Heptagonal-great rhombicuboctahedral duoprism Rank 5 Type Uniform Notation Bowers style acronym Hegirco Coxeter diagram x7o x4x3x ( ) Elements Tera 12 square-heptagonal duoprisms , 8 hexagonal-heptagonal duoprisms , 6 heptagonal-octagonal duoprisms , 7 great rhombicuboctahedral prisms Cells 84 cubes , 56 hexagonal prisms , 24+24+24 heptagonal prisms , 42 octagonal prisms , 7 great rhombicuboctahedra Faces 84+168+168+168 squares , 56 hexagons , 48 heptagons , 42 octagons , Edges 168+168+168+336 Vertices 336 Vertex figure Mirror-symmetric pentachoron , edge lengths √2 , √3 , √2+√2 (base triangle), 2cos(π/7) (top edge), √2 (side edges) Measures (edge length 1) Circumradius ${\frac {\sqrt {13+6{\sqrt {2}}+{\frac {1}{\sin ^{2}{\frac {\pi }{7}}}}}}{2}}\approx 2.58830$ Hypervolume $7{\frac {11+7{\sqrt {2}}}{2\tan {\frac {\pi }{7}}}}\approx 151.89387$ Diteral angles Squahedip–hep–haheddip: $\arccos \left(-{\frac {\sqrt {6}}{3}}\right)\approx 144.73561^{\circ }$ Squahedip–hep–heodip: 135° Gircope–girco–gircope: ${\frac {5\pi }{7}}\approx 128.57143^{\circ }$ Haheddip–hep–heodip: $\arccos \left(-{\frac {\sqrt {3}}{3}}\right)\approx 125.26439^{\circ }$ Squahedip–cube–gircope: 90° Haheddip–hip–gircope: 90° Heodip–op–gircope: 90° Central density 1 Number of external pieces 33 Level of complexity 60 Related polytopes Army Hegirco Regiment Hegirco Dual Heptagonal-disdyakis dodecahedral duotegum Conjugates Heptagrammic-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 characteristic 2 Orientable Yes Properties Symmetry B_{3} ×I_{2} (7) , order 672Convex Yes Nature Tame

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.

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:

$\left(1,\,0,\,\pm (1+2{\sqrt {2}})\sin {\frac {\pi }{7}},\,\pm (1+{\sqrt {2}})\sin {\frac {\pi }{7}},\,\pm \sin {\frac {\pi }{7}}\right),$
$\left(\cos \left({\frac {j\pi }{7}}\right),\,\pm \sin \left({\frac {j\pi }{7}}\right),\,\pm (1+2{\sqrt {2}})\sin {\frac {\pi }{7}},\,\pm (1+{\sqrt {2}})\sin {\frac {\pi }{7}},\,\pm \sin {\frac {\pi }{7}}\right),$
where j = 2, 4, 6.

Klitzing, Richard. "n-girco-dip" .