Obs:

1
A generalized Möbius strip (or Möbius torus)
Florian Nichitiu
2000
Obs Consider a torus with a cross-section a
regular polygon with Nf faces Consider a cut.
Now, one end can be rotated (relative to
another) in such a way that path nr1 of one end
will be in border with path nr NfNs1 (Shift
number is Ns). (For Nf2 (with Ns only
1) it is normal Möbius strip) Beginning anywhere,
a walk around center of the torus on different
paths will be Nt Nf / Ns times. If
Nt is not an integer, the walk around center
is Nf times (maxim). Of course, if Nf is a
prime number, for any shift number Ns (any nr of
rotation of one end relative to another of the
cut torus), the continuous walk is done on all
paths, so will be Nf times walk around the
center of the tour.
 
2
Multi face torus with n different paths
2
3

Example rotation shift 5 paths forward (or 2
paths backward) for a tour with 7 paths (or faces)
in
1
3


2
out
1
1 2 3 4 5 6 7
3 4 5 6 7 1 2
Cut and Rotation of one end
1 2 3 4 n-1 n
Rotation shift 3 paths forward (or n-3 paths
backward) for a torus with n paths (or faces)
n-1 n 1 2
3
Example
Np (nr of paths) 9 Ns (nr of shifts) 3
1 2 3 4 5 6 7 8 9
7 8 9 1 2 3 4 5 6
Here the path nr 1 is in border with path nr 7,
so walking over that border and continue on path
nr 7 you arrive to the new border which will be
with path nr 4. Continuing on path nr 4 the next
border will be with path nr 1, and the trip is
finished. The total nr of rotations is
therefore 3 (we walked over 3 paths, over path
nr 1, 7 and 4). This is of course independent of
the path number with which the trip begin.