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# Pentatope

The pentatope is the simplest regular figure in four dimensions, representing the four-dimensional analog of the solid tetrahedron. It is also called the 5-cell, since it consists of five vertices, or pentachoron. The pentatope is the four-dimensional simplex, and can be viewed as a regular tetrahedron in which a point along the fourth dimension through the center of is chosen so that . The pentatope has Schläfli symbol .

It is one of the six regular polychora.

The skeleton of the pentatope is isomorphic to the complete graph , known as the pentatope graph.

The pentatope is self-dual, has five three-dimensional facets (each the shape of a tetrahedron), 10 ridges (faces), 10 edges, and five vertices. In the above figure, the pentatope is shown projected onto one of the four mutually perpendicular three-spaces within the four-space obtained by dropping one of the four vertex components (R. Towle).

16-Cell, 24-Cell, 120-Cell, 600-Cell, Hypercube, Pentatope Graph, Polytope, Simplex, Tetrahedron

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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.Cvetković, D. M.; Doob, M.; and Sachs, H. Spectra of Graphs: Theory and Applications, 3rd rev. enl. ed. New York: Wiley, 1998.Gardner, M. The Sixth Book of Mathematical Games from Scientific American. Chicago, IL: University of Chicago Press, pp. 187-188, 1984.Nikolić, S.; Trinajstić, N.; and Mihalić, A. "The Detour Matrix and the Detour Index." Ch. 6 in Topological Indices and Related Descriptors in QSAR and QSPR (Ed. J. Devillers A. T. and Balaban). Amsterdam, Netherlands: Gordon and Breach, pp. 279-306, 2000.Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 179-180 and 210, 1991.

## Cite this as:

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