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?

  • 1
    $\begingroup$ "how may continuous tracks are produced in each case" doesn't mean anything. $\endgroup$
    – Dan Asimov
    Nov 20 at 20:57
  • $\begingroup$ 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. $\endgroup$
    – Al.G.
    Nov 21 at 0:04
  • $\begingroup$ 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. $\endgroup$
    – Narasimham
    Nov 21 at 10:16

1 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

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

enter image description here

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.


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