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Maurits Cornelis Escher (; 17 June 1898 – 27 March 1972) was a Dutch graphic artist who made mathematically inspired
woodcut Woodcut is a relief printing technique in printmaking. An artist carves an image into the surface of a block of wood—typically with gouges—leaving the printing parts level with the surface while removing the non-printing parts. Areas tha ...
s, lithographs, and mezzotints. Despite wide popular interest, Escher was for most of his life neglected in the art world, even in his native Netherlands. He was 70 before a retrospective exhibition was held. In the late twentieth century, he became more widely appreciated, and in the twenty-first century he has been celebrated in exhibitions around the world. His work features mathematical objects and operations including
impossible object An impossible object (also known as an impossible figure or an undecidable figure) is a type of optical illusion that consists of a two-dimensional figure which is instantly and naturally understood as representing a projection of a three-di ...
s, explorations of infinity, reflection,
symmetry Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...
, perspective, truncated and stellated polyhedra,
hyperbolic geometry In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai–Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: :For any given line ''R'' and point ''P ...
, and
tessellation A tessellation or tiling is the covering of a surface, often a plane, using one or more geometric shapes, called ''tiles'', with no overlaps and no gaps. In mathematics, tessellation can be generalized to higher dimensions and a variety of ...
s. Although Escher believed he had no mathematical ability, he interacted with the mathematicians George Pólya,
Roger Penrose Sir Roger Penrose (born 8 August 1931) is an English mathematician, mathematical physicist, philosopher of science and Nobel Laureate in Physics. He is Emeritus Rouse Ball Professor of Mathematics in the University of Oxford, an emeritus f ...
, Harold Coxeter and crystallographer Friedrich Haag, and conducted his own research into
tessellation A tessellation or tiling is the covering of a surface, often a plane, using one or more geometric shapes, called ''tiles'', with no overlaps and no gaps. In mathematics, tessellation can be generalized to higher dimensions and a variety of ...
. Early in his career, he drew inspiration from nature, making studies of insects, landscapes, and plants such as
lichen A lichen ( , ) is a composite organism that arises from algae or cyanobacteria living among filaments of multiple fungi species in a mutualistic relationship.Alhambra The Alhambra (, ; ar, الْحَمْرَاء, Al-Ḥamrāʾ, , ) is a palace and fortress complex located in Granada, Andalusia, Spain. It is one of the most famous monuments of Islamic architecture and one of the best-preserved palaces of ...
and the Mezquita of Cordoba, and became steadily more interested in their
mathematical structure In mathematics, a structure is a set endowed with some additional features on the set (e.g. an operation, relation, metric, or topology). Often, the additional features are attached or related to the set, so as to provide it with some additiona ...
. Escher's art became well known among scientists and mathematicians, and in popular culture, especially after it was featured by Martin Gardner in his April 1966 Mathematical Games column in ''
Scientific American ''Scientific American'', informally abbreviated ''SciAm'' or sometimes ''SA'', is an American popular science magazine. Many famous scientists, including Albert Einstein and Nikola Tesla, have contributed articles to it. In print since 1845, it ...
''. Apart from being used in a variety of technical papers, his work has appeared on the covers of many books and albums. He was one of the major inspirations of
Douglas Hofstadter Douglas Richard Hofstadter (born February 15, 1945) is an American scholar of cognitive science, physics, and comparative literature whose research includes concepts such as the sense of self in relation to the external world, consciousness, a ...
's
Pulitzer Prize The Pulitzer Prize () is an award for achievements in newspaper, magazine, online journalism, literature, and musical composition within the United States. It was established in 1917 by provisions in the will of Joseph Pulitzer, who had made ...
-winning 1979 book ''
Gödel, Escher, Bach ''Gödel, Escher, Bach: an Eternal Golden Braid'', also known as ''GEB'', is a 1979 book by Douglas Hofstadter. By exploring common themes in the lives and works of logician Kurt Gödel, artist M. C. Escher, and composer Johann Sebastian Bach, t ...
''.


