6-orthoplex

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6-orthoplex
Hexacross
6-cube t5.svg
Orthogonal projection
inside Petrie polygon
Type Regular 6-polytope
Family orthoplex
Schläfli symbols {3,3,3,3,4}
{3,3,3,31,1}
Coxeter-Dynkin diagrams CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png
5-faces 64 {34}5-simplex t0.svg
4-faces 192 {33}4-simplex t0.svg
Cells 240 {3,3}3-simplex t0.svg
Faces 160 {3}2-simplex t0.svg
Edges 60
Vertices 12
Vertex figure 5-orthoplex
Petrie polygon dodecagon
Coxeter groups B6, [3,3,3,3,4]
D6, [33,1,1]
Dual 6-cube
Properties convex

In geometry, a 6-orthoplex, or 6-cross polytope, is a regular 6-polytope with 12 vertices, 60 edges, 160 triangle faces, 240 tetrahedron cells, 192 5-cell 4-faces, and 64 5-faces.

It has two constructed forms, the first being regular with Schläfli symbol {34,4}, and the second with alternately labeled (checkerboarded) facets, with Schläfli symbol {34,1,1} or Coxeter symbol 411.

Alternate names[edit]

Related polytopes[edit]

It is a part of an infinite family of polytopes, called cross-polytopes or orthoplexes. The dual polytope is the 6-hypercube, or hexeract.

Construction[edit]

There are three Coxeter groups associated with the 6-orthoplex, one regular, dual of the hexeract with the C6 or [4,3,3,3,3] Coxeter group, and a half symmetry with two copies of 5-simplex facets, alternating, with the D6 or [33,1,1] Coxeter group. A lowest symmetry construction is based on a dual of a 6-orthotope, called a 6-fusil.

Name Coxeter diagram Schläfli symbol Symmetry Order Vertex figure
Alternate 6-orthoplex CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png {3,3,3,3,4} [3,3,3,3,4] 46080 CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
regular 6-orthoplex CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png {3,3,3,31,1} [3,3,3,31,1] 23040 CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png
6-fusil CDel node f1.pngCDel 2.pngCDel node f1.pngCDel 2.pngCDel node f1.pngCDel 2.pngCDel node f1.pngCDel 2.pngCDel node f1.pngCDel 2.pngCDel node f1.png {}+{}+{}+{}+{}+{} [25] 64 CDel node f1.pngCDel 2.pngCDel node f1.pngCDel 2.pngCDel node f1.pngCDel 2.pngCDel node f1.pngCDel 2.pngCDel node f1.png

Cartesian coordinates[edit]

Cartesian coordinates for the vertices of a 6-orthoplex, centered at the origin are

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

Every vertex pair is connected by an edge, except opposites.

Images[edit]

orthographic projections
Coxeter plane B6 B5 B4
Graph 6-cube t5.svg 6-cube t5 B5.svg 6-cube t5 B4.svg
Dihedral symmetry [12] [10] [8]
Coxeter plane B3 B2
Graph 6-cube t5 B3.svg 6-cube t5 B2.svg
Dihedral symmetry [6] [4]
Coxeter plane A5 A3
Graph 6-cube t5 A5.svg 6-cube t5 A3.svg
Dihedral symmetry [6] [4]

Related polytopes[edit]

The 6-orthoplex can be projected down to 3-dimensions into the vertices of a regular icosahedron, as seen in this 2D projection:

Icosahedron t0 H3.png
Icosahedron
H3 Coxeter plane
6-cube t5 B5.svg
6-orthoplex
D6 Coxeter plane
This construction can be geometrically seen as the 12 vertices of the 6-orthoplex projected to 3 dimensions. This represents a geometric folding of the D6 to H3 Coxeter groups: Geometric folding Coxeter graph D6 H3.png

Seen by these 2D Coxeter plane orthogonal projections, the two overlapping central vertices define the third axis in this mapping.

It is in a dimensional series of uniform polytopes and honeycombs, expressed by Coxeter as 3k1 series. (A degenerate 4-dimensional case exists as 3-sphere tiling, a tetrahedral hosohedron.)

