Horseman

The study includes an analysis of writings by Bruno Ernst, Doris Schattschneider, Wayne Kollinger, Mickey Piller, M. C. Escher, Robert Ferreol et al, Miranda Fellows, Marjorie Senechel, Jeffrey Price, and Gina Davey (8) 

Bruno Ernst

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...

Bruno Ernst, Magic Mirror, pp. 100–101

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 (to me at least) it is unclear what he is saying.

Interestingly, he also makes a tapestry analogy, as does Fellows, q.v. 

Robert Ferréol et al

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).

Robert Ferréol et al, Mathcurve

Doris Schattschneider

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, which I retain for the sake of convenience]

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…

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

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