Early life

Maurits Cornelis Escher was born on 17 June 1898 in
Leeuwarden Leeuwarden (; fy, Ljouwert, longname=yes /; Town Frisian: ''Liwwadden''; Leeuwarder dialect: ''Leewarden'') is a city and municipality in Friesland, Netherlands, with a population of 123,107 (2019). It is the provincial capital and seat of th ...
,
Friesland Friesland (, ; official fry, Fryslân ), historically and traditionally known as Frisia, is a province of the Netherlands located in the country's northern part. It is situated west of Groningen, northwest of Drenthe and Overijssel, north of ...
, the Netherlands, in a house that forms part of the Princessehof Ceramics Museum today. He was the youngest son of the civil engineer George Arnold Escher and his second wife, Sara Gleichman. In 1903, the family moved to
Arnhem Arnhem ( or ; german: Arnheim; South Guelderish: ''Èrnem'') is a city and municipality situated in the eastern part of the Netherlands about 55 km south east of Utrecht. It is the capital of the province of Gelderland, located on both ban ...
, where he attended primary and secondary school until 1918. Known to his friends and family as "Mauk", he was a sickly child and was placed in a special school at the age of seven; he failed the second grade. Although he excelled at drawing, his grades were generally poor. He took
carpentry Carpentry is a skilled trade and a craft in which the primary work performed is the cutting, shaping and installation of building materials during the construction of buildings, ships, timber bridges, concrete formwork, etc. Carpenters t ...
and piano lessons until he was thirteen years old. In 1918, he went to the Technical College of Delft. From 1919 to 1922, Escher attended the
Haarlem Haarlem (; predecessor of ''Harlem'' in English) is a city and municipality in the Netherlands. It is the capital of the province of North Holland. Haarlem is situated at the northern edge of the Randstad, one of the most populated metropoli ...
School of Architecture and Decorative Arts, learning drawing and the art of making
woodcut Woodcut is a relief printing technique in printmaking. An artist carves an image into the surface of a block of wood—typically with gouges—leaving the printing parts level with the surface while removing the non-printing parts. Areas tha ...
s. He briefly studied
architecture Architecture is the art and technique of designing and building, as distinguished from the skills associated with construction. It is both the process and the product of sketching, conceiving, planning, designing, and constructing buildings ...
, but he failed a number of subjects (due partly to a persistent skin infection) and switched to
decorative arts ] The decorative arts are arts or crafts whose object is the design and manufacture of objects that are both beautiful and functional. It includes most of the arts making objects for the interiors of buildings, and interior design, but not usua ...
, studying under the graphic artist Samuel Jessurun de Mesquita.Locher, 1974. p. 5


