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7-cube

{{Short description|7-dimensional hypercube}}

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!bgcolor=#e7dcc3 colspan=2|7-cube
Hepteract

bgcolor=#ffffff align=center colspan=2|280px
Orthogonal projection
inside Petrie polygon
The central orange vertex is doubled
bgcolor=#e7dcc3|TypeRegular 7-polytope
bgcolor=#e7dcc3|Familyhypercube
bgcolor=#e7dcc3|Schläfli symbol{4,35}
bgcolor=#e7dcc3|Coxeter-Dynkin diagrams{{CDD|node_1|4|node|3|node|3|node|3|node|3|node|3|node}}

{{CDD|node_1|2c|node_1|4|node|3|node|3|node|3|node|3|node

}

{{CDD|node_1|2c|node_1|2c|node_1|4|node|3|node|3|node|3|node|}}

{{CDD|node_1|2c|node_1|2c|node_1|2c|node_1|4|node|3|node|3|node|}}

{{CDD|node_1|2c|node_1|2c|node_1|2c|node_1|2c|node_1|4|node|3|node|}}

{{CDD|node_1|2c|node_1|2c|node_1|2c|node_1|2c|node_1|2c|node_1|4|node|}}

{{CDD|node_1|2c|node_1|2c|node_1|2c|node_1|2c|node_1|2c|node_1|2c|node_1}}

|-

|bgcolor=#e7dcc3|6-faces||14 {4,34} 25px

|-

|bgcolor=#e7dcc3|5-faces||84 {4,33} 25px

|-

|bgcolor=#e7dcc3|4-faces||280 {4,3,3} 25px

|-

|bgcolor=#e7dcc3|Cells||560 {4,3} 25px

|-

|bgcolor=#e7dcc3|Faces||672 {4} 25px

|-

|bgcolor=#e7dcc3|Edges||448

|-

|bgcolor=#e7dcc3|Vertices||128

|-

|bgcolor=#e7dcc3|Vertex figure||6-simplex 25px

|-

|bgcolor=#e7dcc3|Petrie polygon||tetradecagon

|-

|bgcolor=#e7dcc3|Coxeter group||C7, [35,4]

|-

|bgcolor=#e7dcc3|Dual||7-orthoplex

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|bgcolor=#e7dcc3|Properties||convex, Hanner polytope

|}

In geometry, a 7-cube is a seven-dimensional hypercube with 128 vertices, 448 edges, 672 square faces, 560 cubic cells, 280 tesseract 4-faces, 84 penteract 5-faces, and 14 hexeract 6-faces.

It can be named by its Schläfli symbol {4,35}, being composed of 3 6-cubes around each 5-face. It can be called a hepteract, a portmanteau of tesseract (the 4-cube) and hepta for seven (dimensions) in Greek. It can also be called a regular tetradeca-7-tope or tetradecaexon, being a 7 dimensional polytope constructed from 14 regular facets.

Related polytopes

The 7-cube is 7th in a series of hypercube:

{{Hypercube polytopes}}

The dual of a 7-cube is called a 7-orthoplex, and is a part of the infinite family of cross-polytopes.

Applying an alternation operation, deleting alternating vertices of the hepteract, creates another uniform polytope, called a demihepteract, (part of an infinite family called demihypercubes), which has 14 demihexeractic and 64 6-simplex 6-faces.

As a configuration

This configuration matrix represents the 7-cube. 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-cube. 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}

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

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

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

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

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

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

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

\end{matrix}\end{bmatrix}

Cartesian coordinates

Cartesian coordinates for the vertices of a hepteract centered at the origin and edge length 2 are

: (±1,±1,±1,±1,±1,±1,±1)

while the interior of the same consists of all points (x0, x1, x2, x3, x4, x5, x6) with -1 < xi < 1.

{{-}}

Projections

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valign=top

|240px
This hypercube graph is an orthogonal projection. This orientation shows columns of vertices positioned a vertex-edge-vertex distance from one vertex on the left to one vertex on the right, and edges attaching adjacent columns of vertices. The number of vertices in each column represents rows in Pascal's triangle, being 1:7:21:35:35:21:7:1.

{{7-cube Coxeter plane graphs|t0|150}}

References

  • H.S.M. Coxeter:
  • Coxeter, Regular Polytopes, (3rd edition, 1973), Dover edition, {{ISBN|0-486-61480-8}}, p. 296, Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n≥5)
  • H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973, p. 296, Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n≥5)
  • 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}} [http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html]
  • (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)|o3o3o3o3o3o4x - hept}}

External links

  • {{MathWorld|title=Hypercube|urlname=Hypercube}}
  • {{MathWorld|title=Hypercube graph|urlname=HypercubeGraph}}
  • {{GlossaryForHyperspace | anchor=Measure | title=Measure polytope }}
  • [http://tetraspace.alkaline.org/glossary.htm Multi-dimensional Glossary: hypercube] Garrett Jones
  • [http://www.4d-screen.de/related-space/4d-space/7d-cube.htm Rotation of 7D-Cube] www.4d-screen.de

{{Polytopes}}

Category:7-polytopes

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