Swans

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.

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

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. Escher on Escher. Exploring the Infinite, p. 29.

Escher neither confirms nor denies a Möbius strip intent, rather discussing the print in looser terms, i.e. as a 'figure of 8', in so many words. As I discuss, using such terms is ambiguous and leads to uncertainty about meaning. A 'figure of 8' can be formed as either a Möbius strip or a non-Möbius strip, with remarkably similar appearances at first glance!

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,...
Marjorie Senechal. 'Escher Designs on Surfaces'. In M.C. Escher: Art and Science, pp. 103–105.

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

Same principle [as Horseman] for these swans with black-and-white faces. (i.e. they are a topological Möbius strip)
Robert Ferréol et al. Mathcurve Website

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

Summary

The consensus is that Swans is not a Möbius strip. My opinion of this (after making paper models) is that it is not a Möbius strip, as it is not possible to traverse both sides and so is rather a topological cylinder. Explicit naysayers include Ernst and Senechal.

Created 15 May 2024. Last Updated 15 May 2024