# 5-orthoplex

Regular 5-orthoplex
(pentacross)

Orthogonal projection
inside Petrie polygon
Type Regular 5-polytope
Family orthoplex
Schläfli symbol {3,3,3,4}
{3,3,31,1}
Coxeter-Dynkin diagrams
4-faces 32 {33}
Cells 80 {3,3}
Faces 80 {3}
Edges 40
Vertices 10
Vertex figure
16-cell
Petrie polygon decagon
Coxeter groups BC5, [3,3,3,4]
D5, [32,1,1]
Dual 5-cube
Properties convex

In five-dimensional geometry, a 5-orthoplex, or 5-cross polytope, is a five-dimensional polytope with 10 vertices, 40 edges, 80 triangle faces, 80 tetrahedron cells, 32 5-cell 4-faces.

It has two constructed forms, the first being regular with Schläfli symbol {33,4}, and the second with alternately labeled (checkerboarded) facets, with Schläfli symbol {32,1,1} or Coxeter symbol 211.

It is a part of an infinite family of polytopes, called cross-polytopes or orthoplexes. The dual polytope is the 5-hypercube or 5-cube.

## Alternate names

• pentacross, derived from combining the family name cross polytope with pente for five (dimensions) in Greek.
• Triacontaditeron (or triacontakaiditeron) - as a 32-facetted 5-polytope (polyteron).

## Cartesian coordinates

Cartesian coordinates for the vertices of a 5-orthoplex, centered at the origin are

(±1,0,0,0,0), (0,±1,0,0,0), (0,0,±1,0,0), (0,0,0,±1,0), (0,0,0,0,±1)

## Construction

There are three Coxeter groups associated with the 5-orthoplex, one regular, dual of the penteract with the C5 or [4,3,3,3] Coxeter group, and a lower symmetry with two copies of 5-cell facets, alternating, with the D5 or [32,1,1] Coxeter group, and the final one as a dual 5-orthotope, called a 5-fusil which can have a variety of subsymmetries.

Name Coxeter diagram Schläfli symbol Symmetry Order Vertex figure(s)
regular 5-orthoplex {3,3,3,4} [3,3,3,4] 3840
Alternate 5-orthoplex {3,3,31,1} [3,3,31,1] 1920
5-fusil
{3,3,3,4} [4,3,3,3] 3840
{3,3,4}+{} [4,3,3,2] 768
{3,4}+{4} [4,3,2,4] 384
{3,4}+2{} [4,3,2,2] 192
2{4}+{} [4,2,4,2] 128
{4}+3{} [4,2,2,2] 64
5{} [2,2,2,2] 32

## Other 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]
 The perspective projection (3D to 2D) of a stereographic projection (4D to 3D) of the Schlegel diagram (5D to 4D) of the 5-orthoplex. 10 sets of 4 edges form 10 circles in the 4D Schlegel diagram: two of these circles are straight lines in the stereographic projection because they contain the center of projection.

## Related polytopes and honeycombs

2k1 figures in n dimensions
Space Finite Euclidean Hyperbolic
n 3 4 5 6 7 8 9 10
Coxeter
group
E3=A2A1 E4=A4 E5=D5 E6 E7 E8 E9 = ${\tilde{E}}_{8}$ = E8+ E10 = ${\bar{T}}_8$ = E8++
Coxeter
diagram
Symmetry [3-1,2,1] [30,2,1] [[31,2,1]] [32,2,1] [33,2,1] [34,2,1] [35,2,1] [36,2,1]
Order 12 120 384 51,840 2,903,040 696,729,600
Graph - -
Name 2-1,1 201 211 221 231 241 251 261

This polytope is one of 63 uniform polypeta generated from the B6 Coxeter plane, including the regular 6-cube or 6-orthoplex.

 β5 t1β5 t2γ5 t1γ5 γ5 t0,1β5 t0,2β5 t1,2β5 t0,3β5 t1,3γ5 t1,2γ5 t0,4γ5 t0,3γ5 t0,2γ5 t0,1γ5 t0,1,2β5 t0,1,3β5 t0,2,3β5 t1,2,3γ5 t0,1,4β5 t0,2,4γ5 t0,2,3γ5 t0,1,4γ5 t0,1,3γ5 t0,1,2γ5 t0,1,2,3β5 t0,1,2,4β5 t0,1,3,4γ5 t0,1,2,4γ5 t0,1,2,3γ5 t0,1,2,3,4γ5

## 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. (1966)
• Richard Klitzing, 5D uniform polytopes (polytera), x3o3o3o4o - tac