How could Escher create “mathematically impossible” works if he knew nothing about mathematics?

The Dutch artist Maurits Cornelis Escher (1898-1972) is one of graphic artists and most famous engravers in the world, also particularly linked to Italy as it had a long stay of 11 years in Rome. Very well known, for example, are the “impossible stairs” of the work Relativity, or the black and white paintings with the figures that blend and confuse. On the artist's official website there are all of his works, including woodcuts, wood engravings, over 400 lithographs and over 2000 drawings (all works protected by copyright and unfortunately not replicable in this article). His hypnotic style, influenced byart nouveauhas become famous thanks to the fact that his productions are a continuous homage to geometry and to mathematics, evoking absurd and imaginative worlds. One would expect, therefore, that Escher was a mathematical genius: this was not the case. And not only: he wasn't really good at scientific subjects.

Born in Leeuwarden, NE Netherlands, Escher enrolled (after failing his high school diploma) at the School of Architecture and Decorative Arts in Haarlem, not far from Amsterdam. Already a week after starting his studies, he told his father that he wanted to abandon the courses and concentrate on study of graphic arts: indeed his drawings and linocuts were already promising, but he never finished school.

Working and traveling, Escher created works from complex perspectives and applied to drawing (self-taught) some complex ramifications of mathematicsstarting with tessellations, those structures that produce infinitely repeatable graphic patterns. A fundamental role, in this sense, was played by his meeting withAlhambrathe splendid Arab palace built in the 14th century BC Granadain Spain, which the artist visited twice: the repeated tiles on the floor and walls inspired him to create new interlocking shapes that were increasingly perfect and difficult.

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Alhambra Granada
Glimpse of the Alhambra in Granada

But how could Escher create such technical works if he didn't have a particular predisposition for scientific subjects? With a let's help professional. She is the professor emerita of Mathematics at the Moravian College of Pennsylvania who tells us, Doris Schattschneider, who wrote several books on mathematical application in Escher's work. It is she who tells us that the artist was actually helped by Donald Coxeterat the time professor of Mathematics at the University of Toronto. The two had met at the prestigious International Congress of Mathematicians in Amsterdam in 1954: this meeting started a correspondence that lasted many years and influenced the work of both.

Alhambra geometric pattern
Alhambra geometric pattern

It was particularly important in this exchange a diagram present in an article by Coexter, which changed Escher's perspective on tessellation, taking it to the next level, that is, its application in geometry hyperbolic (a non-Euclidean geometry). This technique, to put it in summary, succeeded in make a two-dimensional figure three-dimensional.

As seen below, Coexter's drawing allowed for the tracing of circular arches, each of which met the outer circle at a 90-degree angle. Escher wrote to Coxeter asking for an explanation of how to make these circles, which were based on the model of disk by the mathematician Poincaré, “whose centers gradually approach from the outside until they reach the limit.” Coxeter responded with some suggestions, and the artist began to work with these new ideas, producing the famous ones woodcuts Circle Limit. With these works, Escher “brought” the infinite inside a circle, expanding it.

The hyperbolic form that inspired Escher

Escher Gresham College