M. C. Escher

The Study of the Möbius Strip in Regard of Escher’s Four Themed Prints 

A study of M. C. Escher’s four Möbius strip-themed prints (or are said to be), namely Horsemen, Swans, Möbius Strip I, and Möbius Strip II. Each print is examined in turn to determine if this is so.

The study was occasioned by an email query from the Escher authority Jeffrey Price, querying if Escher’s Horseman print was, indeed, a Möbius strip, having looked at the matter in some depth, and going to the trouble of a paper model recreation. After an initial investigation on Horsemen, I expanded the study to better understand the issues, hence the inclusion of all the prints above.

To show how the study arose, I show the initial email query (27 December 2023) from Jeffrey Price. Accompanying this was a picture (with commentary) of a 3D actual paper model (Fig. 1), to try and recreate the portrayal as given by Escher:

Perhaps you can help me understand Escher's Horsemen [print].

I wanted to get a better grasp on the structure of "Horsemen" so I made a band and taped it together. see photo. No twists were involved.

I don't recall this being discussed in this way. DeRijk writes in Magic Mirror "In Horsemen, a three-colored woodcut made in 1946, we see a Moebius strip with two half-turns. If you make one for yourself you will find that it automatically forms itself into a figure-eight."

I'm missing something. I trust DeRijk but I don't see a double-half-twist Moebius strip!

What do you think? Is it a Moebius or a Nobius?

Some say M. C. Escher’s “Horsemen” are on a Mobius strip. I’m not so sure. I made a model by creating a loop printed with different colors of horsemen on each side, one side striding in the opposite direction from the other. The two sides can be laid flat on a table with just enough room between them for two pairs of horsemen to march across from one side to the other. Perhaps the horsemen can be said to travel on a “Nobius”, defined as “an odd non-twisted band”. 

A ‘fair representation’ can be said to be given by the recreation, albeit it is not exact; the lower extremity noticeably curls; compare with the top. Escher’s print is symmetrical, whilst Price’s recreation is not exact. But from trying this myself, it is next to near impossible to recreate this as an actual model. Price’s attempt is about as good as it can get. But how does this advance the main point; is it Möbius or Nobius? A point of concern is that of the ‘central insertion’, which is not seen on a ‘basic’ (½ twist) Möbius strip. Does this invalidate the supposed Möbius strip premise? Likewise, two ‘holes’. From attempting to answer this ‘simple’ query, the study expanded way beyond the original remit into an analysis of the four (or are said to be) Möbius-themed prints, and then again, of the Möbius strip as an entity in itself, in its purest form, without ‘anything Escher’, and then yet again, of historical aspects, this time out of general interest. All this was with the intention of better grasping the underlying aspects of the initial query. 

An Examination of the Four Prints

I now examine each of the four prints in turn, in chronological order, without any preconceived assumptions from the authorities. The format is as follows: A discussion by the authorities, in books, articles, websites, and correspondence, placed in order of length. Generally, but not invariably, a lengthy treatment will be an indication of substance. Immediately below each entry, I add comments of my own in which I assess their viewpoints. Following this, I then add my own observations and thoughts in an overall sense.

An Examination of Horseman
Contributors include Bruno Ernst, Doris Schattschneider, Wayne Kollinger, Mickey Piller, M. C. Escher, Robert Ferréol et al, Miranda Fellows, Marjorie Senechel, and Jeffrey Price (9).

Bruno Ernst, Magic Mirror, pp. 100–101

In Horseman, a three-colored woodcut made in 1946, we see a Moebius strip with two half-turns. If you make one for yourself you will find that it automatically forms itself into a figure-eight. This strip definitely has two sides and two edges. Escher has colored one side red and the other blue. He conceives of it as a strip of material with a woven-in pattern of horsemen. The warp and woof are of red and blue thread, so that one horseman comes out blue and the other red. The front and the rear of a horseman are mirror images of each other, and there is nothing unusual in this, for it could be said of any figure one cares to choose. But now Escher starts manipulating the strip so that an entirely different topological figure is produced. In the center of the figure-eight he joins the two parts of the band together in such a way that the front and the rear sides become united. We can copy this in our paper model if we use Scotch tape to turn the middle of the figure-eight into a single plane surface. From a purely topological point of view, we ought at this point to drop one of the two colors, but this is not really what Escher intended. He wishes to show how the little red horsemen on the underside of the print combine with the blue ones, which are their mirror image, to fill up the surface completely. This is achieved in the center of the print.

