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Hypercube
The hypercube is a generalization of a 3-cube to 
dimensions, also
called an 
-cube or measure polytope. It is a regular
polytope with mutually perpendicular
sides, and is therefore an orthotope. It is denoted
and has Schläfli
symbol 
.
The following table summarizes the names of 
-dimensional hypercubes.
 | object |
| 1 | line segment |
| 2 | square |
| 3 | cube |
| 4 | tesseract |
The number of 
-cubes contained in an 
-cube can be found
from the coefficients of 
, namely
, where 
is a binomial
coefficient. The number of nodes in the 
-hypercube is therefore
(Sloane's A000079),
the number of edges is 
(Sloane's A001787),
the number of squares is 
(Sloane's A001788),
the number of cubes is 
(Sloane's A001789),
etc.
The numbers of distinct nets for the 
-hypercube for 
, 2, ... are 1, 11, 261, ... (Sloane's A091159;
Turney 1984-85).
The above figure shows a projection of the tesseract in three-space. A tesseract has 16 polytope vertices, 32 polytope edges, 24 squares, and eight cubes.
The dual of the tesseract is known as the 16-cell. For all dimensions, the dual of the hypercube is the cross polytope (and vice versa).
An isometric projection of the 5-hypercube appears together with the great rhombic triacontahedron on the cover of Coxeter's well-known book on polytopes (Coxeter 1973).
Wilker (1996) considers the point in an 
-cube that maximizes
the products of distances to its vertices (Trott 2004, p. 104). The following
table summarizes results for small 
.
 |  | maximal point |
| 2 |  |  |
| 3 |  |  |
| 4 |  |  |
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hypercube
