Rectified pentachoron

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Rectified pentachoron
Schlegel half-solid rectified 5-cell.png
Rank4
TypeUniform
SpaceSpherical
Bowers style acronymRap
Coxeter diagramo3x3o3o (CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png)
Elements
Vertex figureTriangular prism, edge length 1
Cells5 tetrahedra, 5 octahedra
Faces10+20 triangles
Edges30
Vertices10
Measures (edge length 1)
Circumradius
Hypervolume
Dichoral anglesOct–3–tet:
 Oct-3-oct:
Height
Central density1
Euler characteristic0
Number of pieces10
Level of complexity3
Related polytopes
ArmyRap
RegimentRap
DualJoined pentachoron
ConjugateNone
Topological properties
OrientableYes
Properties
SymmetryA4, order 120
ConvexYes
NatureTame

The rectified pentachoron, or rap, also commonly called the rectified 5-cell, is a convex uniform polychoron that consists of 5 regular tetrahedra and 5 regular octahedra. Two tetrahedra and three octahedra join at each triangular prismatic vertex. As the name suggests, it can be constructed by rectification of the pentachoron.

It is the vertex figure of the demipenteract.

It is also a convex segmentochoron (designated K-4.5 in Richard Klitzing's list), formed as a tetrahedron atop an octahedron. If the edge lengths of this subsymmetrical variation are changed, the result is various non-uniform tetrahedron atop octahedron polychora with 4 triangle antipodiums and 4 triangle pyramids as sides.

It is also isogonal under the 5-2 step prism subsymmetry, and can be considered to be a 5-2 double step prism.

Gallery[edit | edit source]

Rap sections Bowers.png Rectified pentachoron net.png

Vertex coordinates[edit | edit source]

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

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

Representations[edit | edit source]

A rectified pentachoron has the following Coxeter diagrams:

  • o3x3o3o (full symmetry)
  • xo3ox3oo&#x (A3 axial, as tetrahedron atop octahedron
  • oxo oxo3oox&#xt (A2×A1 axial, vertex-first)
  • oxoo3xoxo&#xr (A2 axial)
  • oxoox oxoxo&#xr (A1×A1 axial)

Segmentochoron display[edit | edit source]

Related polychora[edit | edit source]

The rectified pentachoron is the colonel of a three-member regiment that also includes the facetorectified pentachoron and the prismatointercepted pentachoron.

The rectified pentachoron can be diminished by cutting off triangular prismatic pyramids, with each removing 2 tetrahedra and diminishing the octahedra down to square pyramids. If one pyramid is removed, the result is the triangular antifastegium. If two non-adjacent pyramids are removed, such that one of the octahedra gets reduced down to an equatorial square only, the result is the bidiminished rectified pentachoron.

An octahedral pyramid can be attached to one of the rectified pentachoron's octahedral cells to create the augmented rectified pentachoron.

o3o3o3o truncations
Name OBSA CD diagram Picture
Pentachoron pen CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
Schlegel wireframe 5-cell.png
Truncated pentachoron tip CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
Schlegel half-solid truncated pentachoron.png
Rectified pentachoron rap CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
Schlegel half-solid rectified 5-cell.png
Decachoron deca CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
Schlegel half-solid bitruncated 5-cell.png
Rectified pentachoron rap CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
Schlegel half-solid rectified 5-cell.png
Truncated pentachoron tip CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
Schlegel half-solid truncated pentachoron.png
Pentachoron pen CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
Schlegel wireframe 5-cell.png
Small rhombated pentachoron srip CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
Schlegel half-solid cantellated 5-cell.png
Great rhombated pentachoron grip CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
Schlegel half-solid cantitruncated 5-cell.png
Small rhombated pentachoron srip CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
Schlegel half-solid cantellated 5-cell.png
Great rhombated pentachoron grip CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
Schlegel half-solid cantitruncated 5-cell.png
Small prismatodecachoron spid CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
Schlegel half-solid runcinated 5-cell.png
Prismatorhombated pentachoron prip CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
Schlegel half-solid runcitruncated 5-cell.png
Prismatorhombated pentachoron prip CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
Schlegel half-solid runcitruncated 5-cell.png
Great prismatodecachoron gippid CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
Schlegel half-solid omnitruncated 5-cell.png


Isogonal derivatives[edit | edit source]

Substitution by vertices of these following elements will produce these convex isogonal polychora:

External links[edit | edit source]

  • Klitzing, Richard. "rap".