Of course, we can also find this in a very fine page from Escher’s space-filling sketchbook (figure 225), but in this particular print it is presented in a most dramatic way, for here we can see the filling process actually taking place before our eyes.

DB. Ernst’s account is somewhat confused (to my mind, at least), with many contradictions. This begins somewhat inexactly with:

In Horseman, a three-colored woodcut made in 1946, we see a Moebius strip with two half-turns.

A Möbius strip is defined as an odd number of twists, or if of an even number is a topological cylinder. Ernst effectively contradicts himself. He also states that it (the print portrayal) has two sides and two edges and so is thus not a Möbius strip, and yet at the start he says it is a Möbius strip! ‘This strip definitely has two sides and two edges’. He then states:

‘But now Escher starts manipulating the strip so that an entirely different topological figure is produced’.

So it is unclear what he intends. Interestingly, he also makes a tapestry analogy, as does Fellows, q.v. 

Doris Schattschneider. Visions of Symmetry, pp. 241, 243

Regular division of the plane was, for Escher, a means to capture infinity. His challenge was to capture infinity in a ‘’closed” composition; he could not accept the abrupt cutoff of the theoretically infinite repetition seen in the periodic drawings. His description of how his regular division drawing 67 of a horseman came into being (quoted on page 110) does not end with the creation of that working drawing. It continues: [She then quotes Escher]

The horseman has been born, but although his contour is closed, the plane in which he moves is limitless in all directions, and for this reason we can also work him as a fragment into a picture.

What is done next is present him as a complete entity on a limited surface. Many solutions to this problem are possible and they are all poor. My chief goal was to show quite clearly that the congruent shapes moving in alternate rows to the left and to the right are indeed each other’s mirror images. To achieve this I presented the two processions on one strip which, except in the center of the print, has a plastic appearance as if it were a woven fabric: the “‘pattern’” on the front side is of the same color as the “background” on the reverse side, and vice versa. Thus the two processions now form a closed circuit. In the center where the three-dimensional illusion merges into a flat surface, they have obviously been integrated—become part of one single regular division of the plane. [end of Escher’s quote]

Escher's concept sketches for the print clearly show his exploration of this central idea for presenting the processions of riders.

The device of a cycle, a closed loop, or a suggestion of a repeated procession of figures doubling back on each other are all used by Escher in several other graphic works which incorporate his periodic drawings as fragments. In these prints, he also mixes the illusion of three dimensions with two: the representation is in the plane, the regular division is in the plane, but the figures take on life, like the playing cards in Alice in Wonderland, and move about to complete their cycles…

DB. A prescient background comment was made by Schattschneider. This admirably details the background to the print as to Escher’s intentions, in general, and in particular, and greatly aids the interpretation. In short, Escher here is capturing infinity in a closed composition. 

Of note here is that Schattschneider and the Escher quotes do not mention Möbius strips at all, but rather ‘one strip’ and a ‘closed circuit’. An open question here is how careful Schattschneider (and Escher) were with their words here. Did he purposely exclude a Möbius reference (on the grounds of a non-Möbius strip), or was it just an inadvertent oversight? It’s hard (if not impossible) to tell. There’s very little more that I can add.

Wayne Kollinger

Horsemen depicts an interesting non-orientable surface.

It is not a Mobius strip.

A Mobius strip has an odd number of half-twists, or an odd number of bends (skewed folds).

In more general terms, a Mobius strip has an odd number of front/back inversions.

The Horseman has two bends (or folds) which is an even number.

Moreover, it is not a strip.

However, it can be made from a strip that is joined to itself.

It is a surface that can be divided in two halves. Each of which is a Mobius strip with a single fold.

It is these Mobius strips that result in it being a non-orientable surface.

