Euler's formula for polyhedra
- From: "George Szpiro" <george@xxxxxxxxxxxxxxxx>
- Date: Tue, 23 Aug 2005 20:45:26 +0300
Hello everybody:
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.
Now I bore two tunnels through the cube, such that they enter and exit at
two opposite sides. V=24, e=36, f=10. Euler's formula gives -2, because the
genus of a cube with two holes is 2.
NOW MY QUESTION:
If I bore two PARALLEL tunnels through the cube, I get v=24, e=36 and f=12
and Euler's formula gives 0. But the genus of a cube with two holes should
be 2?
Where did I go wrong?
Would you kindly send any answer to my private email address:
george@xxxxxxxxxxxxxxxx
Thank you,
George
--
--------
George Szpiro
Neue Zürcher Zeitung (Switzerland)
POB 6278
Jerusalem 91060
Israel
.
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