M. C. Escher

• 1. Introduction

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  M. C. Escher was born on the 17th of June 1898 in Leeuwarden, Friesland, the Netherlands, in a house that now forms part of the Princessehof Ceramics Museum. For most of his life, the art world barely noticed him. He was 70 years old before his native Netherlands held a retrospective of his work. Yet by the twenty-first century, a single exhibition of his prints in Rio de Janeiro drew more than 573,000 visitors in 2011, making its daily count of 9,677 the highest of any museum show anywhere in the world that year. How did a sickly child who failed the second grade become one of the most recognized graphic artists of the modern era? And why did the art establishment take so long to catch up with everyone else? The answers run through the tiled walls of a Moorish palace, a chance exchange of letters with a Cambridge mathematician, and a sustained, obsessive investigation into the nature of infinity itself.

• 2. An Unlikely Beginning

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  Known to family and friends as "Mauk", Escher struggled from the start in ways that gave little hint of what was coming. He was placed in a special school at the age of seven, and he failed the second grade outright. His grades were generally poor, though he excelled at drawing. He took carpentry and piano lessons until he was thirteen.

  In 1918 he enrolled at the Technical College of Delft, then moved to the Haarlem School of Architecture and Decorative Arts from 1919 to 1922. He started in architecture but failed several subjects, partly because of a persistent skin infection, and switched to decorative arts. There he studied under the graphic artist Samuel Jessurun de Mesquita, learning woodcut-making, a technique that would define much of his mature output.

  His father was the civil engineer George Arnold Escher. That practical, structural sensibility seems to have passed to the youngest son not as a gift for engineering but as an instinct for order, for pattern, for the hidden geometry that holds things together.

• 3. The Alhambra Obsession

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  In 1922, an important year by Escher's own reckoning, he traveled through Italy and Spain, visiting Florence, San Gimignano, Volterra, Siena, Ravello, Madrid, Toledo, and Granada. In Granada, the Moorish architecture of the fourteenth-century Alhambra stopped him cold. Its interlocking repeating patterns in coloured tiles and sculpted walls and ceilings set off something he could not then fully articulate.

  He settled in Rome from 1923 to 1935, married a Swiss woman named Jetta Umiker in 1924, and traveled widely across Italy throughout the late 1920s and early 1930s. But the Alhambra kept pulling at him. In May and June of 1936 he went back to Spain expressly to make detailed drawings of its mosaic patterns, spending days at a stretch on the work. He described tessellation in words that do not sound like a detached observer: "It remains an extremely absorbing activity, a real mania to which I have become addicted, and from which I sometimes find it hard to tear myself away."

  That same 1936 journey took him to the Mezquita, the Moorish mosque of Cordoba. It would be his last long study trip abroad. After 1937, his art was made in the studio, not the field, and the shift showed immediately: his work turned from careful observation of the physical world to the systematic working-out of geometric ideas that existed entirely in his mind.

• 4. Tessellation as Research

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  Before the Alhambra journeys, Escher's brother Berend, a geologist, sent him papers by the mathematician George Pólya and the crystallographer Friedrich Haag on plane symmetry groups. Escher studied the 17 canonical wallpaper groups carefully and produced periodic tilings documented in 43 drawings of different symmetry types.

  His 1939 pencil, India ink, and watercolour Study of Regular Division of the Plane with Reptiles, built on a hexagonal grid, shows exactly what he was doing: the heads of red, green, and white reptiles meet at a single vertex, while their tails, legs, and sides interlock without gap or overlap. He used it as the basis for his 1943 lithograph Reptiles.

  In 1941 and 1942 he compiled a private sketchbook labeled, following Haag, Regelmatige vlakverdeling in asymmetrische congruente veelhoeken, meaning "Regular division of the plane with asymmetric congruent polygons." The mathematician Doris Schattschneider later described this notebook as recording "a methodical investigation that can only be termed mathematical research." The questions Escher was investigating, she wrote, were formal ones: what shapes can tile a plane such that every tile is surrounded in exactly the same way, and how do the edges of such a tile relate to each other by isometries? His 1958 book, titled Regular Division of the Plane, published reproductions of woodcuts built from these ideas and described the systematic buildup of mathematical designs in his work.

