david@cvl.UUCP (David Harwood) (06/17/85)
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>From: graner@ut-ngp.UTEXAS (Nicolas Graner)
Newsgroups: net.math
Subject: Euler's(?) formula
Message-ID: <1832@ut-ngp.UTEXAS>
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
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There is an intuitive constructive 'proof': Verify that the
formula is true for a trivial 'object'; supposing that any object may
be constructed from this with the addition of vertices, we observe that
if the added vertex splits an edge, then F + (V+1) = (E+1) + 2, else
if it splits a face, then (F+2) + (V+1) = (E+3).
(Of course, I am ignoring the holomogy of these 'objects'.)