the Dehn-Sommerville relations
/ DAYN SOM-er-vil /
When you count the faces of a nicely-behaved polytope — one whose facets are all triangles, triangles-of-triangles, and so on (a simplicial polytope) — the counts cannot be anything you like. They obey a tight web of linear equations that force the high-dimensional face counts to mirror the low-dimensional ones. The Dehn-Sommerville relations are exactly that web: a set of symmetry equations among the entries of a simplicial polytope's f-vector, generalizing the single Euler relation to a whole family.
They read cleanest in the h-vector, a linear repackaging of the f-vector defined by sum over i of f_{i-1} * (x - 1)^{n-i} = sum over i of h_i * x^{n-i} (with f_{-1} = 1 for the empty face). The Dehn-Sommerville relations then say simply h_i = h_{n - i} for all i: the h-vector of a simplicial n-polytope is palindromic, a mirror symmetry. Translated back to face counts this gives roughly n/2 independent linear relations among the f_k, so for a simplicial polytope only about half the f-vector is free — the rest is determined. Euler's relation V - E + F = 2 is the single relation you get in dimension 3 (where it reduces to h_0 = h_3 and h_1 = h_2).
These relations are the structural backbone of polytope combinatorics. They are what make the upper-bound theorem provable (Stanley's proof uses the h-vector and its algebraic meaning as Hilbert-function data of a graded ring), and the palindromic h-vector is the shadow of Poincare duality for the underlying sphere a simplicial polytope's boundary triangulates. An honest caveat: Dehn-Sommerville holds for SIMPLICIAL polytopes (and more generally Eulerian simplicial complexes); a general polytope whose facets are not simplices satisfies only the single Euler relation, not the full palindromic symmetry. To use the relations on an arbitrary polytope you must first triangulate or dualize.
Take any simplicial 3-polytope (all facets triangles), say the octahedron with f-vector (6, 12, 8). Its h-vector comes out (1, 3, 3, 1) — palindromic, as Dehn-Sommerville requires: h_0 = h_3 = 1 and h_1 = h_2 = 3. The palindrome packs both Euler's relation and the triangle-count constraint into one symmetry, and it predicts that f_1 is forced once you know f_0: here 3*f_0 - 6 = 3*6 - 6 = 12 edges.
The octahedron's h-vector (1,3,3,1) is palindromic — the Dehn-Sommerville mirror.
The relations are a property of SIMPLICIAL polytopes (or their duals, simple polytopes). A cube is simple but not simplicial, so apply the relations to its dual the octahedron, or to a simplicial polytope; using them on a non-simplicial polytope directly gives wrong constraints.