Hypercube

n-hypercube
Rank	n
Type	Regular
Space	Spherical
Coxeter diagram	x4o3o3...3o (...)
Schläfli symbol	{4,3,3, ..., ,3}
Elements
Vertices	$2^n$
Vertex figure	(n − 1)-simplex, edge length √2
Measures (edge length 1)
Circumradius	$\frac{\sqrt{n}}{2}$
Inradius	$\frac12 = 0.5$
Volume	1
Height	1
Central density	1
Number of pieces	$2n$
Level of complexity	1
Related polytopes
Dual	n-orthoplex
Conjugate	None
Abstract properties
Flag count	$n!\times 2^n$
Euler characteristic	0 if n even 2 if n odd
Topological properties
Orientable	Yes
Properties
Symmetry	BCn, order $n!\times 2^n$
Convex	Yes
Nature	Tame

For the specific polytope sometimes called the hypercube, see tesseract.

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.

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

Elements

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 2^n-dC(n, n). This is because for each choice of n-d of the hypercube’s n directions, and for each of the subsequent 2^n–d 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

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

Hypercubes by dimension

Rank	Name	Picture		Rank	Name	Picture
1	Dyad			6	Hexeract
2	Square			7	Hepteract
3	Cube			8	Octeract
4	Tesseract			9	Enneract
5	Penteract			10	Dekeract

Vertex coordinates

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

• The circumradius of an n-dimensional hypercube of unit edge length is given by $\frac{\sqrt{n}}{2}$.
• Its inradius is $\frac12$, regardless of n.
• Its height from a facet to the opposite facet is twice the inradius, that is $1$.
• Its hypervolume is $1$, regardless of n.
• The angle between two facet hyperplanes is $\frac{\pi}2 = 90°$, regardless of n.

External links

• Klitzing, Richard. The Prism Product.
• Wikipedia Contributors. "Hypercube".
• Hi.gher.Space Wiki Contributors. "Hypercube".