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Further information: Mathematics and art
Forerunner of Escher's curved perspectives, geometries, and reflections: Parmigianino's Self-portrait in a Convex Mirror, 1524
Forerunner of Escher's fantastic endless stairs: Piranesi's Carceri Plate VII – The Drawbridge, 1745, reworked 1761
Further information: Tessellation
In his early years, Escher sketched landscapes and nature. He also sketched insects such as ants, bees, grasshoppers and mantises, 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 "Mathematicians have opened the gate leading to an extensive domain."
Hexagonal tessellation with animals: Study of Regular Division of the Plane with Reptiles (1939). Escher reused the design in his 1943 lithograph Reptiles.
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 the properties and possibilities of 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 exactly interlock. 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 (known as Beer). He carefully studied the 17 wallpaper groups, and created periodic tilings with 43 drawings of different types of symmetry.[c] From this point on he developed a mathematical approach to expressions of symmetry in his art works using his own notation. Starting in 1937, he created woodcuts based on the 17 groups. His Metamorphosis I (1937) began a series of designs that told a story through the use of pictures. In Metamorphosis I, he transformed convex polygons into regular patterns in a plane to form a human motif. He extended the approach in his piece Metamorphosis III, which is four metres long.
In 1941 and 1942, Escher summarized 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
(1) What are the possible shapes for a tile that can produce a regular division of the plane, that is, a tile that can fill the plane with its congruent images such that every tile is surrounded in the same manner?
(2) Moreover, in what ways are the edges of such a tile related to each other by isometries?
Further information: perspective (geometry) and curvilinear perspective
Multiple viewpoints and impossible stairs: Relativity, 1953
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 which 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 also 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 recreate part of its right-hand panel, Hell, as a lithograph in 1935. He reused the figure of a Mediaeval 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 thus not only interested in possible or impossible geometry, but was in his own words a "reality enthusiast"; he combined "formal astonishment with a vivid and idiosyncratic vision."
Perpetual motion machine with Penrose triangles: Waterfall, 1961
Escher worked primarily in the media of lithographs and woodcuts, though the few mezzotints he made are considered to be masterpieces of the technique. In his graphic art, he portrayed mathematical relationships among shapes, figures and space. Integrated into his prints were mirror images of cones, spheres, cubes, rings and spirals.
Escher was also fascinated by mathematical objects like the Möbius strip, which has only one surface. His wood engraving Möbius Strip II (1963) depicts a chain of ants marching for ever around 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
An endless ring-shaped band usually has two distinct surfaces, one inside and one outside. Yet on this strip nine red ants crawl after each other and travel the front side as well as the reverse side. Therefore the strip has only one surface.
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" alongside Piero della Francesca, Leonardo da Vinci, Albrecht Dürer, Wenzel Jamnitzer, Abraham Bosse, Girard Desargues, and Père Nicon. Flocon was delighted by Escher's Grafiek en tekeningen ("Graphics in Drawing"), which he read in 1959. This stimulated Flocon and André Barre to correspond with Escher, and to write the book La Perspective curviligne ("Curvilinear perspective").
Sculpture of the small stellated dodecahedron as in Escher's 1952 work Gravitation. University of Twente
Escher often incorporated three-dimensional objects such as the Platonic solids such as spheres, tetrahedons and cubes into his works, as well as mathematical objects like 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:
The flat shape irritates me - I feel like telling my objects, you are too fictitious, lying there next to each other static and frozen: do something, come off the paper and show me what you are capable of! ... So I make them come out of the plane. ... My objects ... may finally return to the plane and disappear into their place of origin.
Escher's artwork is especially well liked by mathematicians like Doris Schattschneider and scientists like 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's impossible building are topped with compound polyhedra, one a compound of three cubes, the other a stellated rhombic dodecahedron known as Escher's solid. Escher had used this solid in his 1948 woodcut Stars, which also 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.
Drawing Hands, 1948
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
It is a neat depiction of one of Escher's enduring fascinations: the contrast between the two-dimensional flatness of a sheet of paper and the illusion of three-dimensional volume that can be created with certain marks. In Drawing Hands, space and the flat plane coexist, each born from and returning to the other, the black magic of the artistic illusion made creepily manifest.
Doris Schattschneider's reconstruction of the diagram of hyperbolic tiling sent by Escher to the mathematician H. S. M. Coxeter
In 1954, the International Congress of Mathematicians met in Amsterdam, and N. G. de Bruin organized a display of Escher's work at the Stedelijk Museum for the participants. Both Roger Penrose and H. S. M. Coxeter were deeply impressed with Escher's intuitive grasp of mathematics. Inspired by Relativity, Penrose devised his tribar, and his father, Lionel Penrose, devised an endless staircase. Roger Penrose sent sketches of both objects to Escher, and the cycle of invention was closed when Escher then created the perpetual motion machine of Waterfall and the endless march of the monk-figures of Ascending and Descending. In 1957, Coxeter obtained Escher's permission to use two of his drawings in his paper "Crystal symmetry and its generalizations". 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[d] 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". Among the results were the series of wood engravings Circle Limit I–IV. In 1959, Coxeter published his finding that these works were extraordinarily accurate: "Escher got it absolutely right to the millimeter."
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