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.
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 (OEIS A000079),
the number of edges is (OEIS
A001787), the number of squares is
(OEIS A001788), the number of cubes is (OEIS A001789),
etc.
The numbers of distinct nets for the -hypercube
for , 2, ... are 1, 11, 261, ... (OEIS A091159;
Turney 1984-85).
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 .