I suspect it has been mistaken for a Mobius strip because it resembles a strip and because it is a non-orientable surface.

Wayne Kollinger, email 26 February 2024

DB. An insightful analysis is given by Wayne Kollinger. This rings true. He states that it is not a Möbius strip. 

Mickey Piller, Perpetual Motion. Escher in het Paleis, not dated.
In Horseman the relationship between the elements of the central section of the print differs from the three-dimensional figure of 8 that forms the basic structure of the Möbius strip. In Horseman the central section of the print is a one-dimensional tessellation. Escher plays with our perception in this section. And this is what initially gives us the impression that we are looking at a Möbius strip.
DB. A small piece on Horseman appears within the theme of perpetual motion. Not dated, but must be after 2015, when she retired as the article states ‘former’.

Piller appears to imply that this is not a Möbius strip, despite the impression.

Robert Ferréol et al, Mathcurve
Escher's famous horsemen are represented on a strip with two half-twists, hence with two faces (colored in grey and beige) and two boundaries, but the central fusion that simulates the identification between two segment lines of each boundary (indicated in green) makes it a topological Möbius strip with only one face, and only one boundary, indicated in red (cf. Möbius shorts). 

Miranda Fellows, The Life and Works of Escher, p. 45
ESCHER’S UNDERSTANDING OF weaving techniques is manifest in this work. The horsemen are portrayed on a fabric-like ‘band’. On the outer surface their tone is dark and strong. On the inner surface, or the ‘back’ of the fabric, they are far paler, where on a fabric the reverse of the weave would be shown. Where the pale horsemen meet the darker ones, the front and back of the band merge, as do the figures, until, in the centre of the image, one can no longer tell which is which, or where one side ends and the other begins.
DB. Fellows is a general art writer and artist. She is not particularly known as an Escher authority, and I recall errors in her book. But this does not necessarily preclude her from having something to say on the matter. Of note here is the tapestry analogy, somewhat reminiscent of Ernst’s earlier account, q.v. Perhaps this put her in mind? These are the only two such references to weaving/tapestry I know. I am hindered here in that I have only layman’s knowledge. I will accept it at face value. She is categorical about Escher’s understanding of weaving, although perhaps a little overstated. Escher was familiar with Edmond de Cneudt’s hand-loom and carpet-weaving firm in Hilversum, but that does not presuppose a deep interest, see Escher the Complete Graphic Work, p. 67.

Of note is that Fellows describes the portrayal as a (standard) band and not a Möbius strip.

Marjorie Senechal. ‘Escher Designs on Surfaces’. In M.C. Escher Art and Science, pp. 103–105.
His famous woodcut "Swans" (which he adapted to the school pillar) is sometimes said to be a Möbius band but in fact it is a cylinder, and the ribbon-like "Horseman" is a figure eight.
DB. Senechal’s statement (to me at least) is ambiguous. She appears to be implying that Horseman is not a Möbius strip. 

Jeffrey Price, 23 February 2024 email (in the context of a wider discussion of the four prints)
Horsemen is not a Möbius, but a band, as my original illustration showed. Escher's tessellation fits brilliantly since it has two colors of horsemen traveling in two directions. These qualities allow them to march across from one part (or side) of the band to the other. Problematically, the marching horsemen don't fit exactly, hence Escher's horses asses require clarifying 'tabs.'

DB. Categorically states that this is not a Möbius strip.

John Fletcher

John Hey pointed out that the third picture [Horseman] is not really a Mobius strip. It has two sides (not one).

The silver side is separate from the gold side except where they intersect in the middle, so perhaps you could call it one side!

Concept Sketches in the Notebooks and Microfiches

For possible clues, I now examine the concept sketches in the notebooks and microfiches (Fig. 3). Incidentally, the sketches are not shown in the other major examination of the print, in Magic Mirror.

Understandably, given their nature (essentially of ‘background interest’), these are little discussed.

Again, I make use of Kollinger’s insight.

Wayne Kollinger, email 26 February 2024

The first illustration on the top left shows how a non-orientable surface can be made from an ordinary ring.

Start with a strip and bringing the two ends together in the front.

Join the bottom front edge and the top back edge.

