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).
See also
Polyhedral Graph, Schlegel Graph
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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."
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Cite this as:
Weisstein, Eric W. "Skeleton." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Skeleton.html