Enneagrammic antiprism

Enneagrammic antiprism
(3D model) (OFF file)
Rank	3
Type	Uniform
Notation
Bowers style acronym	Steap
Coxeter diagram	s2s18/2o ()
Elements
Faces	18 triangles, 2 enneagrams
Edges	18+18
Vertices	18
Vertex figure	Isosceles trapezoid, edge lengths 1, 1, 1, 2cos(2π/9)
Measures (edge length 1)
Circumradius	${\sqrt {\frac {3-2\cos {\frac {2\pi }{9}}}{8-8\cos {\frac {2\pi }{9}}}}}\approx 0.88560$
Volume	${\frac {3{\sqrt {12\cos ^{2}{\frac {\pi }{9}}-3}}}{8\sin ^{2}{\frac {2\pi }{9}}}}\approx 2.50148$
Dihedral angles	3–3: $\arccos \left({\frac {1-4\cos {\frac {2\pi }{9}}}{3}}\right)\approx 133.47668^{\circ }$
	9/2–3: $\arccos \left(-{\frac {{\sqrt {3}}\tan {\frac {\pi }{9}}}{3}}\right)\approx 102.13046^{\circ }$
Height	${\sqrt {\frac {1+2\cos {\frac {2\pi }{9}}}{2+2\cos {\frac {2\pi }{9}}}}}\approx 0.84669$
Central density	2
Number of external pieces	56
Level of complexity	11
Related polytopes
Army	Semi-uniform Ep, edge lengths ${\frac {1}{2\cos {\frac {\pi }{9}}}}$ (base), ${\sqrt {\frac {1+2\cos {\frac {2\pi }{9}}}{2+2\cos {\frac {2\pi }{9}}}}}$ (sides)
Regiment	Steap
Dual	Enneagrammic antitegum
Conjugate	Great enneagrammic antiprism
Convex core	Enneagonal bifrustum
Abstract & topological properties
Flag count	144
Euler characteristic	2
Orientable	Yes
Genus	0
Properties
Symmetry	I2(9)×A1, order 36
Convex	No
Nature	Tame

The enneagrammic antiprism, or steap, is a prismatic uniform polyhedron. It consists of 18 triangles and 2 enneagrams. Each vertex joins one enneagram and three triangles. As the name suggests, it is an antiprism based on an enneagram. It is one of three ennegrammic antiprisms, the other two being the great enneagrammic antiprism and the great enneagrammic retroprism.

Vertex coordinates

The vertex coordinates of an enneagrammic antiprism with edge length 2sin(2π/9) are given by:

• $\left(1,\,0,\,\pm H\sin {\frac {2\pi }{9}}\right)$,
• $\left(\cos \left({\frac {2\pi }{9}}\right),\,\pm \sin \left({\frac {2\pi }{9}}\right),\,\pm H\sin {\frac {2\pi }{9}}\right)$,
• $\left(\cos \left({\frac {4\pi }{9}}\right),\,\pm \sin \left({\frac {4\pi }{9}}\right),\,\pm H\sin {\frac {2\pi }{9}}\right)$,
• $\left(-{\frac {1}{2}},\,\pm {\frac {\sqrt {3}}{2}},\,\pm H\sin {\frac {2\pi }{9}}\right)$,
• $\left(\cos \left({\frac {8\pi }{9}}\right),\,\pm \sin \left({\frac {8\pi }{9}}\right),\,\pm H\sin {\frac {2\pi }{9}}\right)$,

where $H={\sqrt {\frac {1+2\cos {\frac {2\pi }{9}}}{2+2\cos {\frac {2\pi }{9}}}}}$.

External links

• Wikipedia contributors. "Enneagrammic antiprism".