This is not the surface Escher decide to go with.

The illustrations that follow it depict the surface he used.

Start with a strip.

Fold the left end to the front toward the center.

Fold the right end to the back toward the center.

Join the two ends underneath the center of the strip.

Join the bottom front edge and the top back edge.

What happens If you try this with a Mobius strip?

A traditional Mobius strip has three bends.

When the top front edge and the bottom back edge are joined the surface has a single bend on one side and two bends on the other side.

If Escher’s horsemen were to start traveling upright along the top of the back, after they pass through the two bends they end up upside down along the bottom of the front.

In this position they do not tessellate.

If Escher consider this option, it’s easy to see why he didn’t use it.

DB. Nonetheless, despite little discussion, they remain of interest. To try and at least give some basic detail, I now examine each sketch in turn, likely in order of creation, despite being non-sequenced. Indeed, a different interpretation is possible; the left and right-hand pages with their opposite drawings bear similarities, albeit perhaps coincidental. Whether these interpretations echo Escher’s thoughts will never be known, but nothing is lost by a circumspect examination.

1. A pencil sketch seeingly establishing the tessellation principle of glide reflection with kites.

2. A coloured pencil sketch, in red and blue, more or less repeating 1.

3. A pencil sketch, with the strip as an abstract entity, without the kites. It possesses 180° rotational symmetry. Note the two ‘gaps’ lacking in 1 and 2.

4. A pencil sketch of unclear purpose. This has echoes of 1.

5. A pencil sketch of unclear purpose. This has echoes of 2.

6. A coloured pencil sketch, in blue and orange, that closely resembles the print (albeit without the horsemen).

A few general observations (overall):

1. Although a Möbius strip is (generally, by others) said to underlay this print, it cannot be said to be an ‘obvious’ presentation of a Möbius strip (contrast with, say, Escher’s Möbius Strip II, where it is made more obvious and is unambiguous, discussed below). Or indeed, of an actual paper model, with any number of half-twists. Certainly, it gives the ‘general impression’ of a Möbius strip but is not clear cut. As an aside, Möbius Strip I, despite its name, is not strictly a Möbius strip - see the discussion.

2. Of note is the tessellation used, periodic drawing 67, of June 1946 (incidentally preceding the print by just a month). An open question is the significance of the selection of the periodic drawing, based on kites, or more exactly its symmetry, consisting of a glide reflection. A glide reflection shows motifs in two orientations only. An open question is which, if any, of the four isometries (translations, rotations, reflections, and glide reflections) will permit a simple straightforward Escher-like tiling Möbius strip artefact. I used to think that glide reflection was a prerequisite, but this is not so. There are various nuances here.
(i) A motif with the same colour for ‘both sides’ is not possible

(ii) A motif with a different colour for ‘both sides’ is possible (but surely negates the premise)

(iii) A wireframe motif, i.e. without colour, is possible, but without colour would not be distinct

3, Save for the centre section (of eight horsemen), the tessellation element is not fully shown, with a single strip of horses and riders, top and bottom. Is this significant? 

4. Curiously, the title, Horsemen, does not mention a Möbius strip, although, of course, this does not preclude it being so. But it may be thought that if a Möbius strip was the underlying feature, or of a prime concept, such a description would have been naturally incorporated into the title. Of interest is to compare the other (four) prints on (or are said to be) the theme in this regard. The next, Swans, of 1956, also adopts the same (simple single) title convention being named after the periodic drawing motif. In many ways, in terms of the presentation, this is like Horsemen, with a central tessellating ‘panel’. Then with Möbius Strip I, of 1961, the connection is made explicit for the first time, albeit it will be seen that this is not strictly of a Möbius strip. And finally, Möbius Strip II, of 1963, follows similarly in the same vein but this time is so, without any quibbles whatsoever. So a Möbius strip connection is not made explicit here, which suggests at the least a quibble.

5. The strip is obviously ‘simple’ in terms of the number of twists, i.e. it is clearly not one of multiple twists. This should, in theory at least, be easy to recreate as a paper model. However, it does not seem to be possible. It could even be considered a stylised representation of such a model. 

