# Heptagonal-snub cubic duoprism

Heptagonal-snub cubic duoprism
Rank5
TypeUniform
Notation
Bowers style acronymHesnic
Coxeter diagramx7o s4s3s ()
Elements
Tera
Cells
Faces
Edges84+168+168+168
Vertices168
Vertex figureMirror-symmetric pentagonal scalene, edge lengths 1, 1, 1, 1, 2 (base pentagon), 2cos(π/7) (top edge), 2 (side edges)
Measures (edge length 1)
Hypervolume≈ 28.66968
Diteral anglesTheddip–hep–theddip: ≈ 153.23459°
Theddip–hep–squahedip: ≈ 142.98343°
Sniccup–snic–sniccup: ${\displaystyle {\frac {5\pi }{7}}\approx 128.57143^{\circ }}$
Theddip–trip–sniccup: 90°
Squahedip–cube–sniccup: 90°
Central density1
Number of external pieces45
Level of complexity50
Related polytopes
ArmyHesnic
RegimentHesnic
DualHeptagonal-pentagonal icositetrahedral duotegum
ConjugatesHeptagrammic-snub cubic duoprism, Great heptagrammic-snub cubic duoprism
Abstract & topological properties
Flag count33600
Euler characteristic2
OrientableYes
Properties
SymmetryB3+×I2(7), order 336
Flag orbits100
ConvexYes
NatureTame

The heptagonal-snub cubic duoprism (OBSA: hesnic) is a convex uniform duoprism that consists of 7 snub cubic prisms, 6 square-heptagonal duoprisms, and 32 triangular-heptagonal duoprisms of two kinds. Each vertex joins 2 snub cubic prisms, 4 triangular-heptagonal duoprisms, and 1 square-heptagonal duoprism.

## Vertex coordinates

The vertices of a heptagonal-snub cubic duoprism of edge length 1 are given by by all even permutations with an even number of sign changes, plus all odd permutations with an odd amount of sign changes, of the last three coordinates of:

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

where

• j = 0, 2, 4, 6,
• ${\displaystyle c_{1}={\sqrt {{\frac {1}{12}}\left(4-{\sqrt[{3}]{17+3{\sqrt {33}}}}-{\sqrt[{3}]{17-3{\sqrt {33}}}}\right)}}}$,
• ${\displaystyle c_{2}={\sqrt {{\frac {1}{12}}\left(2+{\sqrt[{3}]{17+3{\sqrt {33}}}}+{\sqrt[{3}]{17-3{\sqrt {33}}}}\right)}}}$,
• ${\displaystyle c_{3}={\sqrt {{\frac {1}{12}}\left(4+{\sqrt[{3}]{199+3{\sqrt {33}}}}+{\sqrt[{3}]{199-3{\sqrt {33}}}}\right)}}}$.