Heptagonal antiprism

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Heptagonal antiprism
Rank3
TypeUniform
Notation
Bowers style acronymHeap
Coxeter diagrams2s14o
Elements
Faces14 triangles, 2 heptagons
Edges14+14
Vertices14
Vertex figureIsosceles trapezoid, edge lengths 1, 1, 1, 2cos(π/7)
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {\frac {3-2\cos {\frac {\pi }{7}}}{8-8\cos {\frac {\pi }{7}}}}}\approx 1.22973}$
Volume${\displaystyle {\frac {7{\sqrt {4\cos ^{2}{\frac {\pi }{14}}-1}}\sin {\frac {3\pi }{14}}}{12\sin ^{2}{\frac {\pi }{7}}}}\approx 3.23392}$
Dihedral angles3–3: ${\displaystyle \arccos \left({\frac {1-4\cos {\frac {\pi }{7}}}{3}}\right)\approx 150.22226^{\circ }}$
7–3: ${\displaystyle \arccos \left(-{\frac {{\sqrt {3}}\tan {\frac {\pi }{14}}}{3}}\right)\approx 97.57226^{\circ }}$
Height${\displaystyle {\sqrt {\frac {1+2\cos {\frac {\pi }{7}}}{2+2\cos {\frac {\pi }{7}}}}}\approx 0.85847}$
Central density1
Number of external pieces16
Level of complexity4
Related polytopes
ArmyHeap
RegimentHeap
DualHeptagonal antitegum
ConjugatesGreat heptagrammic antiprism, heptagrammic retroprism
Abstract & topological properties
Flag count112
Euler characteristic2
SurfaceSphere
OrientableYes
Genus0
Properties
Symmetry(I2(14)×A1)/2, order 28
ConvexYes
NatureTame

The heptagonal antiprism, or heap, is a prismatic uniform polyhedron. It consists of 14 triangles and 2 heptagons. Each vertex joins one heptagon and three triangles. As the name suggests, it is an antiprism based on a heptagon.

Vertex coordinates

The vertices of a heptagonal antiprism, centered at the origin and with edge length 2sin(π/7), are given by the following points, as well as their central inversions:

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

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