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Simple Polyhedron


A simple polyhedron, also called a simplicial polyhedron, is a polyhedron that is topologically equivalent to a sphere (i.e., if it were inflated, it would produce a sphere) and whose faces are simple polygons.

The number of simple polyhedra on n=1, 2, ... nodes are 0, 0, 1, 1, 1, 2, 5, 14, 50, 233, 1249, ... (OEIS A000109).

SimplePolyhedra

The skeletons of the simple polyhedra correspond to the triangulated graphs, the smallest of which are illustrated above.


See also

Cubic Polyhedral Graph, Planar Graph, Simple Polygon, Triangulated Graph

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References

Bokowski, J. and Schuchert, P. "Equifacetted 3-Spheres as Topes of Nonpolytopal Matroid Polytopes." Disc. Comput. Geom. 13, 347-361, 1995.Bowen, R. and Fisk, S. "Generation of Triangulations of the Sphere." Math. Comput. 21, 250-252, 1967.Dillencourt, M. B. "Polyhedra of Small Orders and Their Hamiltonian Properties." Tech. Rep. 92-91, Info. and Comput. Sci. Dept., Univ. Calif. Irvine, 1992.Federico, P. J. "Enumeration of Polyhedra: The Number of 9-Hedra." J. Combin. Th. 7, 155-161, 1969.Gardner, M. "Mathematical Games: On the Remarkable Császár Polyhedron and Its Applications in Problem Solving." Sci. Amer. 232, 102-107, May 1975.Grünbaum, B. Convex Polytopes. New York: Wiley, p. 424, 1967.Lederberg, J. "Hamilton Circuits of Convex Trivalent Polyhedra (up to 18 Vertices)." Amer. Math. Monthly 74, 522-527, 1967.Sloane, N. J. A. Sequence A000109/M1469 in "The On-Line Encyclopedia of Integer Sequences."

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Simple Polyhedron

Cite this as:

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

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