Rectified pentachoric prism

Rectified pentachoric prism
File:Rectified pentachoric prism.png (OFF file)
Rank	5
Type	Uniform
Notation
Bowers style acronym	Rappip
Coxeter diagram	x o3x3o3o ()
Elements
Tera	5 tetrahedral prisms, 2 rectified pentachora, 5 octahedral prisms
Cells	10 tetrahedra, 10+20 triangular prisms, 10 octahedra
Faces	20+40 triangles, 30 squares
Edges	10+60
Vertices	20
Vertex figure	Triangular prismatic pyramid, edge lengths 1 (base), √2 (legs)
Measures (edge length 1)
Circumradius	${\frac {\sqrt {85}}{10}}\approx 0.92195$
Hypervolume	${\frac {11{\sqrt {5}}}{96}}\approx 0.25622$
Diteral angles	Tepe–trip–ope: $\arccos \left(-{\frac {1}{4}}\right)\approx 104.47751^{\circ }$
	Rap–oct–ope: 90°
	Rap–tet–tepe: 90°
	Ope–trip–ope: $\arccos \left({\frac {1}{4}}\right)\approx 75.52249^{\circ }$
Heights	Rap atop rap: 1
	Tepe atop ope: ${\frac {\sqrt {10}}{4}}\approx 0.79057$
Central density	1
Number of external pieces	12
Level of complexity	15
Related polytopes
Army	Rappip
Regiment	Rappip
Dual	Joined pentachoric tegum
Conjugate	None
Abstract & topological properties
Flag count	3600
Euler characteristic	2
Orientable	Yes
Properties
Symmetry	A4×A1, order 240
Convex	Yes
Nature	Tame

The rectified pentachoric prism or rappip is a prismatic uniform polyteron that consists of 2 rectified pentachora, 5 octahedral prisms and 5 tetrahedral prisms. 1 rectified pentachoron, 2 tetrahedral prisms, and 3 octahedral prisms join at each vertex. As the name suggests, it is a prism based on the rectified pentachoron. As such it is also a convex segmentoteron.

Vertex coordinates

The vertices of a rectified pentachoric prism of edge length 1 are given by:

• $\left(-{\frac {3{\sqrt {10}}}{20}},\,-{\frac {\sqrt {6}}{4}},\,0,\,0,\,\pm {\frac {1}{2}}\right),$
• $\left(-{\frac {3{\sqrt {10}}}{20}},\,{\frac {\sqrt {6}}{12}},\,-{\frac {\sqrt {3}}{3}},\,0,\,\pm {\frac {1}{2}}\right),$
• $\left(-{\frac {3{\sqrt {10}}}{20}},\,{\frac {\sqrt {6}}{12}},\,{\frac {\sqrt {3}}{6}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}}\right),$
• $\left({\frac {\sqrt {10}}{10}},\,{\frac {\sqrt {6}}{6}},\,{\frac {\sqrt {3}}{3}},\,0,\,\pm {\frac {1}{2}}\right),$
• $\left({\frac {\sqrt {10}}{10}},\,-{\frac {\sqrt {6}}{6}},\,-{\frac {\sqrt {3}}{3}},\,0,\,\pm {\frac {1}{2}}\right),$
• $\left({\frac {\sqrt {10}}{10}},\,{\frac {\sqrt {6}}{6}},\,-{\frac {\sqrt {3}}{6}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}}\right),$
• $\left({\frac {\sqrt {10}}{10}},\,-{\frac {\sqrt {6}}{6}},\,{\frac {\sqrt {3}}{6}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}}\right).$

Representations

A rectified pentachoric prism has the following Coxeter diagrams:

• x o3x3o3o (full symmetry)
• oo3xx3oo3oo&#x (A4 symmetry, as segmentoteron)
• xx xo3ox3oo&#x (A3×A1 axial, tetrahedral prism atop octahedral prism)
• xxx oxo oxo3oox&#xt (A2×A1×A1 symmetry, edge-first)

External links

• Klitzing, Richard. "Rappip".