Rectified 5-orthoplexes

5-cube	Rectified 5-cube	Birectified 5-cube Birectified 5-orthoplex
5-orthoplex	Rectified 5-orthoplex
Orthogonal projections in A5 Coxeter plane

In five-dimensional geometry, a rectified 5-orthoplex is a convex uniform 5-polytope, being a rectification of the regular 5-orthoplex.

There are 5 degrees of rectifications for any 5-polytope, the zeroth here being the 5-orthoplex itself, and the 4th and last being the 5-cube. Vertices of the rectified 5-orthoplex are located at the edge-centers of the 5-orthoplex. Vertices of the birectified 5-orthoplex are located in the triangular face centers of the 5-orthoplex.

Rectified 5-orthoplex

Rectified pentacross
Type	uniform 5-polytope
Schläfli symbol	t1{3,3,3,4}
Coxeter-Dynkin diagrams
Hypercells	42 total: 10 {3,3,4} 32 t1{3,3,3}
Cells	240 total: 80 {3,4} 160 {3,3}
Faces	400 total: 80+320 {3}
Edges	240
Vertices	40
Vertex figure	Octahedral prism
Petrie polygon	Decagon
Coxeter groups	BC5, [3,3,3,4] D5, [32,1,1]
Properties	convex

Its 40 vertices represent the root vectors of the simple Lie group D₅. The vertices can be seen in 3 hyperplanes, with the 10 vertices rectified 5-cells cells on opposite sides, and 20 vertices of a runcinated 5-cell passing through the center. When combined with the 10 vertices of the 5-orthoplex, these vertices represent the 50 root vectors of the B₅ and C₅ simple Lie groups.

E. L. Elte identified it in 1912 as a semiregular polytope, identifying it as Cr₅¹ as a first rectification of a 5-dimensional cross polytope.

Alternate names

• rectified pentacross
• rectified triacontiditeron (32-faceted 5-polytope)

Construction

There are two Coxeter groups associated with the rectified pentacross, one with the C₅ or [4,3,3,3] Coxeter group, and a lower symmetry with two copies of 16-cell facets, alternating, with the D₅ or [3^2,1,1] Coxeter group.

Cartesian coordinates

Cartesian coordinates for the vertices of a rectified pentacross, centered at the origin, edge length [math]\displaystyle{ \sqrt{2}\ }[/math] are all permutations of:

(±1,±1,0,0,0)

Images

orthographic projections

Coxeter plane	B5	B4 / D5	B3 / D4 / A2
Graph
Dihedral symmetry	[10]	[8]	[6]
Coxeter plane	B2	A3
Graph
Dihedral symmetry	[4]	[4]

Related polytopes

The rectified 5-orthoplex is the vertex figure for the 5-demicube honeycomb:

or

This polytope is one of 31 uniform 5-polytope generated from the regular 5-cube or 5-orthoplex.

References

• H.S.M. Coxeter:
  • H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
  • Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 [1]
    • (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10]
    • (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591]
    • (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
• Norman Johnson Uniform Polytopes, Manuscript (1991)
  • N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D.
• Klitzing, Richard. "5D uniform polytopes (polytera)". https://bendwavy.org/klitzing/dimensions/polytera.htm.  o3x3o3o4o - rat

External links

• Polytopes of Various Dimensions
• Multi-dimensional Glossary