Polarity in a magnetized Möbius strip When a flat iron or Alnico washer is magnetized one of the faces develops a north polarity and the other, south. The geometric shape here is simple.
However, when a standard Möbius strip (or one of given thickness, radii of curvature and torsion of  edges) is magnetized, which regions develop a north polarity and which regions south and according to which geometrical or other mass distribution criterion/law?
I am curious to know because such a Möbius strip (of rectangular section) has only one surface and only one edge.
Is magnetic polarity and strength distribution after magnetization influenced by changed geometry ( by homeomorphism ) ?
It may be easy to make a flattened thin Möbius strip looking like a recycling symbol to apply a magnetizing current. Thanks in advance for references.

 A: There is an electrical component called a Möbius resistor that takes advantage of this weird geometry.

The current flows in through the wire marked (+) and out through the wire marked (-). Because current flows in both directions around the ring, only a negligible magnetic field is generated. This can be important in high-power, high-frequency electrical circuits such as radar systems.
More mathematically, a Möbius strip is an example of a non-orientable surface. Magnetic field can be represented by arrows (vectors). So, you can place an arrow on a point of the surface pointing away from the surface (perpendicularly) to represent the magnetic field coming out of that surface. If you continue putting arrows along the surface, you will eventually loop back around and put an arrow on the surface facing the opposite way. There is no consistent way to assign a magnetic field to the surface of a Möbius strip, hence the advantageous properties--no stray magnetic fields, hence no inductance--in the electrical component above.
You can take a Möbius strip and magnetize it using an external magnetic field, but the resulting magnetic field will not keep a consistent angle with the surface. In fact, there has to be at least one part of the strip where the magnetic field is parallel to the surface.
A: For a 'uniformly' magnetized material, the longitudinal (down the length) component can be continuous around the loop, but both the transverse (across the width) and perpendicular (through the thickness) components must encounter a 180-degree change somewhere around the loop.  
If the magnetic anisotropy is high, the magnetization may make just a single sharp transition (domain wall).  The type of domain wall will be a Neel wall for transverse magnetization or a Bloch wall for perpendicular magnetization.
A: The answer is easily obtained by remembering what creates the permanent magnetic field. All subatomic particles are magnetic dipoles and in some materials - natural or man-made - subatomic particles are aligned by their magnetic dipoles in such a way that a stable macroscopic magnetic field is present.
The stability of permanent magnets is a relative thing. An external magnetic field can destroy or "remap" the magnetic field of a permanent magnet. And the same thing happens to the areas of a magnetised flat material when it is joined to a Möbius strip.  Initially, the electrons at the junction surface are reoriented by their magnetic dipoles, but since a Möbius strip has no beginning and no end, the reordering should occur over the entire strip.
Theoretically, therefore, the magnetic dipoles should all be oriented along the stripe. In practice, however, the outer electrons of the atoms are part of the magnetic interactions with the other electrons and the nucleus (and also intermolecular interactions) and it would be an interesting area of research - if not done so far - to see what happens in reality in different materials and at different temperatures.
A: 
When a flat iron or Alnico washer is magnetized one of the faces develops a north polarity and the other, south.

That's one way that a flat washer could be magnetized. But, it also could be magnetized in other directions (E.g., with one edge north and the opposite edge south. Or, it could be magnetized with several alternating pairs of north and south poles.
Magnetization doesn't know about "faces" or "surfaces" or "topology." Magnetization happens in the bulk of the material.  Imagine starting with a solid sphere of material, magnetizing the sphere, and then carving material away until a Möbius strip-shaped piece remains. The carving would not change the way that the material was magnetized.*

* assuming that the process doesn't generate too much heat!