Study journeys

In 1922, an important year of his life, Escher traveled through Italy, visiting
Florence Florence ( ; it, Firenze ) is a city in Central Italy and the capital city of the Tuscany region. It is the most populated city in Tuscany, with 383,083 inhabitants in 2016, and over 1,520,000 in its metropolitan area.Bilancio demografico ...
,
San Gimignano San Gimignano () is a small walled medieval hill town in the province of Siena, Tuscany, north-central Italy. Known as the Town of Fine Towers, San Gimignano is famous for its medieval architecture, unique in the preservation of about a dozen of ...
,
Volterra Volterra (; Latin: ''Volaterrae'') is a walled mountaintop town in the Tuscany region of Italy. Its history dates from before the 8th century BC and it has substantial structures from the Etruscan, Roman, and Medieval periods. History Volt ...
,
Siena Siena ( , ; lat, Sena Iulia) is a city in Tuscany, Italy. It is the capital of the province of Siena. The city is historically linked to commercial and banking activities, having been a major banking center until the 13th and 14th centur ...
, and
Ravello Ravello (Campanian: ) is a town and '' comune'' situated above the Amalfi Coast in the province of Salerno, Campania, Southern Italy, with approximately 2,500 inhabitants. Its scenic location makes it a popular tourist destination, and earned it ...
. In the same year, he traveled through Spain, visiting
Madrid Madrid ( , ) is the capital and most populous city of Spain. The city has almost 3.4 million inhabitants and a metropolitan area population of approximately 6.7 million. It is the second-largest city in the European Union (EU), and ...
, Toledo, and
Granada Granada (,, DIN: ; grc, Ἐλιβύργη, Elibýrgē; la, Illiberis or . ) is the capital city of the province of Granada, in the autonomous community of Andalusia, Spain. Granada is located at the foot of the Sierra Nevada mountains, at the c ...
. He was impressed by the Italian countryside and, in Granada, by the
Moorish architecture Moorish architecture is a style within Islamic architecture which developed in the western Islamic world, including al-Andalus (on the Iberian Peninsula, Iberian peninsula) and what is now Morocco, Algeria, and Tunisia (part of the Maghreb). The ...
of the fourteenth-century
Alhambra The Alhambra (, ; ar, الْحَمْرَاء, Al-Ḥamrāʾ, , ) is a palace and fortress complex located in Granada, Andalusia, Spain. It is one of the most famous monuments of Islamic architecture and one of the best-preserved palaces of ...
. The intricate decorative designs of the Alhambra, based on
geometrical Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
symmetries featuring interlocking repetitive patterns in the coloured tiles or sculpted into the walls and ceilings, triggered his interest in the mathematics of
tessellation A tessellation or tiling is the covering of a surface, often a plane, using one or more geometric shapes, called ''tiles'', with no overlaps and no gaps. In mathematics, tessellation can be generalized to higher dimensions and a variety of ...
and became a powerful influence on his work. Escher returned to Italy and lived in
Rome , established_title = Founded , established_date = 753 BC , founder = King Romulus ( legendary) , image_map = Map of comune of Rome (metropolitan city of Capital Rome, region Lazio, Italy).svg , map_caption ...
from 1923 to 1935. While in Italy, Escher met Jetta Umiker – a Swiss woman, like himself attracted to Italy – whom he married in 1924. The couple settled in Rome where their first son, Giorgio (George) Arnaldo Escher, named after his grandfather, was born. Escher and Jetta later had two more sons – Arthur and Jan. He travelled frequently, visiting (among other places)
Viterbo Viterbo (; Viterbese: ; lat-med, Viterbium) is a city and ''comune'' in the Lazio region of central Italy, the capital of the province of Viterbo. It conquered and absorbed the neighboring town of Ferento (see Ferentium) in its early history ...
in 1926, the
Abruzzi , population_note = , population_blank1_title = , population_blank1 = , demographics_type1 = , demographics1_footnotes = , demographics1_title1 = , demographics1 ...
in 1927 and 1929,
Corsica Corsica ( , Upper , Southern ; it, Corsica; ; french: Corse ; lij, Còrsega; sc, Còssiga) is an island in the Mediterranean Sea and one of the 18 regions of France. It is the fourth-largest island in the Mediterranean and lies southeast of ...
in 1928 and 1933,
Calabria , population_note = , population_blank1_title = , population_blank1 = , demographics_type1 = , demographics1_footnotes = , demographics1_title1 = , demographics1_info1 = , demographics1_title2 ...
in 1930, the
Amalfi Amalfi (, , ) is a town and '' comune'' in the province of Salerno, in the region of Campania, Italy, on the Gulf of Salerno. It lies at the mouth of a deep ravine, at the foot of Monte Cerreto (1,315 metres, 4,314 feet), surrounded by dramati ...
coast in 1931 and 1934, and
Gargano Gargano (, Gargano Apulian Italo-Romance arˈgæːnə is a historical and geographical sub-region in the province of Foggia, Apulia, southeast Italy, consisting of a wide isolated mountain massif made of highland and several peaks and forming ...
and
Sicily (man) it, Siciliana (woman) , population_note = , population_blank1_title = , population_blank1 = , demographics_type1 = Ethnicity , demographics1_footnotes = , demographi ...
in 1932 and 1935. The townscapes and landscapes of these places feature prominently in his artworks. In May and June 1936, Escher travelled back to Spain, revisiting the Alhambra and spending days at a time making detailed drawings of its mosaic patterns. It was here that he became fascinated, to the point of obsession, with tessellation, explaining: The sketches he made in the Alhambra formed a major source for his work from that time on. He also studied the architecture of the Mezquita, the Moorish mosque of Cordoba. This turned out to be the last of his long study journeys; after 1937, his artworks were created in his studio rather than in the field. His art correspondingly changed sharply from being mainly observational, with a strong emphasis on the realistic details of things seen in nature and architecture, to being the product of his geometric analysis and his visual imagination. All the same, even his early work already shows his interest in the nature of space, the unusual, perspective, and multiple points of view.