6. Disregarding the motifs, the strip has 180° rotational symmetry.

7. The motifs echo each other in terms of symmetry. For example, a distorted horseman at the extremities has an echo on the opposite side. The same applies to all the other motifs.

8. The strip has a stylised, if not idealised Möbius strip presentation, of a broad figure of 8, merged in the centre. Price has made a fair effort to recreate this.

9. A (non) Möbius strip of one full twist (as above) also has connections in that when viewed at a suitable presentation, it has two ‘holes’ as here, but against that, it is not a Möbius strip.

10. A curiosity is two ‘tabs’ adjoining the central panel, not present in the sketches. What is their purpose or reason? Jeff Price’s thoughts
You have probably noticed the grey and red tabs in the center of the print that have nothing to do with the bands and little to do with the horsemen. I attach an illustration showing the tabs circled in blue.

They exist solely to define the horsemen's shapes. If the tabs did not exist, as I've tried to show in the second illustration, a little bit of the red horseman would be set against a red background and the grey horseman's back and the horse's behind would be on grey, both of which do not align with showing tessellations 'properly.'

Although I was aware of this, I had not really thought about it. Price’s examination seems as good as any.

Summary

So what is it, a Möbius strip, or in Jeff Price's delightful description, a Nobius? I am still not sure! Much better minds than mine have looked at this, albeit the print is seldom discussed as an entity in its own right. And when it is discussed, it can be rather confusing and hard to interpret at times, as in Bruno Ernst’s long discussion. That being said, I much prefer Robert Ferréol's explanation (and illustration). 

Not only is it logical and clear, but he also gives a diagram to give credence to his points. 

In short, Horseman is not a Möbius strip, but is rather a ‘manipulated band’, albeit with a superficial appearance of a Möbius strip. Naysayers include Price, Kollinger, Ferréol et al, Hey, Senechal, and Piller (the last two with reservations). Other people are less than precise, and so I decline to fully decide their interpretations, leaving this open ended. 

An Examination of Swans
Contributors include Bruno Ernst, Maurits Escher, Wayne Kollinger, Marjorie Senechal, Robert Ferréol, Erik Kersten, and Jeffrey Price (7) 

Another print sometimes said to be (or is based upon) a Möbius strip is Swans, of 1956, this being after Horseman (1946), of which it shares some similarities; both have a ‘central panel’ and two ‘holes’ in the composition. As with Horsemen, opinions are divided as to a Möbius strip portrayal. Again, there is, perhaps surprisingly, little discussion, with what there is largely in passing. Perhaps more may be found upon a dedicated search of another day.

Maurits Escher.
“In 1960 I was exhorted by an English mathematician (whose name I do not call to mind) to make a print of a Moebius strip. At that time I scarcely knew what this was.”

In view of the fact that, even as early as 1946 (in his colored woodcut Horseman, figure 91), then again in 1956 (the wood engraving Swans), Escher had brought into play some figures of considerable topological interest and closely related to the Moebius strip, we do not need to take this statement of his too literally.

Bruno Ernst, part quoting Escher, Magic Mirror, p. 99

Ernst effectively states that Swans is closely related to the Möbius strip, but is not actually one.

Here is an example of glide reflection [Swans periodic drawing and print side by side]. The swans on the left appear again in the woodcut on the right. My purpose in making this print was to give a demonstration of that same glide-reflection principle. Aiming to prove that the white swans are mirror reflections of the black ones, I let them fly around, in a path having the form of a recumbent 8. Each bird rises from plane to space, like a flat biscuit sprinkled with sugar on one side and with chocolate on the other. In the center of the 8, the white and the black streams of swans intersect and form together a pattern without gaps.

Maurits Escher. Quoted in Escher on Escher. Exploring the Infinite, p. 29. 

Escher neither confirms nor denies the Möbius strip, rather discussing the print in looser terms, i.e. a fig. of 8.

Wayne Kollinger email 22 February 2024

Horseman and Swans illustrate two surfaces that are topologically the same.

By that I mean they are homeomorphic.

I am assuming that the center section of Swans is common to both halves. That is to say, the halves are connected in the centre rather than overlapping.

