# Heptagonal-truncated tetrahedral duoprism

Heptagonal-truncated tetrahedral duoprism
Rank5
TypeUniform
Notation
Bowers style acronymHetut
Coxeter diagramx7o x3x3o ()
Elements
Tera4 triangular-heptagonal duoprisms, 7 truncated tetrahedral prisms, 4 hexagonal-heptagonal duoprisms
Cells28 triangular prisms, 28 hexagonal prisms, 7 truncated tetrahedra, 6+12 heptagonal prisms
Faces28 triangles, 42+84 squares, 28 hexagons, 12 heptagons
Edges42+84+84
Vertices84
Vertex figureDigonal disphenoidal pyramid, edge lengths 1, 3, 3 (base triangle), 2cos(π/7) (top), 2 (side edges)
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {{\frac {11}{8}}+{\frac {1}{4\sin ^{2}{\frac {\pi }{7}}}}}}\approx 1.64408}$
Hypervolume${\displaystyle {\frac {161{\sqrt {2}}}{48\tan {\frac {\pi }{7}}}}\approx 9.85000}$
Diteral anglesTuttip–tut–tuttip: ${\displaystyle {\frac {5\pi }{7}}\approx 128.57143^{\circ }}$
Theddip-hep-haheddip: ${\displaystyle \arccos \left(-{\frac {1}{3}}\right)\approx 109.47122^{\circ }}$
Theddip–trip–tuttip: 90°
Haheddip-hip-tuttip: 90°
Haheddip–hep–haheddip: ${\displaystyle \arccos \left({\frac {1}{3}}\right)\approx 70.52877^{\circ }}$
Central density1
Number of external pieces15
Level of complexity30
Related polytopes
ArmyHetut
RegimentHetut
DualHeptagonal-triakis tetrahedral duotegum
ConjugatesHeptagrammic-truncated tetrahedral duoprism, Great heptagrammic-truncated tetrahedral duoprism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryA3×I2(7), order 336
ConvexYes
NatureTame

The heptagonal-truncated tetrahedral duoprism or hetut is a convex uniform duoprism that consists of 7 truncated tetrahedral prisms, 4 hexagonal-heptagonal duoprisms, and 4 triangular-heptagonal duoprisms. Each vertex joins 2 truncated tetrahedral prisms, 1 triangular-heptagonal duoprism, and 2 hexagonal-heptagonal duoprisms.

## Vertex coordinates

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

• ${\displaystyle \left(1,\,0,\,{\frac {3{\sqrt {2}}\sin {\frac {\pi }{7}}}{2}},\,{\frac {{\sqrt {2}}\sin {\frac {\pi }{7}}}{2}},\,{\frac {{\sqrt {2}}\sin {\frac {\pi }{7}}}{2}}\right),}$
• ${\displaystyle \left(\cos \left({\frac {j\pi }{7}}\right),\,\pm \sin \left({\frac {j\pi }{7}}\right),\,{\frac {3{\sqrt {2}}\sin {\frac {\pi }{7}}}{2}},\,{\frac {{\sqrt {2}}\sin {\frac {\pi }{7}}}{2}},\,{\frac {{\sqrt {2}}\sin {\frac {\pi }{7}}}{2}}\right),}$

where j = 2, 4, 6.