Hypercube
convex polytope, the n-dimensional analogue of a square and a cube
In geometry, a hypercube is an n-dimensional analogue of a square (n = 2) and a cube (n = 3); the special case for n = 4 is known as a tesseract. It is a closed, compact, convex figure whose 1-skeleton consists of groups of opposite parallel line segments aligned in each of the space's dimensions, perpendicular to each other and of the same length. A unit hypercube's longest diagonal in n dimensions is equal to
n
{\displaystyle {\sqrt {n}}}
.
An n-dimensional hypercube is more commonly referred to as an n-cube or sometimes as an n-dimensional cube. The term measure polytope (originally from Elte, 1912) is also used, notably in the work of H. S. M. Coxeter who also labels the hypercubes the γn polytopes.
The hypercube is the special case of a hyperrectangle (also called an n-orthotope).
A unit hypercube is a hypercube whose side has length one unit. Often, the hypercube whose corners (or vertices) are the 2n points in Rn with each coordinate equal to 0 or 1 is called the unit hypercube.
Construction
By the number of dimensions
A hypercube can be defined by increasing the numbers of dimensions of a shape:
0 – A point is a hypercube of dimension zero.
1 – If one moves this point one unit length, it will sweep out a line segment, which is a unit hypercube of dimension one.
This brief starts where responsible research should: with the source description of “Hypercube” as convex polytope, the n-dimensional analogue of a square and a cube. Everything that follows is an evidence route, not borrowed authority.
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Vocabulary and entity names are the principal evidence signals here, because they determine the precision of every later search. The source revision retrieved here is dated Jul 3, 2026. The linked authority identifier is Q213723. None of the 0 selected statements returned an explicit reference. The first chronological checks are 1912.
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This entry incorporates text from “Hypercube” on English Wikipedia. Contributors are listed in the page history. Text is available under the Creative Commons Attribution-ShareAlike 4.0 License. Selected authority identifiers and statements are retrieved from Wikidata under CC0; their references and qualifiers remain part of the verification path.