Great rhombated pentachoron

Great rhombated pentachoron
(OFF file)
Rank	4
Type	Uniform
Space	Spherical
Notation
Bowers style acronym	Grip
Coxeter diagram	x3x3x3o ()
Elements
Cells	10 triangular prisms, 5 truncated tetrahedra, 5 truncated octahedra
Faces	20 triangles, 30 squares, 10+20 hexagons
Edges	30+30+60
Vertices	60
Vertex figure	Sphenoid edge lengths 1 (1), √2 (2), and √3 (3)
Measures (edge length 1)
Circumradius	$\frac{\sqrt{85}}{5} ≈ 1.84391$
Hypervolume	$\frac{287\sqrt5}{32} ≈ 20.05473$
Dichoral angles	Tut–3–trip: $\arccos\left(-\frac{\sqrt6}{4}\right) ≈ 127.76124^\circ$
	Toe–4–trip: $\arccos\left(-\frac{\sqrt6}{6}\right) ≈ 114.09484^\circ$
	Toe–6–tut: $\arccos\left(-\frac14\right) ≈ 104.47751^\circ$
	Toe–6–toe: $\arccos\left(\frac14\right) ≈ 75.52249^\circ$
Central density	1
Number of external pieces	20
Level of complexity	12
Related polytopes
Army	Grip
Regiment	Grip
Dual	Sphenoidal hexecontachoron
Conjugate	None
Abstract & topological properties
Flag count	1440
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	A4, order 120
Convex	Yes
Nature	Tame

The great rhombated pentachoron, or grip, also commonly called the cantitruncated 5-cell or cantitruncated pentachoron, is a convex uniform polychoron that consists of 10 triangular prisms, 5 truncated tetrahedra, and 5 truncated octahedra. 1 triangular prism, 1 truncated tetrahedron, and 2 truncated octahedra join at each vertex. As one of its names suggests, it can be obtained by cantitruncating the pentachoron.

