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From: msb@lsuc.UUCP (Mark Brader)
Newsgroups: net.math
Subject: Re: Euler's(?) formula
Message-ID: <673@lsuc.UUCP>
Date: Wed, 19-Jun-85 12:30:58 EDT
Article-I.D.: lsuc.673
Posted: Wed Jun 19 12:30:58 1985
Date-Received: Wed, 19-Jun-85 12:40:46 EDT
References: <1832@ut-ngp.UTEXAS> <3138@garfield.UUCP>
Reply-To: msb@lsuc.UUCP (Mark Brader|LSUC|Toronto)
Distribution: net
Organization: Law Society of Upper Canada, Toronto
Lines: 12
Summary: Not just convex


The formula applies to concave as well as convex polyhedra, as long as
the faces don't intersect (I'm not sure what happens if they do); after all,
they are topologically equivalent.  What it does not apply to is non-simple
polyhedra, that is, those with "tunnels" in them.  Actually I recall that
there's a simple extension of the formula that even includes those...
I think F + V = E + 4T + 2 where T is the number of tunnels.  The faces
themselves must be polygons, rather than having holes in them.
(At least, the above works for one polyhedron I drew, and I remember it being
something like that.  It was in Martin Gardner's column once upon a time.)

Mark Brader