,,,

The Mobius strips you get by cutting Horseman or Swans in half are topologically annuli (annuluses) that incorporate a single bend that achieves the task of joining the front to the back.

If you cut either Horseman or Swans across the centre, you get two surfaces each of which is a Mobius strip.

Neither surface looks like a traditional Mobius strip.

DB. An insightful analysis by Kollinger.

Marjorie Senechal. Quoted in ‘Escher Designs on Surfaces’. In M.C. Escher: Art and Science, pp. 103–105.

Escher did not design many patterns for this one-sided surface. His famous woodcut "Swans" (which he adapted to the school pillar) is sometimes said to be a Möbius band but in fact it is a cylinder,...

DB. Senechal categorically states that it is not a Möbius strip but rather a [topological] cylinder.

Erik Kersten, Band of Möbius I. Escher in the Palace, 13 March 2017

A Möbius band is a spatial figure that seems impossible but can actually exist. Escher already played with it carefully in his woodcut Rider from 1946 and his wood engraving Swans from 1956. Band of Möbius I from March 1961 is the first complete version… 

DB. This is not an easy interpretation. Kersten merely asserts that Swans is a Möbius strip amid a general discussion of the theme. He then complicates matters by noting ‘complete versions’, implying that this is not true, in contradiction to his first sentence. 

Robert Ferréol et al. Mathcurve Website

Same principle [Horseman] for these swans with black-and-white faces. (i.e. a topological Möbius strip)

DB. In so many words, Ferréol asserts that this is not a Möbius strip.

Jeffrey Price, 23 February 2024 email

Swans uses a similar band to Horsemen, but it crosses over itself in the middle. The middle cross-over must be thought of as flat, so imagine the gaps between the swans as being transparent.

DB. The issue is not directly addressed.

Of interest is to examine the sketches in the notebooks. This may be thought to be a straightforward task, but there are various problems here. Although there are indeed obvious instances, others are less so. There are various abstract diagrams of a ribbon theme that almost certainly have no connection with Swans. Nonetheless, three are clearly identifiable with Swans (Figs. 5–7). All have the same ‘flat panel’ in the centre. Although Horseman also has the same ‘flat panel’, the Möbius strip shown there differs, and so one can be confident that these diagrams specifically refer to Swans. An obvious question to ask is if this can be recreated as an actual (paper) model. A figure of 8 is apparent, with either the strip overlaying or is merged at the centre.

The strips here merge at the centre. Interestingly, the central panel is of squares, which is against the finished print kites. Was Escher planning square-based motifs before changing his mind to a kite? 

An Examination of Möbius Strip I
Contributors include Roosevelt-Escher letters, Escher’s lecture notes, Bruno Ernst, Jeffrey Price, Mickey Piller, Alon Amit, Maurits Escher, and.Wayne Kollinger (8)

Of a prima facie examination, and with the title in mind, it may appear obvious that this is a representation of a Mobius 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.

Roosevelt-Escher Letters, 1961
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

Abandoned US lecture tour notes, 1964, Escher on Escher, p. 64.

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.

DB. 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, this notably being three years after the Roosevelt correspondence. 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?

Mickey Piller, Perpetual Motion

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.

Jeffrey Price, 26 February 2024 email

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.

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?

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.

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?

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]

DB. 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, 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. 

DB. Conclusion

So is Möbius Strip I 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 Roosevelt’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 have (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 connection is obvious. You pays your money and you take your choice! 

An Examination of Möbius Strip II
Contributions are by Maurits Escher (2), Doris Schattschneider, Joseph Myers, Marjorie Senechal, and Jeffrey Price (2). 

This is of the ‘basic model’, of a half twist, albeit of a condition, with the edges touching at right angles to give a symmetrical ‘figure of 8’ composition. It will be seen that the print is intended (by Escher) to be presented in a portrait orientation, rather than landscape.

As sure as anything can be certain of the series, this, unlike the others, is indeed unequivocally a Möbius strip! Nonetheless, I examine the discussions.