Later life

In 1935, the political climate in Italy under
Mussolini Benito Amilcare Andrea Mussolini (; 29 July 188328 April 1945) was an Italian politician and journalist who founded and led the National Fascist Party. He was Prime Minister of Italy from the March on Rome in 1922 until Fall of the Fascist re ...
became unacceptable to Escher. He had no interest in politics, finding it impossible to involve himself with any ideals other than the expressions of his own concepts through his own particular medium, but he was averse to fanaticism and hypocrisy. When his eldest son, George, was forced at the age of nine to wear a Ballila uniform in school, the family left Italy and moved to
Château-d'Œx Château-d'Œx () is a municipality in the canton of Vaud in Switzerland. It is in the district of Riviera-Pays-d'Enhaut. History Château-d'Œx is first mentioned in 1115 as ''Oit'', ''Oyz'', ''Oix'' and ''Oyez''. Prehistoric settlements Du ...
, Switzerland, where they remained for two years. The Netherlands post office had Escher design a semi-postal stamp for the "Air Fund" (Dutch: ''Het Nationaal Luchtvaartfonds'') in 1935, and again in 1949 he designed Dutch stamps. These were for the 75th anniversary of the
Universal Postal Union The Universal Postal Union (UPU, french: link=no, Union postale universelle), established by the Treaty of Bern of 1874, is a specialized agency of the United Nations (UN) that coordinates postal policies among member nations, in addition to ...
; a different design was used by
Suriname Suriname (; srn, Sranankondre or ), officially the Republic of Suriname ( nl, Republiek Suriname , srn, Ripolik fu Sranan), is a country on the northeastern Atlantic coast of South America. It is bordered by the Atlantic Ocean to the nor ...
and the
Netherlands Antilles nl, In vrijheid verenigd"Unified by freedom" , national_anthem = , common_languages = Dutch English Papiamento , demonym = Netherlands Antillean , capital = Willemstad , year_start = 1954 , year_end = 2010 , date_start = 15 December , ...
for the same commemoration. Escher, who had been very fond of and inspired by the landscapes in Italy, was decidedly unhappy in Switzerland. In 1937 the family moved again, to
Uccle Uccle () or Ukkel () is one of the 19 municipalities of the Brussels-Capital Region, Belgium. In common with all of Brussels' municipalities, it is legally bilingual (French–Dutch). It is generally considered an affluent area of the city an ...
(Ukkel), a suburb of
Brussels Brussels (french: Bruxelles or ; nl, Brussel ), officially the Brussels-Capital Region (All text and all but one graphic show the English name as Brussels-Capital Region.) (french: link=no, Région de Bruxelles-Capitale; nl, link=no, Bruss ...
, Belgium.
World War II World War II or the Second World War, often abbreviated as WWII or WW2, was a world war that lasted from 1939 to 1945. It involved the World War II by country, vast majority of the world's countries—including all of the great power ...
forced them to move in January 1941, this time to Baarn, Netherlands, where Escher lived until 1970. Most of Escher's best-known works date from this period. The sometimes cloudy, cold, and wet weather of the Netherlands allowed him to focus intently on his work. After 1953, Escher lectured widely. A planned series of lectures in North America in 1962 was cancelled after an illness, and he stopped creating artworks for a time, but the illustrations and text for the lectures were later published as part of the book ''Escher on Escher''. He was awarded the Knighthood of the
Order of Orange-Nassau The Order of Orange-Nassau ( nl, Orde van Oranje-Nassau, links=no) is a civil and military Dutch order of chivalry founded on 4 April 1892 by the queen regent, Emma of the Netherlands. The order is a chivalric order open to "everyone who has ...
in 1955; he was later made an Officer in 1967. In July 1969 he finished his last work, a large woodcut with threefold
rotational symmetry Rotational symmetry, also known as radial symmetry in geometry, is the property a shape has when it looks the same after some rotation by a partial turn. An object's degree of rotational symmetry is the number of distinct orientations in which ...
called ''
Snakes Snakes are elongated, limbless, carnivorous reptiles of the suborder Serpentes . Like all other squamates, snakes are ectothermic, amniote vertebrates covered in overlapping scales. Many species of snakes have skulls with several more j ...
'', in which snakes wind through a pattern of linked rings. These shrink to infinity toward both the center and the edge of a circle. It was exceptionally elaborate, being printed using three blocks, each rotated three times about the center of the image and precisely aligned to avoid gaps and overlaps, for a total of nine print operations for each finished print. The image encapsulates Escher's love of symmetry; of interlocking patterns; and, at the end of his life, of his approach to infinity. The care that Escher took in creating and printing this woodcut can be seen in a video recording. Escher moved to the Rosa Spier Huis in Laren in 1970, an artists' retirement home in which he had his own studio. He died in a hospital in
Hilversum Hilversum () is a city and municipality in the province of North Holland, Netherlands. Located in the heart of the Gooi, it is the largest urban centre in that area. It is surrounded by heathland, woods, meadows, lakes, and smaller towns. Hilver ...
on 27 March 1972, aged 73. He is buried at the New Cemetery in Baarn.


