Heptagonal-square antiprismatic duoprism

Heptagonal-square antiprismatic duoprism
(OFF file)
Rank	5
Type	Uniform
Notation
Bowers style acronym	Hesquap
Coxeter diagram	x7o s2s8o
Elements
Tera	7 square antiprismatic prisms, 8 triangular-heptagonal duoprisms, 2 square-heptagonal duoprisms
Cells	56 triangular prisms, 14 cubes, 7 square antiprisms, 8+8 heptagonal prisms
Faces	56 triangles, 14+56+56 squares, 8 heptagons
Edges	56+56+56
Vertices	56
Vertex figure	Isosceles-trapezoidal scalene, edge lengths 1, 1, 1, √2 (base trapezoid), 2cos(π/7) (top), √2 (side edges)
Measures (edge length 1)
Circumradius	${\sqrt {\frac {4+{\sqrt {2}}+{\frac {2}{\sin ^{2}{\frac {\pi }{7}}}}}{8}}}\approx 1.41590$
Hypervolume	${\frac {7{\sqrt {4+3{\sqrt {2}}}}}{12\tan {\frac {\pi }{7}}}}\approx 3.47765$
Diteral angles	Squappip–squap–squappip: ${\frac {5\pi }{7}}\approx 128.57143^{\circ }$
	Theddip–hep–theddip: = $\arccos \left({\frac {1-2{\sqrt {2}}}{3}}\right)\approx 127.55160^{\circ }$
	Theddip–hep–squahedip: = $\arccos \left({\frac {{\sqrt {3}}-{\sqrt {6}}}{3}}\right)\approx 103.83616^{\circ }$
	Theddip–trip–squappip: 90°
	Squahedip–cube–squappip: 90°
	${\frac {\sqrt[{4}]{8}}{2}}\approx 0.84090$
Central density	1
Number of external pieces	17
Level of complexity	40
Related polytopes
Army	Hesquap
Regiment	Hesquap
Dual	Heptagonal-square antitegmatic duotegum
Conjugates	Heptagrammic-square antiprismatic duoprism, Great heptagrammic-square antiprismatic duoprism
Abstract & topological properties
Euler characteristic	2
Orientable	Yes
Properties
Symmetry	I2(7)×I2(8)×A1+, order 224
Convex	Yes
Nature	Tame

The heptagonal-square antiprismatic duoprism or hesquap is a convex uniform duoprism that consists of 7 square antiprismatic prisms, 2 square-heptagonal duoprisms, and 8 triangular-heptagonal duoprisms. Each vertex joins 2 square antiprismatic prisms, 3 triangular-heptagonal duoprisms, and 1 square-heptagonal duoprism.

Vertex coordinates

The vertices of a heptagonal-square antiprismatic duoprism of edge length 2sin(π/7) are given by:

• $\left(1,\,0,\,\pm \sin {\frac {\pi }{7}},\,\pm \sin {\frac {\pi }{7}},\,{\frac {{\sqrt[{4}]{8}}\sin {\frac {\pi }{7}}}{2}}\right),$
• $\left(\cos {\frac {j\pi }{7}},\,\pm \sin {\frac {j\pi }{7}},\,\pm \sin {\frac {\pi }{7}},\,\pm \sin {\frac {\pi }{7}},\,{\frac {{\sqrt[{4}]{8}}\sin {\frac {\pi }{7}}}{2}}\right),$
• $\left(1,\,0,\,0,\,\pm {\sqrt {2}}\sin {\frac {\pi }{7}},\,-{\frac {{\sqrt[{4}]{8}}\sin {\frac {\pi }{7}}}{2}}\right),$
• $\left(\cos {\frac {j\pi }{7}},\,\pm \sin {\frac {j\pi }{7}},\,0,\,\pm {\sqrt {2}}\sin {\frac {\pi }{7}},\,-{\frac {{\sqrt[{4}]{8}}\sin {\frac {\pi }{7}}}{2}}\right),$
• $\left(1,\,0,\,\pm {\sqrt {2}}\sin {\frac {\pi }{7}},\,0,\,-{\frac {{\sqrt[{4}]{8}}\sin {\frac {\pi }{7}}}{2}}\right),$
• $\left(\cos {\frac {j\pi }{7}},\,\pm \sin {\frac {j\pi }{7}},\,\pm {\sqrt {2}}\sin {\frac {\pi }{7}},\,0,\,-{\frac {{\sqrt[{4}]{8}}\sin {\frac {\pi }{7}}}{2}}\right),$

where j = 2, 4, 6.

Representations

A heptagonal-square antiprismatic duoprism has the following Coxeter diagrams:

• x7o s2s8o (full symmetry; square antiprisms as alternated octagonal prisms)
• x7o s2s4s (square antiprisms as alternated ditetragonal prisms)