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Cube (3cube)  Tesseract (4cube) 

In geometry, a hypercube is an ndimensional analogue of a square (n = 2) and a cube (n = 3). It is a closed, compact, convex figure whose 1skeleton 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 ndimensions is equal to .
An ndimensional hypercube is also called an ncube. The term "measure polytope" is also used, notably in the work of H.S.M. Coxeter (originally from Elte, 1912^{[1]}), but it has now been superseded.
The hypercube is the special case of a hyperrectangle (also called an orthotope).
A unit hypercube is a hypercube whose side has length one unit. Often, the hypercube whose corners (or vertices) are the 2^{n} points in R^{n} with coordinates equal to 0 or 1 is called "the" unit hypercube.
This can be generalized to any number of dimensions. This process of sweeping out volumes can be formalized mathematically as a Minkowski sum: the ddimensional hypercube is the Minkowski sum of d mutually perpendicular unitlength line segments, and is therefore an example of a zonotope.
The 1skeleton of a hypercube is a hypercube graph.
A unit hypercube of n dimensions is the convex hull of the points given by all sign permutations of the Cartesian coordinates . It has an edge length of 1 and an ndimensional volume of 1.
An ndimensional hypercube is also often regarded as the convex hull of all sign permutations of the coordinates . This form is often chosen due to ease of writing out the coordinates. Its edge length is 2, and its ndimensional volume is 2^{n}.
The hypercubes are one of the few families of regular polytopes that are represented in any number of dimensions.
The hypercube (offset) family is one of three regular polytope families, labeled by Coxeter as γ_{n}. The other two are the hypercube dual family, the crosspolytopes, labeled as β_{n}, and the simplices, labeled as α_{n}. A fourth family, the infinite tessellations of hypercubes, he labeled as δ_{n}.
Another related family of semiregular and uniform polytopes is the demihypercubes, which are constructed from hypercubes with alternate vertices deleted and simplex facets added in the gaps, labeled as hγ_{n}.
Every ncube of n > 0 is composed of elements, or ncubes of a lower dimension, on the (n1)dimensional surface on the parent hypercube. A side is any element of (n1) dimension of the parent hypercube. A hypercube of dimension n has 2n sides (a 1dimensional line has 2 end points; a 2dimensional square has 4 sides or edges; a 3dimensional cube has 6 2dimensional faces; a 4dimensional tesseract has 8 cells). The number of vertices (points) of a hypercube is (a cube has vertices, for instance).
A simple formula to calculate the number of "n2"faces in an ndimensional hypercube is:
The number of mdimensional hypercubes (just referred to as mcube from here on) on the boundary of an ncube is
For example, the boundary of a 4cube (n=4) contains 8 cubes (3cubes), 24 squares (2cubes), 32 lines (1cubes) and 16 vertices (0cubes).
This identity can be proved by combinatorial arguments; each of the vertices defines a vertex in a dimensional boundary. There are ways of choosing which lines ("sides") that defines the subspace that the boundary is in. But, each side is counted times since it has that many vertices, we need to divide with this number. Hence the identity above.
These numbers can also be generated by the linear recurrence relation
For example, extending a square via its 4 vertices adds one extra line (edge) per vertex, and also adds the final second square, to form a cube, giving = 12 lines in total.
m  0  1  2  3  4  5  6  7  8  9  10  

n  γ_{n}  ncube  Names Schläfli symbol CoxeterDynkin  Vertices  Edges  Faces  Cells (3faces)  4faces  5faces  6faces  7faces  8faces  9faces  10faces 
0  γ_{0}  0cube  Point   1  
1  γ_{1}  1cube  Line segment {}  2  1  
2  γ_{2}  2cube  Square Tetragon {4}  4  4  1  
3  γ_{3}  3cube  Cube Hexahedron {4,3}  8  12  6  1  
4  γ_{4}  4cube  Tesseract Octachoron {4,3,3}  16  32  24  8  1  
5  γ_{5}  5cube  Penteract Decateron {4,3,3,3}  32  80  80  40  10  1  
6  γ_{6}  6cube  Hexeract Dodecapeton {4,3,3,3,3}  64  192  240  160  60  12  1  
7  γ_{7}  7cube  Hepteract Tetradeca7tope {4,3,3,3,3,3}  128  448  672  560  280  84  14  1  
8  γ_{8}  8cube  Octeract Hexadeca8tope {4,3,3,3,3,3,3}  256  1024  1792  1792  1120  448  112  16  1  
9  γ_{9}  9cube  Enneract Octadeca9tope {4,3,3,3,3,3,3,3}  512  2304  4608  5376  4032  2016  672  144  18  1  
10  γ_{10}  10cube  Dekeract icosa10tope {4,3,3,3,3,3,3,3,3}  1024  5120  11520  15360  13440  8064  3360  960  180  20  1 
An ncube can be projected inside a regular 2ngonal polygon by a skew orthogonal projection, shown here from the line segment to the 12cube.
The graph of the nhypercube's edges is isomorphic to the Hasse diagram of the (n1)simplex's face lattice. This can be seen by orienting the nhypercube so that two opposite vertices lie vertically, corresponding to the (n1)simplex itself and the null polytope, respectively. Each vertex connected to the top vertex then uniquely maps to one of the (n1)simplex's facets (n2 faces), and each vertex connected to those vertices maps to one of the simplex's n3 faces, and so forth, and the vertices connected to the bottom vertex map to the simplex's vertices.
This relation may be used to generate the face lattice of an (n1)simplex efficiently, since face lattice enumeration algorithms applicable to general polytopes are more computationally expensive.