Mathematically inspired work

Escher's work is inescapably mathematical. This has caused a disconnect between his full-on popular fame and the lack of esteem with which he has been viewed in the art world. His originality and mastery of graphic techniques are respected, but his works have been thought too intellectual and insufficiently lyrical. Movements such as
conceptual art Conceptual art, also referred to as conceptualism, is art in which the concept(s) or idea(s) involved in the work take precedence over traditional aesthetic, technical, and material concerns. Some works of conceptual art, sometimes called ins ...
have, to a degree, reversed the art world's attitude to intellectuality and lyricism, but this did not rehabilitate Escher, because traditional critics still disliked his narrative themes and his use of perspective. However, these same qualities made his work highly attractive to the public.Locher, 1974. p. 13 Escher is not the first artist to explore mathematical themes:
Parmigianino Girolamo Francesco Maria Mazzola (11 January 150324 August 1540), also known as Francesco Mazzola or, more commonly, as Parmigianino (, , ; "the little one from Parma"), was an Italian Mannerist painter and printmaker active in Florence, Rome, B ...
(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 William Hogarth (; 10 November 1697 – 26 October 1764) was an English painter, engraver, pictorial satirist, social critic, editorial cartoonist and occasional writer on art. His work ranges from realistic portraiture to comic strip-like ...
's 1754 '' Satire on False Perspective'' foreshadows Escher's playful exploration of errors in perspective. Another early artistic forerunner is Giovanni Battista Piranesi (1720–1778), whose dark "fantastical" prints such as ''The Drawbridge'' in his ''Carceri'' ("Prisons") sequence depict perspectives of complex architecture with many stairs and ramps, peopled by walking figures. Escher greatly admired Piranesi and had several of Piranesi's prints hanging in his studio. Only with 20th century movements such as
Cubism Cubism is an early-20th-century avant-garde art movement that revolutionized European painting and sculpture, and inspired related movements in music, literature and architecture. In Cubist artwork, objects are analyzed, broken up and reassemble ...
, De Stijl, Dadaism, and
Surrealism Surrealism is a cultural movement that developed in Europe in the aftermath of World War I in which artists depicted unnerving, illogical scenes and developed techniques to allow the unconscious mind to express itself. Its aim was, according to ...
did mainstream art start to explore Escher-like ways of looking at the world with multiple simultaneous viewpoints. However, although Escher had much in common with, for example, Magritte's surrealism and
Op art Op art, short for optical art, is a style of visual art that uses optical illusions. Op artworks are abstract, with many better-known pieces created in black and white. Typically, they give the viewer the impression of movement, hidden image ...
, he did not make contact with any of these movements. File:Parmigianino Selfportrait.jpg, Forerunner of Escher's curved perspectives, geometries, and reflections:
Parmigianino Girolamo Francesco Maria Mazzola (11 January 150324 August 1540), also known as Francesco Mazzola or, more commonly, as Parmigianino (, , ; "the little one from Parma"), was an Italian Mannerist painter and printmaker active in Florence, Rome, B ...
's '' Self-portrait in a Convex Mirror'', 1524 File:William Hogarth - Absurd perspectives.png, Forerunner of Escher's impossible perspectives:
William Hogarth William Hogarth (; 10 November 1697 – 26 October 1764) was an English painter, engraver, pictorial satirist, social critic, editorial cartoonist and occasional writer on art. His work ranges from realistic portraiture to comic strip-like ...
's '' Satire on False Perspective'', 1753 File:Giovanni Battista Piranesi - The Drawbridge - Google Art Project.jpg, Forerunner of Escher's fantastic endless stairs:
Piranesi Giovanni Battista (or Giambattista) Piranesi (; also known as simply Piranesi; 4 October 1720 – 9 November 1778) was an Italian Classical archaeologist, architect, and artist, famous for his etchings of Rome and of fictitious and atmospheri ...
's '' Carceri'' Plate VII – The Drawbridge, 1745, reworked 1761


