Great heptagrammic retroprism

Great heptagrammic retroprism
(3D model) (OFF file)
Rank	3
Type	Uniform
Notation
Bowers style acronym	Gisharp
Coxeter diagram	s2s14/4o
Elements
Faces	14 triangles, 2 great heptagrams
Edges	14+14
Vertices	14
Vertex figure	Crossed isosceles trapezoid, edge lengths 1, 1, 1, 2cos(3π/7)
Measures (edge length 1)
Circumradius	${\sqrt {\frac {3+2\cos {\frac {3\pi }{7}}}{8+8\cos {\frac {3\pi }{7}}}}}\approx 0.59350$
Volume	${\frac {7{\sqrt {4\cos ^{2}{\frac {2\pi }{7}}-1}}\sin {\frac {6\pi }{7}}}{12\sin ^{2}{\frac {4\pi }{7}}}}\approx 0.19837$
Dihedral angles	3–3: $\arccos \left({\frac {1+4\cos {\frac {3\pi }{7}}}{3}}\right)\approx 50.94782^{\circ }$
	7/3–3: $\arccos \left({\frac {{\sqrt {3}}\tan {\frac {2\pi }{7}}}{3}}\right)\approx 43.61641^{\circ }$
Height	${\sqrt {\frac {1-2\cos {\frac {3\pi }{7}}}{2-2\cos {\frac {3\pi }{7}}}}}\approx 0.59741$
Central density	4
Number of external pieces	128
Level of complexity	30
Related polytopes
Army	Semi-uniform Hep, edge lengths $2\cos {\frac {3\pi }{7}}$ (base), ${\sqrt {\frac {1-2\cos {\frac {3\pi }{7}}}{2-2\cos {\frac {3\pi }{7}}}}}$ (sides)
Regiment	Gisharp
Dual	Heptagrammic concave antitegum
Conjugate	Heptagrammic antiprism
Convex core	Heptagonal tegum
Abstract & topological properties
Flag count	112
Euler characteristic	2
Orientable	Yes
Genus	0
Properties
Symmetry	I2(7)×A1, order 28
Convex	No
Nature	Tame

The great heptagrammic retroprism, or gisharp, also called the great heptagrammic crossed antiprism or simply the heptagrammic crossed antiprism, is a prismatic uniform polyhedron. It consists of 14 triangles and 2 great heptagrams. Each vertex joins one great heptagram and three triangles. As the name suggests, it is a crossed antiprism based on a great heptagram, treated as a 7/4-gon rather than 7/3.

Vertex coordinates

The vertices of a great heptagrammic retroprism, centered at the origin and with edge length 2sin(3π/7), are given by:

• $\left(1,\,0,\,\pm H\right),$
• $\left(\cos \left({\frac {2\pi }{7}}\right),\,\pm \sin \left({\frac {2\pi }{7}}\right),\,\pm H\right),$
• $\left(\cos \left({\frac {4\pi }{7}}\right),\,\pm \sin \left({\frac {4\pi }{7}}\right),\,\pm H\right),$
• $\left(\cos \left({\frac {6\pi }{7}}\right),\,\pm \sin \left({\frac {6\pi }{7}}\right),\,\pm H\right),$

where $H={\sqrt {\frac {1-2\cos {\frac {3\pi }{7}}}{2-2\cos {\frac {3\pi }{7}}}}}\sin {\frac {3\pi }{7}}.$

External links

• Wikipedia contributors. "Heptagrammic crossed-antiprism".
• McCooey, David. "Heptagrammic Crossed Antiprism"