Maurits C. Escher, Graphic Work, p. 12, within the category of ‘Spatial Rings and Spirals’.
An endless ring-shaped band usually has two distinct surfaces, one inside and one outside. Yet on this strip nine red ants crawl after each other and travel the front side as well as the reverse side. Therefore the strip has only one surface.

DB. A straightforward, bare bones description of the print, in line with other entries in the book.

M. C. Escher, Escher on Escher, pp. 62, 64.
Let us change the subject once more and see some prints with ribbons…
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 right slide 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.
DB. Escher also commented on the print (in tandem with Möbius Strip I), within the context of ribbons in text taken from his (abandoned) US lecture tour notes, 1964. I omit the distinct discussion on Möbius Strip I to not distract from Möbius Strip II. This is perhaps less than exact. ‘twisted only once’ is open to interpretation. A half twist or full twist? Also, Escher calls it (loosely?) a ribbon. No matter, he is clearly intending a Möbius premise.

Again, another bare-boned description, that is even more sparse than the Graphic Work entry. Although I would want more, the text has to be put into context overall. In both, this is not intended as an all-encompassing discussion of the various nuances, but rather serves as an ‘introduction’. 

Certainly, there is no new insight from the above.

Doris Schattschneider. ‘Escher’s Metaphors’. Scientific American, November 1994, pp. 66–71.
Mobius Strip II harbors a procession of ants crawling in an endless cycle. With a finite number of figures, Escher depicts infinity through the continuous traversal of an endless loop. The ants demonstrate as well that this unusual loop (originally printed vertically) has only one side.

DB. A brief description without delving too deeply into the intricacies.

Joseph Myers, website
Möbius Strip II, 1963

Whilst this behaviour is most popularly exhibited by processionary caterpillars, ants also follow each other to a certain extent. In this woodcut, Escher populates the surface of a Möbius strip with a set of ants. Since the Möbius strip is non-orientable, the ants can see other ants walking on the underside of the surface. With an odd number (in this case, nine) of ants, the ants are directly between their subterranean counterparts; with an even number, equidistant ants form pairs with each ant immediately below its partner. Compare this with star polygons, which are connected if and only if the parameters are coprime.

Whilst reclining on the forbidden grass of our rivals, St. John’s College, a dialogue developed about the one-sided nature of the Möbius strip. Specifically, I was looking through these Escher postcards and explained how to construct a Möbius strip by introducing a half-twist into a strip of paper before connecting the ends. The person to whom I was talking astutely remarked that the surface of an ordinary sheet of paper is also technically one-sided [since it is a topological sphere].

Joseph Myers, website

DB. This is interesting in that Myers posits a reason for there being nine ants, an odd number, as against showing an even number. He also gives a plausible reason for the choice of ants. Not only that, but peripherally, why a sheet of paper is, or can be, considered one-sided.

Marjorie Senechal. Quoted in ‘Escher Designs on Surfaces’. In M.C. Escher: Art and Science, pp. 103–105.
His famous woodcut "Swans" (which he adapted to the school pillar) is sometimes said to be a Möbius band but in fact it is a cylinder, and the ribbon-like "Horseman" is a figure eight. But "Ants" really is a Möbius band; the ants are crawling along a single surface.
DB. Upon an assessment of which of Escher’s prints are or are not a Möbius strip, Senechal categorically asserts that Ants is. 

Jeffrey Price, M. C. Escher Amazing Images, p. 195.
Möbius Strip II (Red Ants)

TEXT TO ADD

DB. An in-depth, dedicated discussion.  A few interesting points are raised, such as why the grid is pierced (to see both sides at once).

An open question is why ants were used, and not some other motif. A plausible explanation is that they naturally follow each other, as in the wild i.e. an ant trail.

Ants is a möbius, though it must be thought of as having no thickness. It is impossible to draw it with no thickness, but an ant crawling along the outside edge as Wayne [Kollinger] suggests is like a magician showing you his trick and in doing so, he makes the magic vanish.

Jeffrey Price, 23 February 2024 email

This clearly shows a Möbius strip is intended, without any quibbles or conditions. Möbius Strip II does not have a single naysayer. 

Created 26-27 May 2025 from existing text.