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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