# Small rhombated pentachoron

Small rhombated pentachoron
Rank4
TypeUniform
SpaceSpherical
Notation
Bowers style acronymSrip
Coxeter diagramx3o3x3o ()
Elements
Cells5 octahedra, 10 triangular prisms, 5 cuboctahedra
Faces10+20+20 triangles, 30 squares
Edges30+60
Vertices30
Vertex figureSquare wedge, edge lengths 1 (base square and top edge) and 2 (side edges)
Measures (edge length 1)
Circumradius${\displaystyle \frac{\sqrt{35}}5 ≈ 1.18322}$
Hypervolume${\displaystyle \frac{73\sqrt5}{48} ≈ 3.40069}$
Dichoral anglesOct–3–trip: ${\displaystyle \arccos\left(-\frac{\sqrt6}4\right) ≈ 127.76124^\circ}$
Co–4–trip: ${\displaystyle \arccos\left(-\frac{\sqrt6}6\right) ≈ 114.09484^\circ}$
Co–3–oct: ${\displaystyle \arccos\left(-\frac14\right) ≈ 104.47751^\circ}$
Co–3–co: ${\displaystyle \arccos\left(\frac14\right) ≈ 75.52249^\circ}$
Central density1
Number of external pieces20
Level of complexity9
Related polytopes
ArmySrip
RegimentSrip
DualNotched triacontachoron
ConjugateNone
Abstract & topological properties
Flag count1080
Euler characteristic0
OrientableYes
Properties
SymmetryA4, order 120
ConvexYes
NatureTame

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.

## Vertex coordinates

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

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

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

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

## 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 ${\displaystyle \sqrt{\frac{2a^2+3b^2+2ab}{5}}}$ and its hypervolume is given by ${\displaystyle (a^4+12a^3b+54a^2b^2+68ab^3+11b^4)\frac{\sqrt5}{96}}$.

## 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:

o3o3o3o truncations
Name OBSA CD diagram Picture
Pentachoron pen
Truncated pentachoron tip
Rectified pentachoron rap
Decachoron deca
Rectified pentachoron rap
Truncated pentachoron tip
Pentachoron pen
Small rhombated pentachoron srip
Great rhombated pentachoron grip
Small rhombated pentachoron srip
Great rhombated pentachoron grip
Small prismatodecachoron spid
Prismatorhombated pentachoron prip
Prismatorhombated pentachoron prip
Great prismatodecachoron gippid