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7-orthoplex

{{Short description|Regular 7- polytope}}

class="wikitable" align="right" style="margin-left:10px" width="250"

!bgcolor=#e7dcc3 colspan=2|Regular 7-orthoplex
Heptacross

bgcolor=#ffffff align=center colspan=2|280px
Orthogonal projection
inside Petrie polygon
bgcolor=#e7dcc3|TypeRegular 7-polytope
bgcolor=#e7dcc3|Familyorthoplex
bgcolor=#e7dcc3|Schläfli symbol{35,4}
{3,3,3,3,31,1}
bgcolor=#e7dcc3|Coxeter-Dynkin diagrams{{CDD|node_1|3|node|3|node|3|node|3|node|3|node|4|node}}
{{CDD|node_1|3|node|3|node|3|node|3|node|split1|nodes}}
bgcolor=#e7dcc3|6-faces128 {35} 25px
bgcolor=#e7dcc3|5-faces448 {34} 25px
bgcolor=#e7dcc3|4-faces672 {33} 25px
bgcolor=#e7dcc3|Cells560 {3,3} 25px
bgcolor=#e7dcc3|Faces280 {3}25px
bgcolor=#e7dcc3|Edges84
bgcolor=#e7dcc3|Vertices14
bgcolor=#e7dcc3|Vertex figure6-orthoplex
bgcolor=#e7dcc3|Petrie polygontetradecagon
bgcolor=#e7dcc3|Coxeter groupsC7, [3,3,3,3,3,4]
D7, [34,1,1]
bgcolor=#e7dcc3|Dual7-cube
bgcolor=#e7dcc3|Propertiesconvex, Hanner polytope

In geometry, a 7-orthoplex, or 7-cross polytope, is a regular 7-polytope with 14 vertices, 84 edges, 280 triangle faces, 560 tetrahedron cells, 672 5-cell 4-faces, 448 5-faces, and 128 6-faces.

It has two constructed forms, the first being regular with Schläfli symbol {35,4}, and the second with alternately labeled (checkerboarded) facets, with Schläfli symbol {3,3,3,3,31,1} or Coxeter symbol 411.

It is a part of an infinite family of polytopes, called cross-polytopes or orthoplexes. The dual polytope is the 7-hypercube, or hepteract.

Alternate names

  • Heptacross, derived from combining the family name cross polytope with hept for seven (dimensions) in Greek.
  • Hecatonicosaoctaexon as a 128-facetted 7-polytope (polyexon). Acronym: zee{{sfn|Klitzing|at=[https://bendwavy.org/klitzing/incmats/zee.htm (x3o3o3o3o3o4o - zee)]}}

As a configuration

This configuration matrix represents the 7-orthoplex. The rows and columns correspond to vertices, edges, faces, cells, 4-faces, 5-faces and 6-faces. The diagonal numbers say how many of each element occur in the whole 7-orthoplex. The nondiagonal numbers say how many of the column's element occur in or at the row's element.Coxeter, Regular Polytopes, sec 1.8 ConfigurationsCoxeter, Complex Regular Polytopes, p.117

\begin{bmatrix}\begin{matrix}

14 & 12 & 60 & 160 & 240 & 192 & 64

\\ 2 & 84 & 10 & 40 & 80 & 80 & 32

\\ 3 & 3 & 280 & 8 & 24 & 32 & 16

\\ 4 & 6 & 4 & 560 & 6 & 12 & 8

\\ 5 & 10 & 10 & 5 & 672 & 4 & 4

\\ 6 & 15 & 20 & 15 & 6 & 448 & 2

\\ 7 & 21 & 35 & 35 & 21 & 7 & 128

\end{matrix}\end{bmatrix}

Images

{{7-cube Coxeter plane graphs|t6|150}}

Construction

There are two Coxeter groups associated with the 7-orthoplex, one regular, dual of the hepteract with the C7 or [4,3,3,3,3,3] symmetry group, and a half symmetry with two copies of 6-simplex facets, alternating, with the D7 or [34,1,1] symmetry group. A lowest symmetry construction is based on a dual of a 7-orthotope, called a 7-fusil.

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!Name

!Coxeter diagram

!Schläfli symbol

!Symmetry

!Order

!Vertex figure

align=center

!regular 7-orthoplex

|{{CDD|node_1|3|node|3|node|3|node|3|node|3|node|4|node}}

|{3,3,3,3,3,4}

|[3,3,3,3,3,4]

645120

|{{CDD|node_1|3|node|3|node|3|node|3|node|4|node}}

align=center

!Quasiregular 7-orthoplex

|{{CDD|node_1|3|node|3|node|3|node|3|node|split1|nodes}}

|{3,3,3,3,31,1}

|[3,3,3,3,31,1]

322560

|{{CDD|node_1|3|node|3|node|3|node|split1|nodes}}

align=center

!7-fusil

|{{CDD|node_f1|2|node_f1|2|node_f1|2|node_f1|2|node_f1|2|node_f1|2|node_f1}}

|7{}

|[26]

128

|{{CDD|node_f1|2|node_f1|2|node_f1|2|node_f1|2|node_f1|2|node_f1}}

Cartesian coordinates

Cartesian coordinates for the vertices of a 7-orthoplex, centered at the origin are

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

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

See also

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, [https://www.wiley.com/en-us/Kaleidoscopes-p-9780471010036 wiley.com], {{isbn|978-0-471-01003-6}}
  • (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)
  • {{KlitzingPolytopes|polyexa.htm|7D uniform polytopes (polyexa) with acronyms}} x3o3o3o3o3o4o - zee {{sfn whitelist| CITEREFKlitzing}}

External links

  • {{GlossaryForHyperspace | anchor=Cross | title=Cross polytope }}
  • [http://www.polytope.net/hedrondude/topes.htm Polytopes of Various Dimensions]
  • [http://tetraspace.alkaline.org/glossary.htm Multi-dimensional Glossary]

{{Polytopes}}

Category:7-polytopes