Re: Euler's formula for polyhedra
- From: Gerry Myerson <gerry@xxxxxxxxxxxxxxxxxxxxxxxxx>
- Date: Wed, 24 Aug 2005 10:10:44 +1000
In article <defnam$44k$1@xxxxxxxxxxxxxxxxxxxxxx>,
"George Szpiro" <george@xxxxxxxxxxxxxxxx> wrote:
> A cube has 8 vertices, 12 edges and 6 faces, so the Euler formula v-e+f
> =2 -2g holds, because the genus of the cube is 0.
>
> Now I bore a square tunnel through the cube. I get v=16, e=24 and f=8 if I
> do not count the sides of the cube where the tunnel exits as faces. (FIRST
> QUESTION: Is this correct?) Euler's formula again holds: v-e+f=0, because
> the genus of a cube with one hole is one.
The Euler formula applies to polyhedra. Polyhedra have faces that
are simply connected. When you bore a hole through the cube, the faces
through which you bore the holes are no longer simply connected - they
are annular. You can't just ignore them when you count faces - what you
have is no longer a polyhedron, and if it looks to you as if Euler
works, it's only a coincidence that your ignoring the non-faces makes
the numbers add up right.
> Would you kindly send any answer to my private email address:
> george@xxxxxxxxxxxxxxxx
No. If I do that, no one else will be able to criticize my answer,
and I'll never learn anything.
--
Gerry Myerson (gerry@xxxxxxxxxxxxxxx) (i -> u for email)
.
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