# Twisted Möbius tubes

May be well-known, asking nevertheless.. A long flexible hollow square tube is cut and rotated by a half turn $$180^{\circ},$$ then glued cut tube ends thus making a twisted Möbius square tube with two continuous neighboring tracks.

If instead we start with an equilateral triangle cross section tube, cut it and rotate by$$120^{\circ},240^{\circ},360^{\circ},480^{\circ}$$ before gluing up cut ends how may continuous tracks are produced in each case?

What is their parametrization when embedded in 3-space?

• "how may continuous tracks are produced in each case" doesn't mean anything. Nov 20 at 20:57
• Allow me to disagree - just like a strip of paper has 2 sides that "turn to one" when the ends are glued twisted, a (loong) parallelepiped has 4 which (it seems to me) turn to 1 when its ends are glued twisted by pi/2. I find this an interesting question. Nov 21 at 0:04
• Thank you very much. In fact, earlier (1997?) the classical way it is put to us that the starting band has two sides overlooking the thickness dimension had perplexed me. I made images similar in appearance N. Owad's images here and posted them on sci.math ; ( twisted square section resulting two continuous Möbius tracks). Practically also a square rod was heated red hot and twisted $180^{\circ}$ combining the section to palpably convince anyone that the neglected side edges are equally important as the main strip faces of the Möbius band... in topological ideation, imho. Nov 21 at 10:16

Below are pictures to make this more clear, but if we did no twisting, we would have each face match up with itself, so there would be 3 "tracks," if I am guessing to your meaning correctly. If we twist by $$120^\circ$$ or $$240^\circ$$, then the faces will match with one face over or two faces over - whichever way you are rotating. In both of these instances, we will get a single track. See this by labeling the edges of your triangle as $$a, b,c$$. Then for $$120^\circ$$, we have

• $$a\to b$$
• $$b\to c$$
• $$c\to a$$

and following this path from $$a$$, we get to each side and then back to $$a$$. Similarly, if we twist $$240^\circ$$, we have

• $$a\to c$$
• $$b\to a$$
• $$c\to b$$ Here again, following through we get to each face of the shape. This pattern continues: twisting $$0$$, $$360^\circ$$, or $$2 \times 360^\circ = 720^\circ, \ldots$$ there are three faces. Twisting any other amount will yield one track.

Below are images of twists for $$120^\circ$$, $$240^\circ$$, $$360^\circ$$. The connection is not complete so you can see the sides which will glue up together.

$$120^\circ$$:

$$240^\circ$$:

$$360^\circ$$: