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Hypercube

The hypercube is a generalization of a 3-cube to n dimensions, also called an n-cube or measure polytope. It is a regular polytope with mutually perpendicular sides, and is therefore an orthotope. It is denoted gamma_n and has Schläfli symbol {4,3,3_()_(n-2)}.

The following table summarizes the names of n-dimensional hypercubes.

The number of k-cubes contained in an n-cube can be found from the coefficients of (2k+1)^n, namely (n; k)2^(n-k), where (n; k) is a binomial coefficient. The number of nodes in the n-hypercube is therefore 2^n (OEIS A000079), the number of edges is 2^(n-1)n (OEIS A001787), the number of squares is 2^(n-3)n(n-1) (OEIS A001788), the number of cubes is 2^(n-4)n(n-1)(n-2)/3 (OEIS A001789), etc.

The numbers of distinct nets for the n-hypercube for n=1, 2, ... are 1, 11, 261, ... (OEIS A091159; Turney 1984-85).

TesseractProjection

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 n-cube that maximizes the products of distances to its vertices (Trott 2004, p. 104). The following table summarizes results for small n.

See also

Cross Polytope, Cube, Cube-Connected Cycle Graph, Glome,Hamiltonian Graph, Hypercube Graph, Hypercube Line Picking, Hypersphere,Orthotope, Parallelepiped,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.umn.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. http://www.mathematicaguidebooks.org/.Trott, M. "The Mathematica Guidebooks Additional Material: Hypercube Projections."http://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: 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

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

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