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 Roman or Romans most often refers to: * Rome, the capital city of Italy * Ancient Rome, Roman civilization from 8th century BC to 5th century AD *Roman people, the people of ancient Rome *''Epistle to the Romans'', shortened to ''Romans'', a lett ...
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, "
Mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History On ...
s have opened the gate leading to an extensive domain". After his 1936 journey to the
Alhambra The Alhambra (, ; ar, الْحَمْرَاء, Al-Ḥamrāʾ, , ) is a palace and fortress complex located in Granada, Andalusia, Spain. It is one of the most famous monuments of Islamic architecture and one of the best-preserved palaces of ...
and to La Mezquita, Cordoba, where he sketched the
Moorish The term Moor, derived from the ancient Mauri, is an exonym first used by Christian Europeans to designate the Muslim inhabitants of the Maghreb, the Iberian Peninsula, Sicily and Malta during the Middle Ages. Moors are not a distinct or s ...
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 interlock exactly. It was used as the basis for his 1943 lithograph ''
Reptiles Reptiles, as most commonly defined are the animals in the class Reptilia ( ), a paraphyletic grouping comprising all sauropsids except birds. Living reptiles comprise turtles, crocodilians, squamates ( lizards and snakes) and rhynchoceph ...
''. His first study of mathematics began with papers by George Pólya and by the crystallographer Friedrich Haag on plane
symmetry group In group theory, the symmetry group of a geometric object is the group of all transformations under which the object is invariant, endowed with the group operation of composition. Such a transformation is an invertible mapping of the amb ...
s, sent to him by his brother Berend, a geologist. He carefully studied the 17 canonical
wallpaper group A wallpaper is a mathematical object covering a whole Euclidean plane by repeating a motif indefinitely, in manner that certain isometries keep the drawing unchanged. To a given wallpaper there corresponds a group of such congruent transformati ...
s and created periodic tilings with 43 drawings of different types of symmetry. From this point on, he developed a mathematical approach to expressions of symmetry in his artworks 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 almost seven metres long. 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 Doris J. Schattschneider (née Wood) is an American mathematician, a retired professor of mathematics at Moravian College. She is known for writing about tessellations and about the art of M. C. Escher,.. for helping Martin Gardner validate and ...
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 Corsica ( , Upper , Southern ; it, Corsica; ; french: Corse ; lij, Còrsega; sc, Còssiga) is an island in the Mediterranean Sea and one of the 18 regions of France. It is the fourth-largest island in the Mediterranean and lies southeast of ...
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 Sir Roger Penrose (born 8 August 1931) is an English mathematician, mathematical physicist, philosopher of science and Nobel Laureate in Physics. He is Emeritus Rouse Ball Professor of Mathematics in the University of Oxford, an emeritus f ...
and his father, the biologist
Lionel Penrose Lionel Sharples Penrose, FRS (11 June 1898 – 12 May 1972) was an English psychiatrist, medical geneticist, paediatrician, mathematician and chess theorist, who carried out pioneering work on the genetics of intellectual disability. Penr ...
. 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 Perpetual motion is the motion of bodies that continues forever in an unperturbed system. A perpetual motion machine is a hypothetical machine that can do work infinitely without an external energy source. This kind of machine is impossible, a ...
machine, ''
Waterfall A waterfall is a point in a river or stream where water flows over a vertical drop or a series of steep drops. Waterfalls also occur where meltwater drops over the edge of a tabular iceberg or ice shelf. Waterfalls can be formed in several ...
'' (1961). Chapter 5 is on Escher. 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
Mediaeval In the history of Europe, the Middle Ages or medieval period lasted approximately from the late 5th to the late 15th centuries, similar to the post-classical period of global history. It began with the fall of the Western Roman Empire a ...
woman in a two-pointed headdress and a long gown in his lithograph ''
Belvedere Belvedere (from Italian, meaning "beautiful sight") may refer to: Places Australia *Belvedere, Queensland, a locality in the Cassowary Coast Region Africa * Belvedere (Casablanca), a neighborhood in Casablanca, Morocco * Belvedere, Harare, Z ...
'' in 1958; the image is, like many of his other "extraordinary invented places", peopled with "
jester A jester, court jester, fool or joker was a member of the household of a nobleman or a monarch employed to entertain guests during the medieval and Renaissance eras. Jesters were also itinerant performers who entertained common folk at fairs ...
s, knaves, and contemplators". Thus, Escher not only was 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". Escher worked primarily in the media of lithographs and
woodcut Woodcut is a relief printing technique in printmaking. An artist carves an image into the surface of a block of wood—typically with gouges—leaving the printing parts level with the surface while removing the non-printing parts. Areas tha ...
s, although 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 such as the
Möbius strip In mathematics, a Möbius strip, Möbius band, or Möbius loop is a surface that can be formed by attaching the ends of a strip of paper together with a half-twist. As a mathematical object, it was discovered by Johann Benedict Listing and A ...
, 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: which cites 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 The Mediterranean Sea is a sea connected to the Atlantic Ocean, surrounded by the Mediterranean Basin and almost completely enclosed by land: on the north by Western and Southern Europe and Anatolia, on the south by North Africa, and on ...
, 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". which cites Escher's interest in
curvilinear perspective Curvilinear perspective, also five-point perspective, is a graphical projection used to draw 3D objects on 2D surfaces. It was formally codified in 1968 by the artists and art historians André Barre and Albert Flocon in the book ''La Perspective c ...
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 Leonardo di ser Piero da Vinci (15 April 14522 May 1519) was an Italian polymath of the High Renaissance who was active as a painter, draughtsman, engineer, scientist, theorist, sculptor, and architect. While his fame initially rested on ...
,
Albrecht Dürer Albrecht Dürer (; ; hu, Ajtósi Adalbert; 21 May 1471 – 6 April 1528),Müller, Peter O. (1993) ''Substantiv-Derivation in Den Schriften Albrecht Dürers'', Walter de Gruyter. . sometimes spelled in English as Durer (without an umlaut) or Due ...
, 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 Curvilinear perspective, also five-point perspective, is a graphical projection used to draw 3D objects on 2D surfaces. It was formally codified in 1968 by the artists and art historians André Barre and Albert Flocon in the book ''La Perspective c ...
").


