# Pentagonal-heptagonal duoprismatic prism

Pentagonal-heptagonal duoprismatic prism
Rank5
TypeUniform
Notation
Bowers style acronymPehep
Coxeter diagramx x5o x7o
Elements
Tera7 square-pentagonal duoprisms, 5 square-heptagonal duoprisms, 2 pentagonal-heptagonal duoprisms
Cells35 cubes, 5+10 heptagonal prisms, 7+14 pentagonal prisms
Faces35+35+70 squares, 14 pentagons, 10 heptagons
Edges35+70+70
Vertices70
Vertex figureDigonal disphenoidal pyramid, edge lengths (1+5)/2 (disphenoid base 1), 2cos(π/7) (disphenoid base 2), 2 (remaining edges)
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {\frac {15+2{\sqrt {5}}+{\frac {5}{\sin ^{2}{\frac {\pi }{7}}}}}{20}}}\approx 1.51710}$
Hypervolume${\displaystyle {\frac {7{\sqrt {25+10{\sqrt {5}}}}}{16\tan {\frac {\pi }{7}}}}\approx 6.25206}$
Diteral anglesSquipdip–pip–squipdip: ${\displaystyle {\frac {5\pi }{7}}\approx 128.57143^{\circ }}$
Squahedip–hep–squahedip: 108°
Squahedip–cube–squipdip: 90°
Pheddip–pip–squipdip: 90°
Squahedip–hep–pheddip: 90°
Height1
Central density1
Number of external pieces14
Level of complexity30
Related polytopes
ArmyPehep
RegimentPehep
DualPentagonal-heptagonal duotegmatic tegum
ConjugatesPentagonal-heptagrammic duoprismatic prism, Pentagonal-great heptagrammic duoprismatic prism, Pentagrammic-heptagonal duoprismatic prism, Pentagrammic-heptagrammic duoprismatic prism, Pentagrammic-great heptagrammic duoprismatic prism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryH2×I2(7)×A1, order 280
ConvexYes
NatureTame

The pentagonal-heptagonal duoprismatic prism or pehep, also known as the pentagonal-heptagonal prismatic duoprism, is a convex uniform duoprism that consists of 2 pentagonal-heptagonal duoprisms, 5 square-heptagonal duoprisms, and 7 square-pentagonal duoprisms. Each vertex joins 2 square-pentagonal duoprisms, 2 square-heptagonal duoprisms, and 1 pentagonal-heptagonal duoprism. Being a prism based on an orbiform polytope, it is also a convex segmentoteron.

## Vertex coordinates

The vertices of a pentagonal-heptagonal duoprismatic prism of edge length 2sin(π/7) are given by:

• ${\displaystyle \left(0,\,2{\sqrt {\frac {5+{\sqrt {5}}}{10}}}\sin {\frac {\pi }{7}},\,1,\,0,\,\pm \sin {\frac {\pi }{7}}\right),}$
• ${\displaystyle \left(\pm {\frac {(1+{\sqrt {5}})\sin {\frac {\pi }{7}}}{2}},\,{\sqrt {\frac {5-{\sqrt {5}}}{10}}}\sin {\frac {\pi }{7}},\,1,\,0,\,\pm \sin {\frac {\pi }{7}}\right),}$
• ${\displaystyle \left(\pm \sin {\frac {\pi }{7}},\,-{\sqrt {\frac {5+2{\sqrt {5}}}{5}}}\sin {\frac {\pi }{7}},\,1,\,0,\,\pm \sin {\frac {\pi }{7}}\right),}$
• ${\displaystyle \left(0,\,2{\sqrt {\frac {5+{\sqrt {5}}}{10}}}\sin {\frac {\pi }{7}},\,\cos {\frac {j\pi }{7}},\,\pm \sin {\frac {j\pi }{7}},\,\pm \sin {\frac {\pi }{7}}\right),}$
• ${\displaystyle \left(\pm {\frac {(1+{\sqrt {5}})\sin {\frac {\pi }{7}}}{2}},\,{\sqrt {\frac {5-{\sqrt {5}}}{10}}}\sin {\frac {\pi }{7}},\,\cos {\frac {j\pi }{7}},\,\pm \sin {\frac {j\pi }{7}},\,\pm \sin {\frac {\pi }{7}}\right),}$
• ${\displaystyle \left(\pm \sin {\frac {\pi }{7}},\,-{\sqrt {\frac {5+2{\sqrt {5}}}{5}}}\sin {\frac {\pi }{7}},\,\cos {\frac {j\pi }{7}},\,\pm \sin {\frac {j\pi }{7}},\,\pm \sin {\frac {\pi }{7}}\right),}$

where j = 2, 4, 6.

## Representations

A pentagonal-heptagonal duoprismatic prism has the following Coxeter diagrams:

• x x5o x7o (full symmetry)
• xx5oo xx7oo&#x (pentagonal-heptagonal duoprism atop pentagonal-heptagonal duoprism)