Let P be a polytope of dimension d. The f-vector of P is the finite integer sequence , where the component in position i is the number of i-dimensional faces of P. For some purposes it is convenient to view the empty face and the polytope itself as improper faces, so .
For example, a cube has 8 vertices, 12 edges, and 6 faces, so its f-vector is (8, 12, 6).
The entries in the f-vector of a convex polytope satisfy the Euler–Poincaré–Schläfli formula:
Consequently, the face lattice of a polytope is Eulerian. For any graded poset with maximum and minimum elements there is an extension of the f-vector called the flag f-vector. For any subset S of , the entry of the flag f-vector of P is the number of chains of faces in with dimensions coming only from S.
The flag f-vector of a three-dimensional cube is given in the following table. For simplicity we drop braces and commas.
For example, because each of the 12 edges meets exactly two faces.
Although the flag f-vector of a d-polytope has entries, most of them are redundant, as they satisfy a collection of identities generalizing the Euler–Poincaré–Schläfli formula and called the generalized Dehn-Sommerville relations. Interestingly, the number of nonredundant entries in the flag -vector of a d-polytope is one less than the Fibonacci number .
- 1 Bayer, M. and L. Billera, Generalized Dehn-Sommerville relations for polytopes, spheres and Eulerian partially ordered sets, Invent. Math. 79 (1985), no. 1, 143–157.
- 2 Bayer, M. and A. Klapper, A new index for polytopes, Discrete Comput. Geom. 6 (1991), no. 1, 33–47.
- 3 Ziegler, G., Lectures on polytopes, Springer-Verlag, 1997.
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|Last modified on||2013-03-22 16:59:10|
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