A generalized Mbius strip or Mbius torus Florian

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 nr 1 of one end will be in border with path nr Nf=Ns+1 (Shift number is Ns). (For Nf=2 (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.

Multi ‘face’ torus with n different ‘paths’ 2 1 3 … … 3 in out Example: rotation : shift 5 paths forward (or 2 paths backward) for a tour with 7 paths (or faces) 2 1 Cut and Rotation of one end 1 2 3 4 … n-1 n 1 2 … … 1 2 3 4 5 6 7 1 2 Rotation : shift 3 paths forward (or n-3 paths backward) for a torus with n paths (or faces)

Example: 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 Np (nr of paths) =9 Ns (nr of shifts) =3 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.