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Coxeter diagramx4o3o3...3o (CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.png...CDel 3.pngCDel node.png)
Schläfli symbol{4,3,3, ..., ,3}
Vertex figure(n − 1)-simplex, edge length 2
Measures (edge length 1)
Central density1
Number of pieces
Level of complexity1
Related polytopes
Abstract properties
Flag count
Euler characteristic0 if n even
2 if n odd
Topological properties
SymmetryBCn, order
A cube, with a square, a dyad, and a point colored differently than the others. These are all examples of hypercubes.

A hypercube is the simplest center-symmetric polytope in each respective dimension, by facet count. Hypercubes are a direct generalization of squares and cubes to higher dimensions. The n-dimensional hypercube, or simply the n-cube, has 2n vertices, such that for every of n directions, half the vertices lie on one side, and half lie on the other. Its facets are the 2'n hypercubes defined by the vertices on each side in each direction. Alternatively, one can construct each hypercube as the prism of the hypercube of the lower dimension. The prism product of an m-hypercube and an n-hypercube is an (m+n)-hypercube.

Every hypercube can be made regular. As such, the hypercubes comprise one of the three infinite families of polytopes that exist in every dimension, the other two being the simplexes and the orthoplexes (the duals of the hypercubes).

The hypercube is also called the measure polytope. This is because a hypercube with unit edge length has a unit hypervolume, and as such, it can be used to “measure” n-dimensional space like a grid.[1]

Hypercubes can always tile their respective spaces, forming the hypercubic honeycombs.

Elements[edit | edit source]

All of the elements of a hypercube are hypercubes themselves. The number of d-dimensional elements of an n-hypercube is given by the binomial coefficient 2n-dC(n, n). This is because for each choice of n-d of the hypercube’s n directions, and for each of the subsequent 2nd choices of sides, the vertices on these sides define a unique d-dimensional simplex. In particular, an n-dimensional hypercube has 2n vertices and 2n facets, and its vertex figure is the simplex of the previous dimension.

In total, an n-hypercube has 3n elements, including the nullitope and excluding the bulk of the polytope.

Examples[edit | edit source]

Excluding the point, the hypercubes up to 10D are the following:

Hypercubes by dimension
Rank Name Picture Rank Name Picture
1 Dyad
Line segment.svg
6 Hexeract
6-cube graph.svg
2 Square
7 Hepteract
7-cube graph.svg
3 Cube
3-cube graph.png
8 Octeract
4 Tesseract
4-cube graph.svg
9 Enneract
5 Penteract
5-cube graph.svg
10 Dekeract

Vertex coordinates[edit | edit source]

Coordinates for the vertices of an n-hypercube with edge length 1 are given by:

  • (±1/2, ±1/2, ..., ±1/2),

with n entries.

Measures[edit | edit source]

  • The circumradius of an n-dimensional hypercube of unit edge length is given by .
  • Its inradius is , regardless of n.
  • Its height from a facet to the opposite facet is twice the inradius, that is .
  • Its hypervolume is , regardless of n.
  • The angle between two facet hyperplanes is , regardless of n.

External links[edit | edit source]

References[edit | edit source]

  1. Coxeter, Harold Scott MacDonald (1948). Regular polytopes (1 ed.). p. 123.