3k1 dimensional figures
n 4 5 6 7 8 9
Coxeter
group
A3×A1 A5 D6 E7 {\tilde{E}}_{7}=E7+ E7++
Coxeter
diagram
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.png CDel nodea 1.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.png CDel nodea 1.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.png CDel nodea 1.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png CDel nodea 1.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png CDel nodea 1.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
Symmetry
(order)
[3-1,3,1]
(48)
[30,3,1]
(720)
[[31,3,1]]
(46,080)
[32,3,1]
(2,903,040)
[33,3,1]
(∞)
[34,3,1]
(∞)
Graph 5-simplex t0.svg 6-cube t5.svg Up2 3 21 t0 E7.svg
Name 31,-1 310 311 321 331 341

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

6-cube t5.svg
β6
6-cube t4.svg
t1β6
6-cube t3.svg
t2β6
6-cube t2.svg
t2γ6
6-cube t1.svg
t1γ6
6-cube t0.svg
γ6
6-cube t45.svg
t0,1β6
6-cube t35.svg
t0,2β6
6-cube t34.svg
t1,2β6
6-cube t25.svg
t0,3β6
6-cube t24.svg
t1,3β6
6-cube t23.svg
t2,3γ6
6-cube t15.svg
t0,4β6
6-cube t14.svg
t1,4γ6
6-cube t13.svg
t1,3γ6
6-cube t12.svg
t1,2γ6
6-cube t05.svg
t0,5γ6
6-cube t04.svg
t0,4γ6
6-cube t03.svg
t0,3γ6
6-cube t02.svg
t0,2γ6
6-cube t01.svg
t0,1γ6
6-cube t345.svg
t0,1,2β6
6-cube t245.svg
t0,1,3β6
6-cube t235.svg
t0,2,3β6
6-cube t234.svg
t1,2,3β6
6-cube t145.svg
t0,1,4β6
6-cube t135.svg
t0,2,4β6
6-cube t134.svg
t1,2,4β6
6-cube t125.svg
t0,3,4β6
6-cube t124.svg
t1,2,4γ6
6-cube t123.svg
t1,2,3γ6
6-cube t045.svg
t0,1,5β6
6-cube t035.svg
t0,2,5β6
6-cube t034.svg
t0,3,4γ6
6-cube t025.svg
t0,2,5γ6
6-cube t024.svg
t0,2,4γ6
6-cube t023.svg
t0,2,3γ6
6-cube t015.svg
t0,1,5γ6
6-cube t014.svg
t0,1,4γ6
6-cube t013.svg
t0,1,3γ6
6-cube t012.svg
t0,1,2γ6
6-cube t2345.svg
t0,1,2,3β6
6-cube t1345.svg
t0,1,2,4β6
6-cube t1245.svg
t0,1,3,4β6
6-cube t1235.svg
t0,2,3,4β6
6-cube t1234.svg
t1,2,3,4γ6
6-cube t0345.svg
t0,1,2,5β6
6-cube t0245.svg
t0,1,3,5β6
6-cube t0235.svg
t0,2,3,5γ6
6-cube t0234.svg
t0,2,3,4γ6
6-cube t0145.svg
t0,1,4,5γ6
6-cube t0135.svg
t0,1,3,5γ6
6-cube t0134.svg
t0,1,3,4γ6
6-cube t0125.svg
t0,1,2,5γ6
6-cube t0124.svg
t0,1,2,4γ6
6-cube t0123.svg
t0,1,2,3γ6
6-cube t12345.svg
t0,1,2,3,4β6
6-cube t02345.svg
t0,1,2,3,5β6
6-cube t01345.svg
t0,1,2,4,5β6
6-cube t01245.svg
t0,1,2,4,5γ6
6-cube t01235.svg
t0,1,2,3,5γ6
6-cube t01234.svg
t0,1,2,3,4γ6
6-cube t012345.svg
t0,1,2,3,4,5γ6

References[edit]

  • 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, 6D uniform polytopes (polypeta), x3o3o3o3o4o - gee

External links[edit]