Prismatorhombated pentachoron

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Prismatorhombated pentachoron
Rank4
TypeUniform
Notation
Bowers style acronymPrip
Coxeter diagramx3x3o3x ()
Elements
Cells10 triangular prisms, 10 hexagonal prisms, 5 truncated tetrahedra, 5 cuboctahedra
Faces20+20 triangles, 30+30 squares, 20 hexagons
Edges30+60+60
Vertices60
Vertex figureSkewed rectangular pyramid, base edge lengths 1, 2, 1, 2; lateral edge lengths 2, 2, 3, 3 (
Measures (edge length 1)
Circumradius
Hypervolume
Dichoral anglesHip–4–trip:
 Co–3–trip:
 Tut–6–hip:
 Co–4–hip:
 Co–3–tut:
Central density1
Number of external pieces30
Level of complexity16
Related polytopes
ArmyPrip
RegimentPrip
DualRhombipyramidal hexecontachoron
ConjugateNone
Abstract & topological properties
Flag count1920
Euler characteristic0
OrientableYes
Properties
SymmetryA4, order 120
ConvexYes
NatureTame

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[edit | edit source]

Vertex coordinates[edit | edit source]

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

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

  • .

Representations[edit | edit source]

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[edit | edit source]

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 .

Related polychora[edit | edit source]

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:

External links[edit | edit source]