Platonic and other solids

Escher often incorporated three-dimensional objects such as the
Platonic solid In geometry, a Platonic solid is a convex, regular polyhedron in three-dimensional Euclidean space. Being a regular polyhedron means that the faces are congruent (identical in shape and size) regular polygons (all angles congruent and all e ...
s such as spheres, tetrahedrons, and cubes into his works, as well as mathematical objects such as
cylinders A cylinder (from ) has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes. In elementary geometry, it is considered a prism with a circle as its base. A cylinder may also be defined as an in ...
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 Doris J. Schattschneider (née Wood) is an American mathematician, a retired professor of mathematics at Moravian College. She is known for writing about tessellations and about the art of M. C. Escher,.. for helping Martin Gardner validate and ...
and scientists such as
Roger Penrose Sir Roger Penrose (born 8 August 1931) is an English mathematician, mathematical physicist, philosopher of science and Nobel Laureate in Physics. He is Emeritus Rouse Ball Professor of Mathematics in the University of Oxford, an emeritus f ...
, who enjoy his use of
polyhedra In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices. A convex polyhedron is the convex hull of finitely many points, not all on ...
and
geometric Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ca ...
distortions. For example, in ''
Gravitation In physics, gravity () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interactions, approximately 1038 times weaker than the stron ...
'', 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 also contains all five of the
Platonic solid In geometry, a Platonic solid is a convex, regular polyhedron in three-dimensional Euclidean space. Being a regular polyhedron means that the faces are congruent (identical in shape and size) regular polygons (all angles congruent and all e ...
s and various stellated solids, representing stars; the central solid is animated by
chameleon Chameleons or chamaeleons (family Chamaeleonidae) are a distinctive and highly specialized clade of Old World lizards with 202 species described as of June 2015. The members of this family are best known for their distinct range of colors, bein ...
s climbing through the frame as it whirls in space. Escher possessed a 6 cm
refracting telescope A refracting telescope (also called a refractor) is a type of optical telescope that uses a lens as its objective to form an image (also referred to a dioptric telescope). The refracting telescope design was originally used in spyglasses an ...
and was a keen-enough amateur
astronomer An astronomer is a scientist in the field of astronomy who focuses their studies on a specific question or field outside the scope of Earth. They observe astronomical objects such as stars, planets, moons, comets and galaxies – in either ...
to have recorded observations of
binary star A binary star is a system of two stars that are gravitationally bound to and in orbit around each other. Binary stars in the night sky that are seen as a single object to the naked eye are often resolved using a telescope as separate stars, in ...
s.


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 ''Drawing Hands'' is a lithograph by the Dutch artist M. C. Escher first printed in January 1948. It depicts a sheet of paper, out of which two hands rise, in the paradoxical act of drawing one another into existence. This is one of the mos ...
'' (1948), where two hands are shown, each drawing the other. The critic Steven Poole commented that


Infinity and hyperbolic geometry

In 1954 the International Congress of Mathematicians met in Amsterdam, and N. G. de Bruin organised a display of Escher's work at the Stedelijk Museum for the participants. Both Roger Penrose and
H. S. M. Coxeter Harold Scott MacDonald "Donald" Coxeter, (9 February 1907 – 31 March 2003) was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century. Biography Coxeter was born in Kensington t ...
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 Perpetual motion is the motion of bodies that continues forever in an unperturbed system. A perpetual motion machine is a hypothetical machine that can do work infinitely without an external energy source. This kind of machine is impossible, a ...
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 In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai– Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: :For any given line ''R'' and point ' ...
, growing rapidly smaller towards the edge of the circle, was precisely what he wanted to allow him to represent
infinity Infinity is that which is boundless, endless, or larger than any natural number. It is often denoted by the infinity symbol . Since the time of the ancient Greeks, the philosophical nature of infinity was the subject of many discussions am ...
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". 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". which cites


Legacy

Escher's special way of thinking and rich graphics have had a continuous influence in mathematics and art, as well as
in popular culture Popular culture (also called mass culture or pop culture) is generally recognized by members of a society as a set of practices, beliefs, artistic output (also known as, popular art or mass art) and objects that are dominant or prevalent in a ...
.


In art collections

The Escher intellectual property is controlled by the M.C. Escher Company, while exhibitions of his artworks are managed separately by the M.C. Escher Foundation. The primary institutional collections of original works by M.C. Escher are the
Escher Museum Escher in Het Paleis (''Escher in The Palace'') is a museum in The Hague, Netherlands, featuring the works of the Dutch graphical artist M. C. Escher. It is housed in the Lange Voorhout Palace since November 2002. In 2015 it was revealed tha ...
in
The Hague The Hague ( ; nl, Den Haag or ) is a city and municipality of the Netherlands, situated on the west coast facing the North Sea. The Hague is the country's administrative centre and its seat of government, and while the official capital o ...
; the
National Gallery of Art The National Gallery of Art, and its attached Sculpture Garden, is a national art museum in Washington, D.C., United States, located on the National Mall, between 3rd and 9th Streets, at Constitution Avenue NW. Open to the public and free of ch ...
(Washington, DC); the
National Gallery of Canada The National Gallery of Canada (french: Musée des beaux-arts du Canada), located in the capital city of Ottawa, Ontario, is Canada's national art museum. The museum's building takes up , with of space used for exhibiting art. It is one of the ...
(Ottawa); the
Israel Museum The Israel Museum ( he, מוזיאון ישראל, ''Muze'on Yisrael'') is an art and archaeological museum in Jerusalem. It was established in 1965 as Israel's largest and foremost cultural institution, and one of the world’s leading encyclopa ...
(Jerusalem); and the
Huis ten Bosch Huis ten Bosch ( nl, Paleis Huis ten Bosch, ; English: "House in the Woods") is a royal palace in The Hague, Netherlands. It is one of three official residences of the Dutch monarch; the two others being the Noordeinde Palace in The Hague a ...
(Nagasaki, Japan).


