# Heptagonal-truncated cubic duoprism

Heptagonal-truncated cubic duoprism
Rank5
TypeUniform
Notation
Bowers style acronymHetic
Coxeter diagramx7o x4x3o ()
Elements
Tera6 triangular-heptagonal duoprisms, 7 truncated cubic prisms, 6 heptagonal-octagonal duoprisms
Cells56 triangular prisms, 12+24 heptagonal prisms, 42 octagonal prisms, 7 truncated cubes
Faces56 triangles, 84+168 squares, 24 heptagons, 42 octagons
Edges84+168+168
Vertices168
Vertex figureDigonal disphenoidal pyramid, edge lengths 1, 2+2, 2+2 (base triangle), 2cos(π/7) (top), 2 (side edges)
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {7+4{\sqrt {2}}+{\frac {1}{\sin ^{2}{\frac {\pi }{7}}}}}}{2}}\approx 2.11948}$
Hypervolume${\displaystyle {\frac {49(3+2{\sqrt {2}})}{12\tan {\frac {\pi }{7}}}}\approx 49.41999}$
Diteral anglesTiccup–tic–ticcup: ${\displaystyle {\frac {5\pi }{7}}\approx 128.57143^{\circ }}$
Theddip–hep–heodip: ${\displaystyle \arccos \left(-{\frac {\sqrt {3}}{3}}\right)\approx 125.26439^{\circ }}$
Theddip–trip–ticcup: 90°
Heodip–op–ticcup: 90°
Heodip–hep–heodip: 90°
Central density1
Number of external pieces21
Level of complexity30
Related polytopes
ArmyHetic
RegimentHetic
DualHeptagonal-triakis octahedral duotegum
ConjugatesHeptagrammic-truncated cubic duoprism, Great heptagrammic-truncated cubic duoprism, Heptagonal-quasitruncated hexahedral duoprism, Heptagrammic-quasitruncated hexahedral duoprism, Great heptagrammic-quasitruncated hexahedral duoprism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryB3×I2(7), order 672
ConvexYes
NatureTame

The heptagonal-truncated cubic duoprism or hetic is a convex uniform duoprism that consists of 7 truncated cubic prisms, 6 heptagonal-octagonal duoprisms, and 8 triangular-heptagonal duoprisms. Each vertex joins 2 truncated cubic prisms, 1 triangular-heptagonal duoprism, and 2 heptagonal-octagonal duoprisms.

## Vertex coordinates

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

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

where j = 2, 4, 6.