Gallery

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  Cross-sections

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  Net

Vertex coordinates

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

• $\left(\frac{\sqrt{10}}{20},\,-\frac{\sqrt6}{4},\,\frac{\sqrt3}{2},\,±\frac32\right),$
• $\left(\frac{\sqrt{10}}{20},\,-\frac{\sqrt6}{4},\,-\sqrt3,\,0\right),$
• $\left(\frac{\sqrt{10}}{20},\,\frac{5\sqrt6}{12},\,-\frac{\sqrt3}{6},\,±\frac32\right),$
• $\left(\frac{\sqrt{10}}{20},\,\frac{5\sqrt6}{12},\,-\frac{2\sqrt3}{3},\,±1\right),$
• $\left(\frac{\sqrt{10}}{20},\,\frac{5\sqrt6}{12},\,\frac{5\sqrt3}{6},\,±\frac12\right),$
• $\left(\frac{\sqrt{10}}{20},\,-\frac{7\sqrt6}{12},\,\frac{\sqrt3}{3},\,±1\right),$
• $\left(\frac{\sqrt{10}}{20},\,-\frac{7\sqrt6}{12},\,-\frac{2\sqrt3}{3},\,0\right),$
• $\left(-\frac{\sqrt{10}}{5},\,0,\,\frac{\sqrt3}{2},\,±\frac32\right),$
• $\left(-\frac{\sqrt{10}}{5},\,0,\,-\sqrt3,\,0\right),$
• $\left(-\frac{\sqrt{10}}{5},\,\frac{\sqrt6}{3},\,\frac{\sqrt3}{6},\,±\frac32\right),$
• $\left(-\frac{\sqrt{10}}{5},\,\frac{\sqrt6}{3},\,\frac{2\sqrt3}{3},\,±1\right),$
• $\left(-\frac{\sqrt{10}}{5},\,\frac{\sqrt6}{3},\,-\frac{5\sqrt3}{6},\,±\frac12\right),$
• $\left(-\frac{\sqrt{10}}{5},\,-\frac{2\sqrt6}{3},\,\frac{\sqrt3}{6},\,±\frac12\right),$
• $\left(-\frac{\sqrt{10}}{5},\,-\frac{2\sqrt6}{3},\,-\frac{\sqrt3}{3},\,0\right),$
• $\left(\frac{3\sqrt{10}}{10},\,-\frac{\sqrt6}{6},\,\frac{\sqrt3}{6},\,±\frac32\right),$
• $\left(\frac{3\sqrt{10}}{10},\,\frac{\sqrt6}{6},\,-\frac{\sqrt3}{6},\,±\frac32\right),$
• $\left(\frac{3\sqrt{10}}{10},\,-\frac{\sqrt6}{6},\,\frac{2\sqrt3}{3},\,±1\right),$
• $\left(\frac{3\sqrt{10}}{10},\,\frac{\sqrt6}{6},\,-\frac{2\sqrt3}{3},\,±1\right),$
• $\left(\frac{3\sqrt{10}}{10},\,-\frac{\sqrt6}{6},\,-\frac{5\sqrt3}{6},\,±\frac12\right),$
• $\left(\frac{3\sqrt{10}}{10},\,\frac{\sqrt6}{6},\,\frac{5\sqrt3}{6},\,±\frac12\right),$
• $\left(\frac{3\sqrt{10}}{10},\,±\frac{\sqrt6}{2},\,0,\,±1\right),$
• $\left(\frac{3\sqrt{10}}{10},\,±\frac{\sqrt6}{2},\,±\frac{\sqrt3}{2},\,±\frac12\right),$
• $\left(-\frac{9\sqrt{10}}{20},\,-\frac{\sqrt6}{12},\,\frac{\sqrt3}{3},\,±1\right),$
• $\left(-\frac{9\sqrt{10}}{20},\,-\frac{\sqrt6}{12},\,-\frac{2\sqrt3}{3},\,0\right),$
• $\left(-\frac{9\sqrt{10}}{20},\,\frac{\sqrt6}{4},\,0,\,±1\right),$
• $\left(-\frac{9\sqrt{10}}{20},\,\frac{\sqrt6}{4},\,±\frac{\sqrt3}{2},\,±\frac12\right),$
• $\left(-\frac{9\sqrt{10}}{20},\,-\frac{5\sqrt6}{12},\,\frac{\sqrt3}{6},\,±\frac12\right),$
• $\left(-\frac{9\sqrt{10}}{20},\,-\frac{5\sqrt6}{12},\,-\frac{\sqrt3}{3},\,0\right).$

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

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

Representations

The great rhombated pentachoron has the following Coxeter diagrams:

• x3x3x3o (full symmetry)
• xuxx3xxux3ooox&#xt (A3 axial, truncated tetrahedron-first)
• xu(xd)uxo xu(dx)uxx3oo(ox)xux&#xt (A2×A1 symmetry, triangular prism-first)

Semi-uniform variant

The great rhombated pentachoron has a semi-uniform variant of the form x3y3z3o that maintains its full symmetry. This variant uses 5 semi-uniform truncated tetrahedra of form y3z3o, 5 great rhombitetratetrahedra of form x3y3z, and 10 triangular prisms of form x z3o as cells, with 3 edge lengths.

With edges of lengths a, b, and c (such that it is given by a3b3c3o), its circumradius is given by $\sqrt{\frac{2a^2+3b^2+3c^2+3ab+2ac+4bc}{5}}$.

Related polychora

Uniform polychoron compounds composed of great rhombated pentachora include:

• Great rhombated stellated decachoron (2)
• Great rhombated medial hexacosichoron (120)

External links

• Bowers, Jonathan. "Category 8: Great Rhombates" (#307).

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

• Klitzing, Richard. "grip".

• Quickfur. "The Cantitruncated 5-cell".

• Wikipedia Contributors. "Cantitruncated 5-cell".