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From: jas@duke.UUCP (Jon A. Sjogren)
Newsgroups: net.math
Subject: Re: Euler's(?) formula
Message-ID: <5940@duke.UUCP>
Date: Sun, 16-Jun-85 22:21:15 EDT
Article-I.D.: duke.5940
Posted: Sun Jun 16 22:21:15 1985
Date-Received: Tue, 18-Jun-85 04:23:41 EDT
References: <1832@ut-ngp.UTEXAS>
Organization: Duke University
Lines: 27

It certainly applies to convex polyhedra in space.  
What others? Maybe the proof gives a clue. Take polyhedron
P, and repeat the following steps.

A) If there is a vertex that is an end-point of exactly two edges,
delete the vertex and consider that those two edges are now 
just *one* edge.  This operation will not change the 
value of the expression F-E+V.
Repeat A until there are no more such vertices.


B) A "face" is now defined to be

i) an ordinary face, or
ii) something abstract which replaces another face in this part.

Find a vertex at which 3 or more edges meet. Choose one of the edges
G.  Two distinct faces meet in G. Delete G and put the two faces 
together into a single "face".  (A face is essentially a particular
collection of edges that have no free vertex.)
If this step is not possible, then it is because there is a "cycle"
of edges on the polyhedron that has no distinct interior and exterior
on the polyhedron. ... such as could occur on a torus-like polyhedron).

Repeat A.
Repeat the above until there is only one vertex and one face, when
you have the formula, since you have not changed F-E+V.