Enneagrammic antiprismatic prism

Enneagrammic antiprismatic prism
Rank	4
Type	Uniform
Notation
Bowers style acronym	Steappip
Coxeter diagram	x2s2s18/2o ()
Elements
Cells	18 triangular prisms, 2 enneagrammic prisms, 2 enneagrammic antiprisms
Faces	36 triangles, 18+18 squares, 4 enneagrams
Edges	18+36+36
Vertices	36
Vertex figure	Isosceles trapezoidal pyramid, edge lengths 1, 1, 1, 2cos(2π/9) (base), √2 (legs)
Measures (edge length 1)
Circumradius	${\sqrt {\frac {5-4\cos {\frac {2\pi }{9}}}{8-8\cos {\frac {2\pi }{9}}}}}\approx 1.01700$
Hypervolume	${\frac {3{\sqrt {12\cos ^{2}{\frac {\pi }{9}}-3}}}{8\sin ^{2}{\frac {2\pi }{9}}}}\approx 2.50148$
Dichoral angles	Trip–4–trip: $\arccos \left({\frac {1-4\cos {\frac {2\pi }{9}}}{3}}\right)\approx 133.47668^{\circ }$
	Trip–4–step: $\arccos \left(-{\frac {{\sqrt {3}}\tan {\frac {\pi }{9}}}{3}}\right)\approx 102.13046^{\circ }$
	Steap–9/2–step: 90°
	Steap–3–trip: 90°
Heights	Steap atop steap: 1
	Step atop step: ${\sqrt {\frac {1+2\cos {\frac {2\pi }{9}}}{2+2\cos {\frac {2\pi }{9}}}}}\approx 0.84669$
Number of external pieces	58
Related polytopes
Army	Semi-uniform sendip
Regiment	Steappip
Dual	Enneagrammic antitegmatic tegum
Conjugate	Great enneagrammic antiprismatic prism
Abstract & topological properties
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	I2(9)×A1×A1, order 72
Convex	No
Nature	Tame

The enneagrammic antiprismatic prism or steappip is a prismatic uniform polychoron that consists of 2 enneagrammic antiprisms, 2 enneagrammic prisms, and 18 triangular prisms. Each vertex joins 1 enneagrammic antiprism, 1 enneagrammic prism, and 3 triangular prisms. As the name suggests, it is a prism based on the enneagrammic antiprism. Being a prism based on an orbiform polytope, it is also a segmentochoron.

Vertex coordinates

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

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

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

Representations

An enneagrammic antiprismatic prism has the following Coxeter diagrams:

• x2s2s18/2o (full symmetry)
• x2s2s9/2s
• xx xo9/2ox&#x (enneagrammic prism atop enneagrammic prism)

External links

• Bowers, Jonathan. "Category B: Antiduoprisms".

• Wikipedia contributors. "Uniform antiprismatic prism".