• 5. Impossible Worlds

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  Escher's first print of an impossible reality was Still Life and Street in 1937. Impossible stairs and multiple gravitational viewpoints appear in works such as Relativity, made in 1953. House of Stairs, from 1951, caught the attention of the mathematician Roger Penrose and his father, the biologist Lionel Penrose. In 1956 they published a paper titled "Impossible Objects: A Special Type of Visual Illusion" and sent Escher a copy. He replied, admiring their continuously rising staircase, and enclosed a print of Ascending and Descending, made in 1960.

  The Penroses' paper contained the tribar, now called the Penrose triangle. Escher used it repeatedly in his lithograph Waterfall, finished in 1961, which depicts a building that appears to function as a perpetual motion machine. The two towers of that building are topped with compound polyhedra: one is a compound of three cubes, the other a stellated rhombic dodecahedron now known as Escher's solid.

  Escher insisted he was not purely a geometer. In his own words he was a "reality enthusiast" who combined "formal astonishment with a vivid and idiosyncratic vision." His lithograph Belvedere, made in 1958, is peopled with "jesters, knaves, and contemplators," and reuses a figure of a medieval woman in a two-pointed headdress that he had first drawn from Hieronymus Bosch's 1500 triptych The Garden of Earthly Delights.

• 6. Coxeter and the Circle Limits

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  In 1954, the International Congress of Mathematicians met in Amsterdam. N. G. de Bruin organised a display of Escher's work at the Stedelijk Museum for the participants, and both Roger Penrose and H. S. M. Coxeter were struck by Escher's intuitive grasp of mathematics.

  In 1957 Coxeter obtained Escher's permission to use two of his drawings in a paper titled "Crystal symmetry and its generalizations." He sent Escher a copy. Escher recorded that Coxeter's figure of a hyperbolic tessellation "gave me quite a shock": the infinite regular repetition of tiles in the hyperbolic plane, growing rapidly smaller toward the edge of a circle, was exactly what he had been searching for to represent infinity on a flat surface.

  Escher marked up Coxeter's figure to work out how the successively smaller circles had been constructed, then sent Coxeter a diagram showing his analysis. Coxeter confirmed it was correct, but his reply was highly technical and disappointed Escher. Escher pressed on regardless, calling the practice "Coxetering." The result was the series of wood engravings Circle Limit I-IV. In 1959, Coxeter published his assessment: "Escher got it absolutely right to the millimeter."

• 7. Recognition and Legacy

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  Escher's fame in popular culture accelerated when Martin Gardner featured his work in the April 1966 "Mathematical Games" column in Scientific American. His prints appeared on album covers including The Scaffold's 1969 L the P, Mott the Hoople's 1969 eponymous record, Beaver and Krause's 1970 In A Wild Sanctuary, and Mandrake Memorial's 1970 Puzzle. He was one of the central inspirations for Douglas Hofstadter's Pulitzer Prize-winning 1979 book Gödel, Escher, Bach, which explores self-reference and strange loops.

  The art establishment was slower. His work was considered too intellectual and insufficiently lyrical by traditional critics, who also disliked his narrative themes and his handling of perspective. Even movements such as Cubism, De Stijl, Dadaism, and Surrealism, which explored multiple simultaneous viewpoints, developed without Escher making contact with any of them. He never aligned himself with any movement.

  Doris Schattschneider identifies eleven strands of mathematical and scientific research that Escher either anticipated or directly inspired, ranging from the classification of regular tilings to the filling of the central void in his lithograph Print Gallery, which was eventually solved mathematically by H. Lenstra and B. de Smit. The asteroid 4444 Escher was named in his honor in 1985. His last work, a large woodcut with threefold rotational symmetry called Snakes, was finished in July 1969; it required nine separate print operations per finished print, using three blocks each rotated three times. Escher moved to the Rosa Spier Huis artists' retirement home in Laren in 1970 and died in a hospital in Hilversum on the 27th of March 1972, aged 73.