Prismatorhombated pentachoron

Prismatorhombated pentachoron
(OFF file)
Rank	4
Type	Uniform
Space	Spherical
Notation
Bowers style acronym	Prip
Coxeter diagram	x3x3o3x ()
Elements
Cells	10 triangular prisms, 10 hexagonal prisms, 5 truncated tetrahedra, 5 cuboctahedra
Faces	20+20 triangles, 30+30 squares, 20 hexagons
Edges	30+60+60
Vertices	60
Vertex figure	Skewed rectangular pyramid, base edge lengths 1, √2, 1, √2; lateral edge lengths √2, √2, √3, √3 (
Measures (edge length 1)
Circumradius	$\frac{\sqrt{65}}{5} ≈ 1.61245$
Hypervolume	$\frac{237\sqrt5}{32} ≈ 16.56088$
Dichoral angles	Hip–4–trip: $\arccos\left(-\frac23\right) ≈ 131.81032^\circ$
	Co–3–trip: $\arccos\left(-\frac{\sqrt6}{4}\right) ≈ 127.76125^\circ$
	Tut–6–hip: $\arccos\left(-\frac{\sqrt6}{4}\right) ≈ 127.76125^\circ$
	Co–4–hip: $\arccos\left(-\frac{\sqrt6}{6}\right) ≈ 114.09484^\circ$
	Co–3–tut: $\arccos\left(-\frac14\right) ≈ 104.47751^\circ$
Central density	1
Number of external pieces	30
Level of complexity	16
Related polytopes
Army	Prip
Regiment	Prip
Dual	Rhombipyramidal hexecontachoron
Conjugate	None
Abstract & topological properties
Flag count	1920
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	A4, order 120
Convex	Yes
Nature	Tame

The prismatorhombated pentachoron, or prip, also commonly called the runcitruncated 5-cell or runcitruncated pentachoron, is a convex uniform polychoron that consists of 10 triangular prisms, 10 hexagonal prisms, 5 truncated tetrahedra, and 5 cuboctahedra. 1 triangular prism, 2 hexagonal prisms, 1 truncated tetrahedron, and 1 cuboctahedron join at each vertex. As one of its names suggests, it can be obtained by runcitruncating the pentachoron.

Gallery

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  Cross-sections

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  Net

Vertex coordinates

The vertices of a prismatorhombated pentachoron of edge length 1 are given by:

