# Heptagonal-icosahedral duoprism

Heptagonal-icosahedral duoprism
Rank5
TypeUniform
Notation
Bowers style acronymHeike
Coxeter diagramx7o o5o3x
Elements
Tera20 triangular-heptagonal duoprisms, 77 icosahedral prisms
Cells140 triangular prisms, 30 heptagonal prisms, 7 icosahedra
Faces140 triangles, 210 squares, 12 heptagons
Edges84+210
Vertices84
Vertex figurePentagonal scalene, edge lengths 1 (base pentagon), 2cos(π/7) (top), 2 (sides)
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {\frac {5+{\sqrt {5}}+{\frac {2}{\sin ^{2}{\frac {\pi }{7}}}}}{8}}}\approx 1.49415}$
Hypervolume${\displaystyle {\frac {35(3+{\sqrt {5}})}{48\tan {\frac {\pi }{7}}}}\approx 7.92809}$
Diteral anglesTheddip–hep–theddip: ${\displaystyle \arccos \left(-{\frac {\sqrt {5}}{3}}\right)\approx 138.18969^{\circ }}$
Ipe–ike–ipe: ${\displaystyle {\frac {5\pi }{7}}\approx 128.57143^{\circ }}$
Theddip–trip–ipe: 90°
Central density1
Number of external pieces27
Level of complexity10
Related polytopes
ArmyHeike
RegimentHeike
DualHeptagonal-dodecahedral duotegum
ConjugatesHeptagrammic-icosahedral duoprism, Great heptagrammic-icosahedral duoprism, Heptagonal-great icosahedral duoprism, Heptagrammic-great icosahedral duoprism, Great heptagrammic-great icosahedral duoprism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryH3×I2(7), order 1680
ConvexYes
NatureTame

The heptagonal-icosahedral duoprism or heike is a convex uniform duoprism that consists of 7 icosahedral prisms and 20 triangular-heptagonal duoprisms. Each vertex joins 2 icosahedral prisms and 5 triangular-heptagonal duoprisms.

## Vertex coordinates

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

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

where j = 2, 4, 6.