7-cube Hepteract
Orthogonal projection insidePetrie polygon The central orange vertex is doubled
Type	Regular7-polytope
Family	hypercube
Schläfli symbol	{4,35}
Coxeter-Dynkin diagrams
6-faces	14{4,34}
5-faces	84{4,33}
4-faces	280{4,3,3}
Cells	560{4,3}
Faces	672{4}
Edges	448
Vertices	128
Vertex figure	6-simplex
Petrie polygon	tetradecagon
Coxeter group	C7, [35,4]
Dual	7-orthoplex
Properties	convex,Hanner polytope

Ingeometry, a7-cube is aseven-dimensionalhypercube with 128vertices, 448edges, 672 squarefaces, 560 cubiccells, 280tesseract4-faces, 84penteract5-faces, and 14hexeract6-faces.

It can be named by itsSchläfli symbol {4,3⁵}, being composed of 36-cubes around each 5-face. It can be called ahepteract, aportmanteau oftesseract (the4-cube) andhepta for seven (dimensions) inGreek. It can also be called a regulartetradeca-7-tope ortetradecaexon, being a7 dimensional polytope constructed from 14 regularfacets.

The7-cube is 7th in a series ofhypercube:

Petrie polygonorthographic projections

Line segment	Square	Cube	4-cube	5-cube	6-cube	7-cube	8-cube	9-cube	10-cube

Thedual of a 7-cube is called a7-orthoplex, and is a part of the infinite family ofcross-polytopes.

Applying analternation operation, deleting alternating vertices of the hepteract, creates anotheruniform polytope, called ademihepteract, (part of an infinite family calleddemihypercubes), which has 14demihexeractic and 646-simplex 6-faces.

Thisconfiguration 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.^[1][2]

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 < x_i < 1.

This hypercube graph is anorthogonal 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 inPascal's triangle, being 1:7:21:35:35:21:7:1.

orthographic projections

Coxeter plane	B7 / A6	B6 / D7	B5 / D6 / A4
Graph
Dihedral symmetry	[14]	[12]	[10]
Coxeter plane	B4 / D5	B3 / D4 / A2	B2 / D3
Graph
Dihedral symmetry	[8]	[6]	[4]
Coxeter plane	A5	A3
Graph
Dihedral symmetry	[6]	[4]

• 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[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 JohnsonUniform Polytopes, Manuscript (1991)
  • N.W. Johnson:The Theory of Uniform Polytopes and Honeycombs, Ph.D. (1966)
• Klitzing, Richard."7D uniform polytopes (polyexa) o3o3o3o3o3o4x - hept".

• Weisstein, Eric W."Hypercube".MathWorld.
• Weisstein, Eric W."Hypercube graph".MathWorld.
• Olshevsky, George."Measure polytope".Glossary for Hyperspace. Archived fromthe original on 4 February 2007.
• Multi-dimensional Glossary: hypercube Garrett Jones
• Rotation of 7D-Cube www.4d-screen.de

• 7-polytopes

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