# Heptagonal-great rhombicuboctahedral duoprism

Heptagonal-great rhombicuboctahedral duoprism
Rank5
TypeUniform
Notation
Bowers style acronymHegirco
Coxeter diagramx7o x4x3x ()
Elements
Tera12 square-heptagonal duoprisms, 8 hexagonal-heptagonal duoprisms, 6 heptagonal-octagonal duoprisms, 7 great rhombicuboctahedral prisms
Cells84 cubes, 56 hexagonal prisms, 24+24+24 heptagonal prisms, 42 octagonal prisms, 7 great rhombicuboctahedra
Faces84+168+168+168 squares, 56 hexagons, 48 heptagons, 42 octagons,
Edges168+168+168+336
Vertices336
Vertex figureMirror-symmetric pentachoron, edge lengths 2, 3, 2+2 (base triangle), 2cos(π/7) (top edge), 2 (side edges)
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {13+6{\sqrt {2}}+{\frac {1}{\sin ^{2}{\frac {\pi }{7}}}}}}{2}}\approx 2.58830}$
Hypervolume${\displaystyle 7{\frac {11+7{\sqrt {2}}}{2\tan {\frac {\pi }{7}}}}\approx 151.89387}$
Diteral anglesSquahedip–hep–haheddip: ${\displaystyle \arccos \left(-{\frac {\sqrt {6}}{3}}\right)\approx 144.73561^{\circ }}$
Squahedip–hep–heodip: 135°
Gircope–girco–gircope: ${\displaystyle {\frac {5\pi }{7}}\approx 128.57143^{\circ }}$
Haheddip–hep–heodip: ${\displaystyle \arccos \left(-{\frac {\sqrt {3}}{3}}\right)\approx 125.26439^{\circ }}$
Squahedip–cube–gircope: 90°
Haheddip–hip–gircope: 90°
Heodip–op–gircope: 90°
Central density1
Number of external pieces33
Level of complexity60
Related polytopes
ArmyHegirco
RegimentHegirco
DualHeptagonal-disdyakis dodecahedral duotegum
ConjugatesHeptagrammic-great rhombicuboctahedral duoprism, Great heptagrammic-great rhombicuboctahedral duoprism, Heptagonal-quasitruncated cuboctahedral duoprism, Heptagrammic-quasitruncated cuboctahedral duoprism, Great heptagrammic-quasitruncated cuboctahedral duoprism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryB3×I2(7), order 672
ConvexYes
NatureTame

The heptagonal-great rhombicuboctahedral duoprism or hegirco is a convex uniform duoprism that consists of 7 great rhombicuboctahedral prisms, 6 heptagonal-octagonal duoprisms, 8 hexagonal-heptagonal duoprisms, and 12 square-heptagonal duoprisms. Each vertex joins 2 great rhombicuboctahedral prisms, 1 square-heptagonal duoprism, 1 hexagonal-heptagonal duoprism, and 1 heptagonal-octagonal duoprism.

## Vertex coordinates

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

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

where j = 2, 4, 6.