Pentagrammic retroprismatic prism

Pentagrammic retroprismatic prism
Rank	4
Type	Uniform
Notation
Bowers style acronym	Starpip
Coxeter diagram	x2s2s10/3o ()
Elements
Cells	10 triangular prisms, 2 pentagrammic prisms, 2 pentagrammic retroprisms
Faces	20 triangles, 10+10 squares, 4 pentagrams
Edges	10+20+20
Vertices	20
Vertex figure	Crossed isosceles trapezoidal pyramid, edge lengths 1, 1, 1, (√5–1)/2 (base), √2 (legs)
Measures (edge length 1)
Circumradius	${\sqrt {\frac {7-{\sqrt {5}}}{8}}}\approx 0.77168$
Hypervolume	${\frac {5-2{\sqrt {5}}}{6}}\approx 0.087977$
Dichoral angles	Starp–5/2–stip: 90°
	Starp–3–trip: 90°
	Trip–4–trip: $\arccos \left({\frac {\sqrt {5}}{3}}\right)\approx 41.81031^{\circ }$
	Trip–4–stip: $\arccos \left({\sqrt {\frac {5+2{\sqrt {5}}}{15}}}\right)\approx 37.37737^{\circ }$
Heights	Starp atop starp: 1
	Stip atop gyro stip: ${\sqrt {\frac {5-{\sqrt {5}}}{10}}}\approx 0.52573$
Number of external pieces	78
Related polytopes
Army	Pappip
Regiment	Starpip
Dual	Pentagrammic concave antitegmatic tegum
Conjugate	Pentagonal antiprismatic prism
Abstract & topological properties
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	(I2(10)×A1)+×A1, order 40
Convex	No
Nature	Tame

The pentagrammic retroprismatic prism or starpip is a prismatic uniform polychoron that consists of 2 pentagrammic retroprisms, 2 pentagrammic prisms, and 10 triangular prisms. Each vertex joins 1 pentagrammic retroprism, 1 pentagrammic prism, and 3 triangular prisms. As the name suggests, it is a prism based on a pentagrammic retroprism. Being a prism based on an orbiform polytope, it is also a segmentochoron.

Vertex coordinates

The vertices of a pentagrammic retroprismatic prism of edge length 1 are given by the following points, as well as the central inversions of their first three coordinates:

• $\left(\pm {\frac {1}{2}},\,-{\sqrt {\frac {5-2{\sqrt {5}}}{20}}},\,{\sqrt {\frac {5-{\sqrt {5}}}{40}}},\,\pm {\frac {1}{2}}\right),$
• $\left(\pm {\frac {{\sqrt {5}}-1}{4}},\,{\sqrt {\frac {5+{\sqrt {5}}}{40}}},\,{\sqrt {\frac {5-{\sqrt {5}}}{40}}},\,\pm {\frac {1}{2}}\right),$
• $\left(0,\,-{\sqrt {\frac {5-{\sqrt {5}}}{10}}},\,{\sqrt {\frac {5-{\sqrt {5}}}{40}}},\,\pm {\frac {1}{2}}\right).$

Representations

A pentagrammic retroprismatic prism has the following Coxeter diagrams:

• x2s2s10/3o (full symmetry)
• x2s2s5/3s ()
• xx xo5/3ox&#x (pentagrammic prism atop gyrated pentagrammic prism)

External links

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

• Wikipedia contributors. "Uniform antiprismatic prism".