A couple of weeks ago I learnt about Umbilic torus thanks to a link in the “See also” section of the Möbius strip article in Wikipedia.

That link was followed by a brief explanation claiming that Umbilic thorus is

a three-dimensional shape with its boundary formed by a Möbius strip, glued to itself along its single edge

At the time I took it for granted but as I thought about it in the background an annoying feeling that something is wrong with that statement started to build up.

Today I finally came to the conclusion that the claim was false and edited it away with the following comment:

See also: Deleted a false statement - Umbilic torus cannot be formed from a Möbius strip by gluing along its single edge

Half an hour later, my edit was reverted with the following feedback:

Incorrect. It is a Möbius strip. It has three half-twists rather than the usual single half-twist but that does not affect its topology.

I was not convinced and started a discussion (copied below) on the Talk page of the Möbius strip article.

Relation to Umbilic torus

The link to “Umbilic torus” in the “See also” section is accompanied with a claim that the Umbilic torus can be obtained from a Möbius strip by gluing the latter along its single edge. IMHO, that statement is false, at least in regular 3D space, where the result would have to be a self-intersecting surface, assuming that the gluing is performed in the 3D space, rather than through some topological magic (sorry, I am not an expert in this field and lack the right terminology; my judgements are restricted to common sense intuition). Note that the article about the umbilic torus doesn’t establish any relation of it with the Möbius strip (the link in the “See also” section doesn’t count as such). I think that the misleading sentence has to be either removed or reformulated in a less confusing way and a more detailed explanation can be provided in the article about umbilic torus. Leon.Manukyan (talk) 08:41, 23 September 2023 (UTC)

  • You are confusing the standard embedding of the Möbius strip (with one half-twist) with the Möbius strip itself (a topological surface independent of how it is embedded into space). If you perform any odd number of half-twists to a strip before gluing it back to itself, you get a Möbius strip, with a different embedding for each different number of half-twists (see the second paragraph of the lead). The umbilic torus has a boundary that, if you cut along the ridges, is a Möbius strip. It is a Möbius strip embedded with three half-twists rather than one, but that is still a Möbius strip. —David Eppstein (talk) 10:15, 23 September 2023 (UTC)

    • You provided the same justification in your reversal of my earlier edit. My feeling was that something was wrong with it. A Möbius strip is single-sided whereas the Umbilic torus apparently has inside and outside. Now thinking about it more carefully, I guess that I found the flaw in your reasoning. The fact that the boundary goes three times around the center of the torus doesn’t mean that the strip makes three half-twists. Actually it only makes a full twist (two half-twists). Hence it is not topologically equivalent to a Möbius strip. Leon.Manukyan (talk) 10:31, 23 September 2023 (UTC)

      • Ok, this time I think you may be correct. If it really were a Möbius strip then (regardless of number of twists) traveling around the boundary (without crossing any ridges) until you return to your start should take you to the inside, but it doesn’t. —David Eppstein (talk) 11:53, 23 September 2023 (UTC)

Thus my non-expert judgement proved to be correct and my edit was reinstated.