Much of Escher's work is inescapably mathematical. This has caused a disconnect between his fame among mathematicians and the general public, and the lack of esteem with which he has been viewed in the art world. Escher is not the first artist to explore mathematical themes: J. L. Locher, a previous director of the
Kunstmuseum in
The Hague, pointed out that
Parmigianino (1503–1540) had explored spherical geometry and reflection in his 1524
Self-portrait in a Convex Mirror, depicting his own image in a curved mirror, while
William Hogarth's 1754
Satire on False Perspective foreshadows Escher's playful exploration of errors in perspective. Escher greatly admired Piranesi and had several of Piranesi's prints hanging in his studio. Only with 20th century movements such as
Cubism,
De Stijl,
Dadaism, and
Surrealism did mainstream art start to explore Escher-like ways of looking at the world with multiple simultaneous viewpoints. File:Parmigianino Selfportrait.jpg|Forerunner of Escher's
curved perspectives, geometries, and reflections:
Parmigianino's
Self-portrait in a Convex Mirror, 1524 which appeared frequently in his later work. His early love of
Roman and Italian landscapes and of nature created an interest in tessellation, which he called
Regular Division of the Plane; this became the title of his 1958 book, complete with reproductions of a series of woodcuts based on tessellations of the plane, in which he described the systematic buildup of mathematical designs in his artworks. He wrote, "
crystallographers have opened the gate leading to an extensive domain". ''. After his 1936 journey to the
Alhambra and to
La Mezquita,
Cordoba, where he sketched the
Moorish architecture and the tessellated mosaic decorations, Escher began to explore tessellation using geometric grids as the basis for his sketches. He then extended these to form complex interlocking designs, for example with animals such as birds, fish, and reptiles. One of his first attempts at a tessellation was his pencil, India ink, and watercolour
Study of Regular Division of the Plane with Reptiles (1939), constructed on a hexagonal grid. The heads of the red, green, and white reptiles meet at a vertex; the tails, legs, and sides of the animals interlock exactly. It was used as the basis for his 1943 lithograph
Reptiles. His first study of mathematics began with papers by
George Pólya and by the crystallographer
Friedrich Haag on plane
symmetry groups, sent to him by his brother
Berend, a geologist. In 1941 and 1942, Escher summarised his findings for his own artistic use in a sketchbook, which he labeled (following Haag)
Regelmatige vlakverdeling in asymmetrische congruente veelhoeken ("Regular division of the plane with asymmetric congruent polygons"). The mathematician
Doris Schattschneider unequivocally described this notebook as recording "a methodical investigation that can only be termed mathematical research." She defined the research questions he was following as
Geometries Although Escher did not have mathematical training – his understanding of mathematics was largely visual and intuitive – his
art had a strong mathematical component, and several of the worlds that he drew were built around impossible objects. After 1924, Escher turned to sketching landscapes in Italy and
Corsica with irregular
perspectives that are impossible in natural form. His first print of an impossible reality was
Still Life and Street (1937); impossible stairs and multiple visual and gravitational perspectives feature in popular works such as
Relativity (1953).
House of Stairs (1951) attracted the interest of the mathematician
Roger Penrose and his father, the biologist
Lionel Penrose. In 1956, they published a paper, "Impossible Objects: A Special Type of Visual Illusion" and later sent Escher a copy. Escher replied, admiring the Penroses'
continuously rising flights of steps, and enclosed a print of
Ascending and Descending (1960). The paper contained the tribar or
Penrose triangle, which Escher used repeatedly in his lithograph of a building that appears to function as a
perpetual motion machine,
Waterfall (1961). Escher was interested enough in
Hieronymus Bosch's 1500 triptych
The Garden of Earthly Delights to re-create part of its right-hand panel,
Hell, as a lithograph in 1935. He reused the figure of a
Medieval woman in a two-pointed headdress and a long gown in his lithograph
Belvedere in 1958; the image is, like many of his other "extraordinary invented places", peopled with "
jesters,
knaves, and contemplators". Escher was fascinated by mathematical objects such as the
Möbius strip, which has only one surface. His wood engraving
Möbius Strip II (1963) depicts a chain of ants marching forever over what, at any one place, are the two opposite faces of the object—which are seen on inspection to be parts of the strip's single surface. In Escher's own words: The mathematical influence in his work became prominent after 1936, when, having boldly asked the Adria Shipping Company if he could sail with them as travelling artist in return for making drawings of their ships, they surprisingly agreed, and he sailed the
Mediterranean, becoming interested in order and symmetry. Escher described this journey, including his repeat visit to the Alhambra, as "the richest source of inspiration I have ever tapped". Escher's interest in
curvilinear perspective was encouraged by his friend and "kindred spirit", the art historian and artist Albert Flocon, in another example of constructive mutual influence. Flocon identified Escher as a "thinking artist"
Platonic and other solids , as in Escher's 1952 work
Gravitation (
University of Twente) Escher often incorporated three-dimensional objects such as the
Platonic solids such as spheres, tetrahedrons, and cubes into his works, as well as mathematical objects such as
cylinders and
stellated polyhedra. In the print
Reptiles, he combined two- and three-dimensional images. In one of his papers, Escher emphasized the importance of dimensionality: Escher's artwork is especially well-liked by mathematicians such as
Doris Schattschneider and scientists such as
Roger Penrose, who enjoy his use of
polyhedra and
geometric distortions. For example, in
Gravitation, animals climb around a
stellated dodecahedron. The two towers of
Waterfall impossible building are topped with compound polyhedra, one a
compound of three cubes, the other a stellated
rhombic dodecahedron now known as
Escher's solid. Escher had used this solid in his 1948 woodcut
Stars, which contains all five of the
Platonic solids and various stellated solids, representing stars; the central solid is animated by
chameleons climbing through the frame as it whirls in space. Escher possessed a 6 cm
refracting telescope and was a keen-enough amateur
astronomer to have recorded observations of
binary stars.
Levels of reality Escher's artistic expression was created from images in his mind, rather than directly from observations and travels to other countries. His interest in the multiple levels of reality in art is seen in works such as
Drawing Hands (1948), where two hands are shown, each drawing the other. The critic Steven Poole commented that
Infinity and hyperbolic geometry 's reconstruction of the diagram of hyperbolic tiling sent by Escher to the mathematician
Donald Coxeter He sent Escher a copy of the paper; Escher recorded that Coxeter's figure of a hyperbolic tessellation "gave me quite a shock": the infinite regular repetition of the tiles in the
hyperbolic plane, growing rapidly smaller towards the edge of the circle, was precisely what he wanted to allow him to represent
infinity on a two-dimensional plane. Escher carefully studied Coxeter's figure, marking it up to analyse the successively smaller circles with which (he deduced) it had been constructed. He then constructed a diagram, which he sent to Coxeter, showing his analysis; Coxeter confirmed it was correct, but disappointed Escher with his highly technical reply. All the same, Escher persisted with
hyperbolic tiling, which he called "Coxetering". == Legacy ==