Horseman

An Examination of Escher's print, Horseman.

Contributors include 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)

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 it is unclear what he intends.

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

So what is it, a Möbius strip, or in Jeff'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 reservation). Other people are less than precise, and so I decline to fully decide their interpretations, leaving this open.

Created 14 May 2024. Last Updated 14 May 2024