# Heptagonal-tetrahedral duoprism

Heptagonal-tetrahedral duoprism
Rank5
TypeUniform
Notation
Bowers style acronymHetet
Coxeter diagramx7o x3o3o
Elements
Tera7 tetrahedral prisms, 4 triangular-heptagonal duoprisms
Cells7 tetrahedra, 28 triangular prisms, 6 heptagonal prisms
Faces28 triangles, 42 squares, 4 heptagons
Edges28+42
Vertices28
Vertex figureTriangular scalene, edge lengths 1 (base triangle), 2cos(π/7) (top), 2 (sides)
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {{\frac {3}{8}}+{\frac {1}{4\sin ^{2}{\frac {\pi }{7}}}}}}\approx 1.30498}$
Hypervolume${\displaystyle {\frac {7{\sqrt {2}}}{48\tan {\frac {\pi }{7}}}}\approx 0.42826}$
Diteral anglesTepe–tet–tepe: ${\displaystyle {\frac {5\pi }{7}}\approx 128.57143^{\circ }}$
Tepe–trip–theddip: 90°
Theddip–hep–theddip: ${\displaystyle \arccos {\left({\frac {1}{3}}\right)}\approx 70.52878^{\circ }}$
HeightsHeg atop theddip: ${\displaystyle {\frac {\sqrt {6}}{3}}\approx 0.81650}$
Hep atop perp hep: ${\displaystyle {\frac {\sqrt {2}}{2}}\approx 0.70711}$
Central density1
Number of external pieces11
Level of complexity10
Related polytopes
ArmyHetet
RegimentHetet
DualHeptagonal-tetrahedral duotegum
ConjugatesHeptagrammic-tetrahedral duoprism, Great heptagrammic-tetrahedral duoprism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryA3×I2(7), order 336
ConvexYes
NatureTame

The heptagonal-tetrahedral duoprism or hetet is a convex uniform duoprism that consists of 7 tetrahedral prisms and 4 triangular-heptagonal duoprisms. Each vertex joins 2 tetrahedral prisms and 3 triangular-heptagonal duoprisms.

## Vertex coordinates

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

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

where j = 2, 4, 6.