Like the shelling proof, this proof interprets a polyhedron or polytope as a complex of polyhedral faces of varying dimensions, with fi denoting the number of faces of dimension i. And like that proof, we add two "extra" cells, one for the whole polytope and one for the "empty face", so f-1=fd=1.
We interpret the subsets of faces of dimension i as a vector space over the two-element field GF(2), with vector addition being performed by symmetric difference of subsets (also known as exclusive or). In this way we get d+1 vector spaces Vi, with dimensions equal to fi. The sets of faces of the polytope can be interpreted as forming bases for these vector spaces.
These spaces also come equipped with a linear mapping b taking each space Vi to the next lower dimensional space, defined on each face to be the set of its facets. When v is a sum of faces, b(v) is the sum of the corresponding sets of facets. (The empty face has no facets, and a vertex is defined to have the empty face as its only facet.) This mapping has two very important properties:
With these preliminaries out of the way we are ready to begin the proof that Sum(-1i fi)=0.
Let Bi denote the subspace of Vi consisting of those vectors for which b(v)=0. Then the map b takes Vi to Bi-1. Whenever we have a linear map of vector spaces, we can use it to decompose the original space into its nullspace (the set of vectors mapped to zero, here Bi) and its orthogonal complement, which is mapped one-to-one onto the image of the map. This tells us that dim(Vi)=dim(Bi)+dim(Bi-1). Plugging this expansion into the sum we are trying to evaluate givesCancelling the two extra parameters f-1 and f3 from the sum gives the usual Euler formula. I suspect there is a more direct, dimensionless way of proving that boundaryless vectors are themselves boundaries, but I don't see it and Lakatos doesn't describe it.Sum(-1i fi) = Sum(-1i (dim(Bi) + dim(Bi-1)))which equals zero, because each term Bi appears twice with opposite signs.
Homology theory, one of the foundation stones of algebraic topology, is devoted to defining similar vector spaces and understanding the extent to which the key equation dim(Vi)=dim(Bi)+dim(Bi-1) fails to be true in other topological spaces.
Proofs of Euler's Formula.
From the Geometry Junkyard, computational and recreational geometry pointers.
David Eppstein, Theory Group, ICS, UC Irvine.
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