- n-skeleton
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- This article is not about the topological skeleton concept of computer graphics
In mathematics, particularly in algebraic topology, the n-skeleton of a topological space X presented as a simplicial complex (resp. CW complex) refers to the subspace Xn that is the union of the simplices of X (resp. cells of X) of dimensions m ≤ n. In other words, given an inductive definition of a complex, the n-skeleton is obtained by stopping at the n-th step.
These subspaces increase with n. The 0-skeleton is a discrete space, and the 1-skeleton a topological graph. The skeletons of a space are used in obstruction theory, to construct spectral sequences by means of filtrations, and generally to make inductive arguments. They are particularly important when X has infinite dimension, in the sense that the Xn do not become constant as n → ∞.
In geometry
In geometry, a k-skeleton of n-polytope P (functionally represented as skelk(P)) consists of all i-polytope elements of dimension up to k.[1]
For example:
- skel0(cube) = 8 vertices
- skel1(cube) = 8 vertices, 12 edges
- skel2(cube) = 8 vertices, 12 edges, 6 square faces
References
- ^ Peter McMullen, Egon Schulte, Abstract Regular Polytopes, Cambridge University Press, 2002. ISBN 0-521-81496-0 (Page 29)
External links
- Weisstein, Eric W., "Skeleton" from MathWorld.
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