Small rhombated pentachoron

Small rhombated pentachoron
(OFF file)
Rank	4
Type	Uniform
Space	Spherical
Notation
Bowers style acronym	Srip
Coxeter diagram	x3o3x3o ()
Elements
Cells	5 octahedra, 10 triangular prisms, 5 cuboctahedra
Faces	10+20+20 triangles, 30 squares
Edges	30+60
Vertices	30
Vertex figure	Square wedge, edge lengths 1 (base square and top edge) and √2 (side edges)
Measures (edge length 1)
Circumradius	$\frac{\sqrt{35}}5 ≈ 1.18322$
Hypervolume	$\frac{73\sqrt5}{48} ≈ 3.40069$
Dichoral angles	Oct–3–trip: $\arccos\left(-\frac{\sqrt6}4\right) ≈ 127.76124^\circ$
	Co–4–trip: $\arccos\left(-\frac{\sqrt6}6\right) ≈ 114.09484^\circ$
	Co–3–oct: $\arccos\left(-\frac14\right) ≈ 104.47751^\circ$
	Co–3–co: $\arccos\left(\frac14\right) ≈ 75.52249^\circ$
Central density	1
Number of external pieces	20
Level of complexity	9
Related polytopes
Army	Srip
Regiment	Srip
Dual	Notched triacontachoron
Conjugate	None
Abstract & topological properties
Flag count	1080
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	A4, order 120
Convex	Yes
Nature	Tame

The small rhombated pentachoron, or srip, also commonly called the cantellated 5-cell or cantellated pentachoron, is a convex uniform polychoron that consists of 5 regular octahedra, 10 triangular prisms, and 5 cuboctahedra. 1 octahedron, 2 triangular prisms, and 2 cuboctahedra join at each vertex. As one of its names suggests, it can be formed by cantellating the pentachoron. It can also be formed by rectification of the rectified pentachoron.

Gallery

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  Cross-sections

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  Net

Vertex coordinates

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

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

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

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

Representations

A small rhombated pentachoron has the following Coxeter diagrams:

• x3o3x3o (full symmetry)
• oxx3xxo3oox&#xt (A₃ axial, octahedron-first)
• x(uo)xo x(ou)xx3o(xo)xo&#xt (A₂×A₁ axial, triangular prism-first)

Semi-uniform variant

The small rhombated pentachoron has a semi-uniform variant of the form x3o3y3o that maintains its full symmetry. This variant uses 5 octahedra of size y, 5 rhombitetratetrahedra of form x3o3y, and 10 triangular prisms of form x y3o as cells, with 2 edge lengths.

With edges of length a (surrounds 2 rhombitetratetrahedra) and b (of octahedra), its circumradius is given by $\sqrt{\frac{2a^2+3b^2+2ab}{5}}$ and its hypervolume is given by $(a^4+12a^3b+54a^2b^2+68ab^3+11b^4)\frac{\sqrt5}{96}$.

Related polychora

The small rhombated pentachoron is the colonel of the largest regiment of uniform polychora with A₄ symmetry, which has a total of 7 members. Its facetings include the retrosphenoverted trispentachoron, small rhombic dispentachoron, pseudorhombic prismatopentachoron, grand rhombic prismatopentachoron, prismatopentintercepted dispentachoron, and prismatointercepted prismatodispentachoron.

When viewed in A₃ axial symmetry, the small rhombated pentachoron can be cut into 2 segmentochora, namely cuboctahedron atop truncated tetrahedron and octahedron atop truncated tetrahedron, join at the truncated tetrahedral bases.

The triangular pucofastegium occurs as the triangle-first cap of the small rhombated pentachoron.

Uniform polychoron compounds composed of small rhombated pentachora include:

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

External links

• Bowers, Jonathan. "Category 6: Sphenoverts" (#133).

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

• Klitzing, Richard. "srip".

• Quickfur. "The Cantellated 5-cell".

• Wikipedia Contributors. "Cantellated 5-cell".