# Heptagonal-small rhombicuboctahedral duoprism

Heptagonal-small rhombicuboctahedral duoprism
Rank5
TypeUniform
Notation
Bowers style acronymHesirco
Coxeter diagramx7o x4o3x
Elements
Tera8 triangular-heptagonal duoprisms, 6+12 square-heptagonal duoprisms, 7 small rhombicuboctahedral prisms
Cells56 triangular prisms, 42+84 cubes, 24+24 heptagonal prisms, 7 small rhombicuboctahedra
Faces56 triangles, 42+84+168+168 squares, 24 heptagons
Edges168+168+168
Vertices168
Vertex figureIsosceles-trapezoidal scalene, edge lengths 1, 2, 2, 2 (base trapezoid), 2cos(π/7) (top), 2 (side edges)
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {5+2{\sqrt {2}}+{\frac {1}{\sin ^{2}{\frac {\pi }{7}}}}}}{2}}\approx 1.81248}$
Hypervolume${\displaystyle 7{\frac {6+5{\sqrt {2}}}{6\tan {\frac {\pi }{7}}}}\approx 31.66608}$
Diteral anglesTheddip–hep–squahedip: ${\displaystyle \arccos \left(-{\frac {\sqrt {6}}{3}}\right)\approx 144.73561^{\circ }}$
Squahedip–hep–squahedip: 135°
Sircope–sirco–sircope: ${\displaystyle {\frac {5\pi }{7}}\approx 128.57143^{\circ }}$
Theddip–trip–sircope: 90°
Squahedip–cube–sircope: 90°
Central density1
Number of external pieces33
Level of complexity40
Related polytopes
ArmyHesirco
RegimentHesirco
DualHeptagonal-deltoidal icositetrahedral duotegum
ConjugatesHeptagrammic-small rhombicuboctahedral duoprism, Great heptagrammic-small rhombicuboctahedral duoprism, Heptagonal-quasirhombicuboctahedral duoprism, Heptagrammic-quasirhombicuboctahedral duoprism, Great heptagrammic-quasirhombicuboctahedral duoprism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryB3×I2(7), order 672
ConvexYes
NatureTame

The heptagonal-small rhombicuboctahedral duoprism or hesirco is a convex uniform duoprism that consists of 7 small rhombicuboctahedral prisms, 18 square-heptagonal duoprisms of two kinds, and 8 triangular-heptagonal duoprisms. Each vertex joins 2 small rhombicuboctahedral prisms, 1 triangular-heptagonal duoprism, and 3 square-heptagonal duoprisms.

## Vertex coordinates

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

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

where j = 2, 4, 6.