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# Tesseract

The tesseract is the hypercube in , also called the 8-cell or octachoron. It has the Schläfli symbol , and vertices . The figure above shows a projection of the tesseract in three-space (Gardner 1977). The tesseract is composed of 8 cubes with 3 to an edge, and therefore has 16 vertices, 32 edges, 24 squares, and 8 cubes. It is one of the six regular polychora.

The tesseract has 261 distinct nets (Gardner 1966, Turney 1984-85, Tougne 1986, Buekenhout and Parker 1998).

In Madeleine L'Engle's novel A Wrinkle in Time, the characters in the story travel through time and space using tesseracts. The book actually uses the idea of a tesseract to represent a fifth dimension rather than a four-dimensional object (and also uses the word "tesser" to refer to movement from one three dimensional space/world to another).

In the science fiction novel Factoring Humanity by Robert J. Sawyer, a tesseract is used by humans on Earth to enter the fourth dimension and contact another civilization on a planet orbiting the star Alpha Centauri A. The hypercube initially exists as a series of connected 3-dimensional cubes, which represent a hypercube that has been unfolded. Refolding the cube in a certain specific manner causes the reformation of the hypercube in 4 dimensions.

In John Mighton's play, Half Life, one of the characters (an aging mathematician) builds a tesseract (or rather, the projection of a tesseract) out of popsicle sticks. In the Season 1 episode "Rampage" of the television crime drama NUMB3RS, main character mathematician Charlie Eppes discovers a popsicle-stick tesseract (projection) he built as a boy.

Cube, Hypercube, Magic Tesseract, Polytope, Simplex, Tesseract Graph

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## References

Buekenhout, F. and Parker, M. "The Number of Nets of the Regular Convex Polytopes in Dimension ." Disc. Math. 186, 69-94, 1998.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, Apr. 1986.Fischer, G. (Ed.). Plate 4 in Mathematische Modelle aus den Sammlungen von Universitäten und Museen, Bildband. Braunschweig, Germany: Vieweg, p. 5, 1986.Gardner, M. "Mathematical Games: Is It Possible to Visualize a Four-Dimensional Figure?." Sci. Amer. 214, 138-143, Nov. 1966.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/.L'Engle, M. A Wrinkle in Time. Yearling, 1973.Sawyer, R. Factoring Humanity. New York: Orb Books, 2004.Smith, H. J. "The Tesseract: A Look into 4-Dimensional Space." http://www.geocities.com/hjsmithh/WireFrame4/tesseract.html.Tougne, P. "Combien de patrons un polyèdre peut-il avoir?" Pour la Science, pp. 99-103, May 1986.Turney, P. D. "Unfolding the Tesseract." J. Recr. Math. 17, No. 1, 1-16, 1984-85.Weimholt, A. "Tesseract Foldout." http://www.weimholt.com/andrew/tesseract.html.

Tesseract

## Cite this as:

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