# Heptagonal-icosidodecahedral duoprism

Heptagonal-icosidodecahedral duoprism
Rank5
TypeUniform
Notation
Bowers style acronymHeid
Coxeter diagramx7o o5x3o
Elements
Tera20 triangular-heptagonal duoprisms, 12 pentagonal-heptagonal duoprisms, 7 icosidodecahedral prisms
Cells140 triangular prisms, 84 pentagonal prisms, 60 heptagonal prisms, 7 icosidodecahedra
Faces140 triangles, 420 squares, 84 pentagons, 30 heptagons
Edges210+420
Vertices210
Vertex figureRectangular scalene, edge lengths 1, (1+5)/2, 1, (1+5)/2 (base rectangle), 2cos(π/7) (top), 2 (side edges)
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {\frac {3+{\sqrt {5}}+{\frac {1}{\sin ^{2}{\frac {\pi }{7}}}}}{2}}}\approx 1.98646}$
Hypervolume${\displaystyle 7{\frac {45+17{\sqrt {5}}}{24\tan {\frac {\pi }{7}}}}\approx 50.27709}$
Diteral anglesTheddip–hep–pheddip: ${\displaystyle \arccos \left(-{\sqrt {\frac {5+2{\sqrt {5}}}{15}}}\right)\approx 142.62263^{\circ }}$
Iddip–id–iddip: ${\displaystyle {\frac {5\pi }{7}}\approx 128.57143^{\circ }}$
Theddip–trip–iddip: 90°
Pheddip–pip–iddip: 90°
Central density1
Number of external pieces39
Level of complexity20
Related polytopes
ArmyHeid
RegimentHeid
DualHeptagonal-rhombic triacontahedral duotegum
ConjugatesHeptagrammic-icosidodecahedral duoprism, Great heptagrammic-icosidodecahedral duoprism, Heptagonal-great icosidodecahedral duoprism, Heptagrammic-great icosidodecahedral duoprism, Great heptagrammic-great icosidodecahedral duoprism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryH3×I2(7), order 1680
ConvexYes
NatureTame

The heptagonal-icosidodecahedral duoprism or heid is a convex uniform duoprism that consists of 7 icosidodecahedral prisms, 12 pentagonal-heptagonal duoprisms, and 20 triangular-heptagonal duoprisms. Each vertex joins 2 icosidodecahedral prisms, 2 triangular-heptagonal duoprisms, and 2 pentagonal-heptagonal duoprisms.

## Vertex coordinates

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

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

as well as all even permutations of the last three coordinates of:

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

where j = 2, 4, 6.