Twisted Möbius tubes

Question

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?

Answer 1

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.

$120^\circ$:

$240^\circ$:

$360^\circ$: