Heptagrammic antiprism

Heptagrammic antiprism
(3D model) (OFF file)
Rank	3
Type	Uniform
Notation
Bowers style acronym	Shap
Coxeter diagram	s2s14/2o ()
Elements
Faces	14 triangles, 2 heptagrams
Edges	14+14
Vertices	14
Vertex figure	Isosceles trapezoid, edge lengths 1, 1, 1, 2cos(2π/7)
Measures (edge length 1)
Circumradius	${\sqrt {\frac {3-2\cos {\frac {2\pi }{7}}}{8-8\cos {\frac {2\pi }{7}}}}}\approx 0.76289$
Volume	${\frac {7{\sqrt {4\cos ^{2}{\frac {\pi }{7}}-1}}\sin {\frac {3\pi }{7}}}{12\sin ^{2}{\frac {2\pi }{7}}}}\approx 1.39464$
Dihedral angles	3–3: $\arccos \left({\frac {1-4\cos {\frac {2\pi }{7}}}{3}}\right)\approx 119.86687^{\circ }$
	7/2–3: $\arccos \left(-{\frac {{\sqrt {3}}\tan {\frac {\pi }{7}}}{3}}\right)\approx 106.14310^{\circ }$
Height	${\sqrt {\frac {1+2\cos {\frac {2\pi }{7}}}{2+2\cos {\frac {2\pi }{7}}}}}\approx 0.83188$
Central density	2
Number of external pieces	44
Level of complexity	11
Related polytopes
Army	Semi-uniform Hep, edge lengths ${\frac {1}{2\cos {\frac {\pi }{7}}}}$ (base), ${\sqrt {\frac {1+2\cos {\frac {2\pi }{7}}}{2+2\cos {\frac {2\pi }{7}}}}}$ (sides)
Regiment	Shap
Dual	Heptagrammic antitegum
Conjugate	Great heptagrammic retroprism
Convex core	Heptagonal bifrustum
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 heptagrammic antiprism, or shap, is a prismatic uniform polyhedron. It consists of 14 triangles and 2 heptagrams. Each vertex joins one heptagram and three triangles. As the name suggests, it is an antiprism based on a heptagram. It is one of three heptagrammic antiprisms, the other two being the great heptagrammic antiprism and the great heptagrammic retroprism.

Vertex coordinates

The vertices of a heptagrammic antiprism, centered at the origin and with edge length $2\sin(2\pi /7)$, 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 $H={\sqrt {\frac {1+2\cos {\frac {2\pi }{7}}}{2+2\cos {\frac {2\pi }{7}}}}}\sin {\frac {2\pi }{7}}$.

External links

• Wikipedia contributors. "Heptagrammic antiprism (7/2)".
• McCooey, David. "Heptagrammic 7/2 Antiprism"