Escher: The paradoxical artist beloved by mathematicians

Escher: the paradoxical artist beloved by mathematicians

A new retrospective of M.C. Escher’s work opens this week. Explore some of his most mind-bending and paradoxical works – and the maths behind them

M.C. Escher's work Waterfall, showing a water flowing to a waterfall which then falls to an aqueduct, before impossibly flowing uphill to reach the top of the waterfall again — Waterfall

Waterfall

Illustrations provided The M.C. Escher Heritage, Baarn, The Netherlands

Words by Alex Wilkins

In 1954, a young Roger Penrose was attending the International Congress of Mathematicians in Amsterdam when he came across the work of Maurits Cornelis Escher, a Dutch artist and print-maker. Escher’s works, which draw heavily on mathematical concepts like infinity and paradox, were rapidly gaining attention outside the artistic community, but Penrose was immediately “spellbound” by their mathematical beauty.

Soon after he returned to England, Penrose and his father, the psychiatrist Lionel Penrose, came up with impossible designs of their own, like paradoxical triangles and staircases, publishing their ideas in an in the British Journal of Psychology.

In an Escher-like act of reciprocity, this article then inspired Escher himself to produce at least two works, Waterfall, left, in which flowing water follows an impossible route, just like a Penrose triangle, and Ascending and Descending, below, the image of a never-ending staircase inspired by the. These went on to be some of his most famous images.

M.C. Escher's work Ascending and Descending, showing a never-ending staircase that appears to lead back into itself — Ascending and Descending

Ascending and Descending

Escher’s Penrose-inspired prints, together with more than 150 other works, will feature in “ in London from 6 June. It is a major retrospective of the artist’s work, including drawings and prints from his entire career.

It wasn’t just Penrose who took inspiration from Escher’s works, but an entire generation of mathematicians and artists.

“Escher is naturally beloved by mathematicians because of his approach to tessellation, his approach to infinity, his approach to paradox, but it is not only this,” says Federico Giudiceandrea, president of the M.C. Escher Foundation.

“The last century was a century with paradoxical scientific theories like relativity and quantum mechanics. People looking at Escher’s art understand in some way that the world is different or that our senses are giving us information, but what is behind it can be different. Your senses are not really capturing reality.”

Visitors to the Escher exhibition look at the artist's work Relativity

M.C. Escher's work Relativity, which shows stairs crossing in a paradoxical architecture — Relativity

Relativity

Penrose remembers this 1953 work, titled Relativity and featuring more of those impossible staircases, as one of the first Escher images he saw and one that left a lasting impression. The drawing, which Escher originally carved into wood, depicts a world in which three separate sources of gravity act on its inhabitants in different directions. The sense of local possibility but overall paradox was what stuck in Penrose’s mind and later inspired his impossible triangles.

Escher is naturally beloved by mathematicians because of his approach to tessellation, his approach to infinity
Federico Giudiceandrea, president of the M.C. Escher Foundation.

The ease with which Escher worked with mathematical concepts in his art was impressive, says Giudiceandrea, given his lack of formal maths education. “He had a huge mathematical intuition,” he says. “I always am amazed and, really, astonished to see how deeply and precisely he transformed these mathematical ideas into artworks.”

In the 1970s, Penrose was working to create new repeating tile patterns called aperiodic tessellations, and Escher was one of the first recipients of his new designs. Giudiceandrea said there were many letters sent between Escher and Penrose. “They really had a strong relationship, and Escher was very grateful to Penrose for these ideas,” he says.

M.C. Escher's work Tower of Babel, depicting the building of a tall tower from above — Tower of Babel

Tower of Babel

M.C. Escher's work Möbius Strip II, showing ants crawling on the continuous loop of a mobius strip — Möbius Strip II

Möbius Strip II

Escher had familiar themes that he returned to over his artistic career, such as unusual and extreme perspectives, like in the Tower of Babel, which he drew early in his artistic career, in 1928, or in 1960’s Ascending and Descending.

The inspiration for the building depicted in the latter probably came from Escher’s time living in Italy, particularly buildings from the town of Atrani on the Amalfi Coast, where he visited twice, and his study of architecture in Rome, where he lived for several years. “Escher was always amazed by this perspective from a bird’s eye view, or from strange perspectives. This is something maybe he learned in Italy,” says Giudiceandrea. “The landscape is completely different than in Holland, where everything is flat.”

Over time, he became more drawn to overt mathematical structures, such as in Mobius Strip II, one of a series of drawings he did of the one-dimensional surfaces whose sides never join up.

While mathematicians were an obvious group of fans for Escher’s geometric drawings, less-likely Escher enthusiasts were hippies. Reptiles, which also became the cover art for band Mott the Hoople’s debut album, features tessellating lizards coming to life. The drawing also contains several elements that some observers thought might signal that Escher had a soft spot for the countercultural revolution, such as JOB cigarette or joint-rolling papers, and the hallucinatory nature of drawings becoming real.

“Hippies were convinced that Escher was taking drugs or something like this, but it's absolutely not true,” says Giudiceandrea. “Escher was a very serious person, not even drinking alcohol.”

Escher was not overtly religious during his lifetime, but his paintings frequently contained spiritual themes, such as the unity and intertwining of things. In this drawing, called Bond of Union, Escher was reportedly inspired by H.G. Wells’s The Invisible Man, which features a scientist who accidentally renders himself transparent, but it is also widely taken to be about Escher and his wife, Jutte. In it, a single strip of paper twists to form two heads.

Reptiles

Bond of Union

M.C. Escher's work Hand with Reflecting Sphere, depicting a hand holding a reflective sphere showing the artist — Hand With Reflecting Sphere

Hand With Reflecting Sphere

While Escher’s images are instantly recognisable, people tend to know less about the artist himself, says Giudiceandrea. Escher produced this self-portrait, above, called Hand with Reflecting Sphere, in 1935, one of 12 self-portraits that he made over his career.

“Most people know Escher’s prints and images, but it is very strange that, if you ask who the artist was that made this kind of image, most people wouldn’t recognise him,” says Giudiceandrea. “We want to show people that may have seen Escher’s prints this exhibition, so you finally understand how this art was developed, and can recognise him as the father of these paradoxical images.”

The retrospective M.C. Escher: The Exhibition runs from 5 June until 6 September 2026.

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Escher: the paradoxical artist beloved by mathematicians

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