# Heptagonal-truncated dodecahedral duoprism

Heptagonal-truncated dodecahedral duoprism
Rank5
TypeUniform
Notation
Bowers style acronymHetid
Coxeter diagramx7o x5x3o
Elements
Tera20 triangular-heptagonal duoprisms, 12 heptagonal-decagonal duoprisms, 7 truncated dodecahedral prisms
Cells140 triangular prisms, 30+60 heptagonal prisms, 84 decagonal prisms, 7 truncated dodecahedra
Faces140 triangles, 210+420 squares, 60 heptagons, 84 decagons
Edges210+420+420
Vertices420
Vertex figureDigonal disphenoidal pyramid, edge lengths 1, (5+5)/2, (5+5)/2 (base triangle), cos(π/7) (top), 2 (side edges)
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {\frac {37+15{\sqrt {5}}+{\frac {2}{\sin ^{2}{\frac {\pi }{7}}}}}{8}}}\approx 3.18522}$
Hypervolume${\displaystyle 35{\frac {99+47{\sqrt {5}}}{48\tan {\frac {\pi }{7}}}}\approx 309.02670}$
Diteral anglesTheddip–hep–hedadip: ${\displaystyle \arccos \left(-{\sqrt {\frac {5+2{\sqrt {5}}}{15}}}\right)\approx 142.62263^{\circ }}$
Tiddip–tid–tiddip: ${\displaystyle {\frac {5\pi }{7}}\approx 128.57143^{\circ }}$
Hedadip–hep–hedadip: ${\displaystyle \arccos \left(-{\frac {\sqrt {5}}{5}}\right)\approx 116.56505^{\circ }}$
Theddip–trip–tiddip: 90°
Central density1
Number of external pieces39
Level of complexity30
Related polytopes
ArmyHetid
RegimentHetid
DualHeptagonal-triakis icosahedral duotegum
ConjugatesHeptagrammic-truncated dodecahedral duoprism, Great heptagrammic-truncated dodecahedral duoprism, Heptagonal-quasitruncated great stellated dodecahedral duoprism, Heptagrammic-quasitruncated great stellated dodecahedral duoprism, Great heptagrammic-quasitruncated great stellated dodecahedral duoprism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryH3×I2(7), order 1680
ConvexYes
NatureTame

The heptagonal-truncated dodecahedral duoprism or hetid is a convex uniform duoprism that consists of 7 truncated dodecahedral prisms, 12 heptagonal-decagonal duoprisms, and 20 triangular-heptagonal duoprisms. Each vertex joins 2 truncated dodecahedral prisms, 1 triangular-heptagonal duoprism, and 2 heptagonal-decagonal duoprisms.

## Vertex coordinates

The vertices of a heptagonal-truncated dodecahedral duoprism of edge length 2sin(π/7) are given by all even permutations of the last three coordinates of:

• ${\displaystyle \left(1,\,0,\,0,\,\pm \sin {\frac {\pi }{7}},\,\pm {\frac {(5+3{\sqrt {5}})\sin {\frac {\pi }{7}}}{2}}\right),}$
• ${\displaystyle \left(1,\,0,\,\pm \sin {\frac {\pi }{7}},\,\pm {\frac {(3+{\sqrt {5}})\sin {\frac {\pi }{7}}}{2}},\,\pm (3+{\sqrt {5}})\sin {\frac {\pi }{7}}\right),}$
• ${\displaystyle \left(1,\,0,\,\pm {\frac {(3+{\sqrt {5}})\sin {\frac {\pi }{7}}}{2}},\,\pm (1+{\sqrt {5}})\sin {\frac {\pi }{7}},\,\pm (2+{\sqrt {5}})\sin {\frac {\pi }{7}}\right),}$
• ${\displaystyle \left(\cos \left({\frac {j\pi }{7}}\right),\,\pm \sin \left({\frac {j\pi }{7}}\right),\,0,\,\pm \sin {\frac {\pi }{7}},\,\pm {\frac {(5+3{\sqrt {5}})\sin {\frac {\pi }{7}}}{2}}\right),}$
• ${\displaystyle \left(\cos \left({\frac {j\pi }{7}}\right),\,\pm \sin \left({\frac {j\pi }{7}}\right),\,\pm \sin {\frac {\pi }{7}},\,\pm {\frac {(3+{\sqrt {5}})\sin {\frac {\pi }{7}}}{2}},\,\pm (3+{\sqrt {5}})\sin {\frac {\pi }{7}}\right),}$
• ${\displaystyle \left(\cos \left({\frac {j\pi }{7}}\right),\,\pm \sin \left({\frac {j\pi }{7}}\right),\,\pm {\frac {(3+{\sqrt {5}})\sin {\frac {\pi }{7}}}{2}},\,\pm (1+{\sqrt {5}})\sin {\frac {\pi }{7}},\,\pm (2+{\sqrt {5}})\sin {\frac {\pi }{7}}\right),}$

where j = 2, 4, 6.