Möbius Strip I

Of a prima facie examination, and with the title in mind, it may appear obvious that this is a representation of a Möbius strip (it being suitably titled); indeed, perhaps it is so obvious that nothing more needs to be said. Indeed, I, along with nearly all contemporary others, have simply accepted this at face value. However, it is strictly not so! This 'anomaly' was apparently first noticed as far back in 1961, in a letter from C.V.S. Roosevelt to Escher. However, this was not in the public domain (in the National Gallery of Art, US archives), of which it has only recently emerged, in February 2024, as a result of Jeffrey Price's investigations (a visit), of which he has kindly made the letter (and two others) available to me. The correspondence thereof, and a 'statement' by Roosevelt, is shown below, along with excerpts from books and articles.

Contributors include Roosevelt-Escher letters, Escher's lecture notes, Bruno Ernst, Jeffrey Price, Mickey Piller, Alon Amit, Maurits Escher, and Wayne Kollinger (8)

I was much intrigued… Möbius strip…
However, I have one question on this piece. Is it indeed a Möbius Strip?...
In the woodcut, there would appear to be gray surface and a reddish surface, but each is distinct and separate. A bug crawling on the gray surface can only arrive at the red surface by crawling over an edge. For the woodcut to be true Moebius Strip, would it not be necessary to introduce a half twist at some point?
Roosevelt-Escher, 19 August 1961

I am very glad that the thing [Möbius strip print] pleases you. Indeed, I think that perhaps the image which I made is no more a real Moebius strip, since with a pair of scissors, I did cut the whole ribbon in two halves, with a white interspace between them.
But that's just the point!: I started to cut at a random point and I continued along the strip till I reached that same point again. So the result should have been: two independent strips. But it happens that there is still only one strip (three fishes, biting each other in their tails).
The original, real Moebius strip, which could and should be rendered in one colour only, (as its underside has disappeared) and two colours are needed now.
Escher-Roosevelt, 24 August 1961

It is true, as Escher says in his letter, that this representation is no longer a true Moebius Strip. It did start out as one. Escher took a long strip and gave it one and one-half twists after which he joined the ends. He then cut it down the middle and wound up with the figure shown in this woodcut.
Roosevelt ['statement'], 3 September 1961

To conclude this ribbon series [which also included Spirals, Sphere Spirals, Rind, Bond of Union], I show you two Möbius strips [Möbius Strip I and Möbius Strip II]. The left one is folded three times and is presented as if it were cut in two parts with a pair of scissors over the whole length. Nevertheless, it is still one uninterrupted slip [sic], consisting of three fishes biting one another's tail.
The right side shows a ribbon that is twisted only once. Nine red ants are walking in a continuous file, one after another, at both sides of the ribbon.
Abandoned US lecture tour notes, 1964, Escher on Escher, p. 64.

Escher also commented on the print (in tandem with Möbius Strip II), in text taken from his lecture notes. I have omitted the distinct discussion on Möbius Strip II to concentrate on Möbius Strip I.

Escher seems quite happy describing it as a Möbius strip, notably three years after the Roosevelt correspondence (qv). Interestingly, he uses the term 'folded' rather than 'twisted'. Was this a careful, considered choice or more of a throwaway nature? That is, is he making a clear distinction?

Möbius I can be pictured as a paper chain link that consists of three stylised fish, each holding the tail of the fish in front of it in its mouth. The eyes of the fish are aligned with the folds in the link. To make it more complicated, the link has been cut in two along its length. It has also been cut along its width and the ends have been twisted and joined together again. See the example above, which is taken from Escher, Magician on Paper by Bruno Ernst.* In Möbius II the Möbius strip is easier to see.…
I would imagine that the Möbius strips, which Escher completed in 1961 and 1963, evolved out of two earlier prints completed in 1955 and 1956, Rind and Bond of Union. In these works Escher also depicts a strip, which he once compared with a strip of orange peel.

Mickey Piller, Perpetual Motion

Möbius I is confusing since the red and green fish-bands appear to be two distinct bands. This brings up the complexities that occur when you cut a möbius, but this I believe to be a red herring or an unnecessary diversion that disguises the simple truth. If you join the bands and think of the shape as a single band that is half green and half red it becomes simpler to visualize the möbius.

Jeffrey Price, 26 February 2024 email

Oh, but it is a Möbius strip, with the suitable perspective.
Escher's Möbius strip I shows a Möbius strip with a threefold symmetry, cut along its midline. When you cut a Möbius strip like that you get a longer doubly twisted strip, which Escher draws with contrasting colors on its two sides. Notice, however, that the coloring doesn't apply to the original, uncut band, since every surface carries both colors.
Since this seems to be confusing: the Möbius strip is the whole surface, without the long cut along the middle and without the cuts for the "eyes". The double-length, half-width, colored surface is not a Möbius strip.

Alon Amit, answering the question on Quora (I omitted the illustrations) Why does the famous "Möbius Strip I" picture by M.C. Escher have this title although what it depicts is not a Möbious [sic] strip?

You order a pizza and it is delivered cut into six or eight pieces. Do you still consider it to be a pizza? Of course you do. It's a pizza that's been cut into pieces.
Escher considered this to be a Mobius strip cut in half down the center. Hence the title.
However, you do have a legitimate concern. When you say it is not a Mobius strip you are right. It is not even a Mobius strip that has been cut in half. It is an illustration of what you get when you cut a model of a Mobius strip in half…
[The text continues but is strictly unrelated to Mobius Strip I, hence omitted]

Wayne Kollinger, answering the question on Quora: Why does the famous "Möbius Strip I" picture by M.C. Escher have this title although what it depicts is not a Möbious [sic] strip?

I now give my own analysis. The print itself is two-coloured with threefold symmetry. A rudimentary creature is discernible, but it is far from clear what it is supposedly representing. Escher calls the motifs 'fishes'.

The origin of the composition becomes clearer when a 'reverse engineering' construction is made, as a complete Möbius strip (a), which is then separated a little down the centre, this resulting in the portrayal as given by Escher (b). Image (and idea) courtesy of Jeffrey Price.

Escher's Möbius Strip I shows what can be described as a faux Möbius strip, cut along its midline and slightly separated, with an even gap throughout.

Conclusion

So is it a Möbius strip or not? Whether it is a true Möbius or something else depends on interpretation. I tried recreating the print as an abstract construct, with three half-twists (in Rosevelt's terminology 'one and one-half' twists) and cut down the middle and slightly separated (as Escher appears to have done). I then carefully kept the resulting 'separation' tight, by sellotaping a parallel gap throughout (Fig. 13). Upon investigating, you cannot traverse this model! I also tried just a half-twist. Again, the same outcome (Fig. 14). So it appears that Roosevelt and Escher are vindicated in their opinions! In short, as pointed out by Roosevelt (3 September 1961), and suitably amended, although it started as a Möbius strip, it didn't end that way. Kudos to Roosevelt for his examination.

That said, it is indeed a Möbius strip in spirit, if not to the letter. It should be noted that Escher's (first) and Roosevelt's viewpoints has (almost certainly independently) recently been noticed by a few others on Quora, as discussed above.

So, it's really a matter of interpretation. Cases both for and against can be made. There's no real right or wrong. In short, it started as a Möbius strip but did not end as one, although the Möbius strip connection is obvious. You pays your money and you takes your choice!

Created 16 May 2024. Last Updated 24 May 2024