• x x7o x6x (dodecagons as dihexagons)
• xx7oo xx12oo&#x (heptagonal-dodecagonal duoprism atop heptagonal-dodecagonal duoprism)
• xx7oo xx6xx&#x