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The hypercube, also called an -cube or measure polytope, is a generalization of a 3-cube to dimensions, where is a positive integer. It is a regular polytope with mutually perpendicular sides, and is therefore an orthotope. It is denoted and for has Schläfli symbol , with entries equal to 3. The square has symbol .

The following table summarizes the names of -dimensional hypercubes.

The number of -cubes contained in an -cube can be found as the coefficient of in , namely , where is a binomial coefficient. The number of polytope vertices in the -hypercube is therefore (OEIS A000079), the number of polytope 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 , 3, ... are 1, 11, 261, ... (OEIS 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 the unit -cube that maximizes the product of distances to its polytope vertices (Trott 2004, p. 104). The following table summarizes results for small .

See also

Cross Polytope, Cube, Cube-Connected Cycle Graph, Glome, Hamiltonian Graph, Hypercube Graph, Hypercube Line Picking, Hypersphere, Orthotope, Parallelepiped, Polytope, Regular Polytope, Simplex, Tesseract Explore this topic in the MathWorld classroom

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References

Born, M. Problems of Atomic Dynamics. Cambridge, MA: MIT Press, 1926.Coxeter, H. S. M. Regular Polytopes, 3rd ed. New York: Dover, p. 123, 1973.Dewdney, A. K. "Computer Recreations: A Program for Rotating Hypercubes Induces Four-Dimensional Dementia." Sci. Amer. 254, 14-23, Mar. 1986.Fischer, G. (Ed.). Plates 3-4 in Mathematische Modelle aus den Sammlungen von Universitäten und Museen, Bildband. Braunschweig, Germany: Vieweg, pp. 4-5, 1986.Gardner, M. "Hypercubes." Ch. 4 in Mathematical Carnival: A New Round-Up of Tantalizers and Puzzles from Scientific American. New York: Vintage Books, pp. 41-54, 1977.Geometry Center. "The Tesseract (or Hypercube)." http://www.geom.uiuc.edu/docs/outreach/4-cube/.Pappas, T. "How Many Dimensions Are There?" The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, pp. 204-205, 1989.Skiena, S. "Hypercubes." §4.2.5 in Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, pp. 148-150, 1990.Sloane, N. J. A. Sequences A000079/M1129, A001787/M3444, A001788/M4161, A001789/M4522, and A091159 in "The On-Line Encyclopedia of Integer Sequences."Trott, M. The Mathematica GuideBook for Programming. New York: Springer-Verlag, 2004. https://www.mathematicaguidebooks.org/.Trott, M. "The Mathematica Guidebooks Additional Material: Hypercube Projections." https://www.mathematicaguidebooks.org/additions.shtml#N_1_04.Turney, P. D. "Unfolding the Tesseract." J. Recr. Math. 17, No. 1, 1-16, 1984-85.Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London, England: Penguin, pp. 113-114 and 210, 1991.Wilker, J. B. "An Extremum Problem for Hypercubes." J. Geom. 55, 174-181, 1996.Williams, R. The Geometrical Foundation of Natural Structure: A Source Book of Design. New York: Dover, 1979.

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Hypercube

Cite this as:

Weisstein, Eric W. "Hypercube." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Hypercube.html

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