Skeleton

In algebraic topology, a -skeleton is a simplicial subcomplex of  that is the collection of all simplices of  of dimension at most , denoted .

The graph obtained by replacing the faces of a polyhedron with its edges and vertices is therefore the skeleton of the polyhedron. The polyhedral graphs corresponding to the skeletons of Platonic solids are illustrated above. The number of topologically distinct skeletons  with  graph vertices for , 5, 6, ... are 1, 2, 7, 18, 52, ... (OEIS A006869).

REFERENCES:

Gardner, M. Martin Gardner's New Mathematical Diversions from Scientific American. New York: Simon and Schuster, p. 233, 1966.

Hatcher, A. Algebraic Topology. Cambridge, England: Cambridge University Press, 2002.

Munkres, J. R. Elements of Algebraic Topology. New York: Perseus Books Pub., 1993.

Sloane, N. J. A. Sequence A006869/M1748 in "The On-Line Encyclopedia of Integer Sequences."