Möbius and Friends (Scrolls V)

I am sure we all have cut a Möbius strip in half, and been irritated by not getting two pieces but instead a doubly twisted strip.

Moebiuscut

The quickest way to parametrize a Möbius strip is as a ruled surface, letting line segments rotate by 180 degrees while moving them around the same circle where we usually cut. This raises the tantalizing question whether we could possible make a book whose pages are Möbius strips. For the moment I don’t know, I haven’t been able to find many explicit bendings of the ruled Möbius strip, except for its Doppelgänger:

MobiusDual0

This, however, is not a closed band but instead continues on periodically. There is a version of the Möbius strip as a minimal surface that also has a circle as a core curve.

Mobius12

As with ruled surfaces, you can twist these minimal surfaces more or less often around the circle. Here for instance is the triply twisted version.

Mobius32

All these minimal surfaces can be bent in their associate family. They stay minimal, and, surprisingly, also closed bands (after two turns), except for the case of a doubly twisted band. The conjugate of the triply twisted band looks like this.

Mobius32conj

Of course three is never enough, so here is the 40 fold twisted version. Amazingly enough, all these surfaces have explicit formulas.

Mobius20b

This would be another possibility for a book project: One long twisted sheet of paper, bent into disk like pages…

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