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From: ptb@ukc.UUCP (P.T.Breuer)
Newsgroups: net.math
Subject: Re: Euler's(?) formula
Message-ID: <5259@ukc.UUCP>
Date: Tue, 18-Jun-85 23:18:49 EDT
Article-I.D.: ukc.5259
Posted: Tue Jun 18 23:18:49 1985
Date-Received: Fri, 21-Jun-85 01:56:23 EDT
References: <1832@ut-ngp.UTEXAS>
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Organization: Computing Laboratory, U of Kent at Canterbury, UK
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In article <1832@ut-ngp.UTEXAS> graner@ut-ngp.UTEXAS (Nicolas Graner) writes:
>I think it was Euler who showed that a polyhedron with F faces,
>V vertices and E edges satisfies the relation: F + V = E + 2.
>
>I have seen a very technical proof, but the result is so simple and
>beautiful that there should be a simple and beautiful proof (i.e.
>accessible to non mathematicians). Does anyone know of such a proof?
>Also, to what kind of polyhedrons does it apply (convex, connected...) ?
>
>Nic.                      {ihnp4,seismo,allegra,...}!ut-ngp!graner
>
>*If Murphy's law can go wrong, it won't*


I assume the technical proof you've seen is Poincare's, which translates the
problem into what's now called homology theory. (The basic idea being to define
vector spaces generated by the faces, edges and vertices, and consider the 
kernels and images of the boundary maps between these spaces.) I don't know of
a simpler, general proof. There are several proofs which look very plausible 
and work for (usually unspecified) special cases.
You can find many of these in a beautiful book "Proofs and Refutations" by
Imre Lakatos. The book uses the history of this problem to illustrate 
Lakatos's ideas on the philosophy and history of mathematics. Thoroughly
recommended to mathematicians and interested bystanders alike.