• $\left(\frac{\sqrt{10}}{10},\,-\frac{\sqrt6}{6},\,\frac{\sqrt3}{6},\,±\frac32\right),$
• $\left(\frac{\sqrt{10}}{10},\,\frac{\sqrt6}{6},\,-\frac{\sqrt3}{6},\,±\frac32\right),$
• $\left(\frac{\sqrt{10}}{10},\,-\frac{\sqrt6}{6},\,\frac{2\sqrt3}{3},\,±1\right),$
• $\left(\frac{\sqrt{10}}{10},\,\frac{\sqrt6}{6},\,-\frac{2\sqrt3}{3},\,±1\right),$
• $\left(\frac{\sqrt{10}}{10},\,-\frac{\sqrt6}{6},\,-\frac{5\sqrt3}{6},\,±\frac12\right),$
• $\left(\frac{\sqrt{10}}{10},\,\frac{\sqrt6}{6},\,\frac{5\sqrt3}{6},\,±\frac12\right),$
• $\left(\frac{\sqrt{10}}{10},\,±\frac{\sqrt6}{2},\,0,\,±1\right),$
• $\left(\frac{\sqrt{10}}{10},\,±\frac{\sqrt6}{2},\,±\frac{\sqrt3}{2},\,±\frac12\right),$
• $\left(\frac{7\sqrt{10}}{20},\,-\frac{\sqrt6}{12},\,\frac{\sqrt3}{3},\,±1\right),$
• $\left(\frac{7\sqrt{10}}{20},\,-\frac{\sqrt6}{12},\,-\frac{2\sqrt3}{3},\,0\right),$
• $\left(\frac{7\sqrt{10}}{20},\,\frac{\sqrt6}{4},\,0,\,±1\right),$
• $\left(\frac{7\sqrt{10}}{20},\,\frac{\sqrt6}{4},\,±\frac{\sqrt3}{2},\,±\frac12\right),$
• $\left(\frac{7\sqrt{10}}{20},\,-\frac{5\sqrt6}{12},\,\frac{\sqrt3}{6},\,±\frac12\right),$
• $\left(\frac{7\sqrt{10}}{20},\,-\frac{5\sqrt6}{12},\,-\frac{\sqrt3}{3},\,0\right),$
• $\left(-\frac{3\sqrt{10}}{20},\,\frac{\sqrt6}{12},\,\frac{\sqrt3}{6},\,±\frac32\right),$
• $\left(-\frac{3\sqrt{10}}{20},\,\frac{\sqrt6}{12},\,\frac{2\sqrt3}{3},\,±1\right),$
• $\left(-\frac{3\sqrt{10}}{20},\,\frac{\sqrt6}{12},\,-\frac{5\sqrt3}{6},\,±\frac12\right),$
• $\left(-\frac{3\sqrt{10}}{20},\,\frac{5\sqrt6}{12},\,\frac{\sqrt3}{3},\,±1\right),$
• $\left(-\frac{3\sqrt{10}}{20},\,\frac{5\sqrt6}{12},\,-\frac{2\sqrt3}{3},\,0\right),$
• $\left(-\frac{3\sqrt{10}}{20},\,-\frac{7\sqrt6}{12},\,-\frac{\sqrt3}{6},\,±\frac12\right),$
• $\left(-\frac{3\sqrt{10}}{20},\,-\frac{7\sqrt6}{12},\,\frac{\sqrt3}{3},\,0\right),$
• $\left(-\frac{2\sqrt{10}}{5},\,0,\,0,\,±1\right),$
• $\left(-\frac{2\sqrt{10}}{5},\,0,\,±\frac{\sqrt3}{2},\,±\frac12\right),$
• $\left(-\frac{2\sqrt{10}}{5},\,-\frac{\sqrt6}{3},\,-\frac{\sqrt3}{6},\,±\frac12\right),$
• $\left(-\frac{2\sqrt{10}}{5},\,\frac{\sqrt6}{3},\,\frac{\sqrt3}{6},\,±\frac12\right),$
• $\left(-\frac{2\sqrt{10}}{5},\,-\frac{\sqrt6}{3},\,\frac{\sqrt3}{3},\,0\right),$
• $\left(-\frac{2\sqrt{10}}{5},\,\frac{\sqrt6}{3},\,-\frac{\sqrt3}{3},\,0\right).$

Much simpler coordinates can be given in five dimensions, as all permutations of:

• $\left(\frac{3\sqrt2}{2},\,\sqrt2,\,\sqrt2,\,\frac{\sqrt2}{2},\,0\right).$

Representations

A prismatorhombated pentachoron has the following coxeter diagrams:

• x3x3o3x (full symmetry)
• xxux3xxoo3oxxx&#xt (A3 symmetry, truncated tetrahedron-first)
• x(ou)(dx)ux x(uo)(ox)xx3x(xu)(xu)xo&#xt (A2×A1 symmetry, hexagonal prism-first)

Semi-uniform variant

The prismatorhombated pentachoron has a semi-uniform variant of the form x3y3o3z that maintains its full symmetry. This variant uses 5 semi-uniform truncated tetrahedra of form x3y3o, 5 rhombitetratetrahedra of form y3o3z, 10 triangular prisms of form x z3o, and 10 ditrigonal prisms of form z x3y as cells, with 3 edge lengths.

With edges of lengths a, b, and c (such that it is given by a3b3o3c), its circumradius is given by $\sqrt{\frac{2a^2+3b^2+2c^2+3ab+ac+2bc}{5}}$.

Related polychora

The prismatorhombated pentachoron is the colonel of a 3-member regiment that also includes the prismatotrispentachoron and the rhombiprismic dispentachoron.

The segmentochoron truncated tetrahedron atop truncated octahedron is a cap of the prismatorhombated pentachoron.

Uniform polychoron compounds composed of prismatorhombated pentachora include:

• Prismatorhombated stellated decachoron (2)
• Prismatorhombated medial hexacosichoron (120)

External links

• Bowers, Jonathan. "Category 10: Prismatorhombates" (#352).

• Bowers, Jonathan. "Pennic and Decaic Isogonals".

• Klitzing, Richard. "prip".

• Quickfur. "The Runcitruncated 5-cell".

• Wikipedia Contributors. "Runcitruncated 5-cell".