Exhibitions

Despite wide popular interest, Escher was for a long time somewhat neglected in the art world; even in his native Netherlands, he was 70 before a retrospective exhibition was held. In the twenty-first century, major exhibitions have been held in cities around the world. An exhibition of his work in Rio de Janeiro attracted more than 573,000 visitors in 2011; its daily visitor count of 9,677 made it the most visited museum exhibition of the year, anywhere in the world. No major exhibition of Escher's work was held in Britain until 2015, when the Scottish National Gallery of Modern Art ran one in
Edinburgh Edinburgh ( ; gd, Dùn Èideann ) is the capital city of Scotland and one of its 32 Council areas of Scotland, council areas. Historically part of the county of Midlothian (interchangeably Edinburghshire before 1921), it is located in Lothian ...
from June to September 2015, moving in October 2015 to the Dulwich Picture Gallery, London. The exhibition poster is based on ''
Hand with Reflecting Sphere ''Hand with Reflecting Sphere'', also known as ''Self-Portrait in Spherical Mirror'', is a lithograph by Dutch artist M. C. Escher, first printed in January 1935. The piece depicts a hand holding a reflective sphere. In the reflection, mos ...
'', 1935, which shows Escher in his house reflected in a handheld sphere, thus illustrating the artist, his interest in levels of reality in art (e.g., is the hand in the foreground more real than the reflected one?), perspective, and
spherical geometry 300px, A sphere with a spherical triangle on it. Spherical geometry is the geometry of the two-dimensional surface of a sphere. In this context the word "sphere" refers only to the 2-dimensional surface and other terms like "ball" or "solid sp ...
. The exhibition moved to Italy in 2015–2016, attracting over 500,000 visitors in Rome and Bologna, and then
Milan Milan ( , , Lombard: ; it, Milano ) is a city in northern Italy, capital of Lombardy, and the second-most populous city proper in Italy after Rome. The city proper has a population of about 1.4 million, while its metropolitan city ...
.


In mathematics and science

Doris Schattschneider Doris J. Schattschneider (née Wood) is an American mathematician, a retired professor of mathematics at Moravian College. She is known for writing about tessellations and about the art of M. C. Escher,.. for helping Martin Gardner validate and ...
identifies eleven strands of mathematical and scientific research anticipated or directly inspired by Escher. These are the classification of regular tilings using the edge relationships of tiles: two-color and two-motif tilings (counterchange symmetry or antisymmetry); color symmetry (in
crystallography Crystallography is the experimental science of determining the arrangement of atoms in crystalline solids. Crystallography is a fundamental subject in the fields of materials science and solid-state physics ( condensed matter physics). The wor ...
); metamorphosis or topological change; covering surfaces with symmetric patterns; Escher's algorithm (for generating patterns using decorated squares); creating tile shapes; local versus global definitions of regularity; symmetry of a tiling induced by the symmetry of a tile; orderliness not induced by symmetry groups; the filling of the central void in Escher's lithograph '' Print Gallery'' by H. Lenstra and B. de Smit. The
Pulitzer Prize The Pulitzer Prize () is an award for achievements in newspaper, magazine, online journalism, literature, and musical composition within the United States. It was established in 1917 by provisions in the will of Joseph Pulitzer, who had made ...
-winning 1979 book ''
Gödel, Escher, Bach ''Gödel, Escher, Bach: an Eternal Golden Braid'', also known as ''GEB'', is a 1979 book by Douglas Hofstadter. By exploring common themes in the lives and works of logician Kurt Gödel, artist M. C. Escher, and composer Johann Sebastian Bach, t ...
'' by
Douglas Hofstadter Douglas Richard Hofstadter (born February 15, 1945) is an American scholar of cognitive science, physics, and comparative literature whose research includes concepts such as the sense of self in relation to the external world, consciousness, a ...
discusses the ideas of self-reference and
strange loop A strange loop is a cyclic structure that goes through several levels in a hierarchical system. It arises when, by moving only upwards or downwards through the system, one finds oneself back where one started. Strange loops may involve self-refer ...
s expressed in Escher's art. The
asteroid An asteroid is a minor planet of the inner Solar System. Sizes and shapes of asteroids vary significantly, ranging from 1-meter rocks to a dwarf planet almost 1000 km in diameter; they are rocky, metallic or icy bodies with no atmosphere. ...
4444 Escher was named in Escher's honor in 1985.


In popular culture

Escher's fame in popular culture grew when his work was featured by Martin Gardner in his April 1966 "Mathematical Games" column in ''
Scientific American ''Scientific American'', informally abbreviated ''SciAm'' or sometimes ''SA'', is an American popular science magazine. Many famous scientists, including Albert Einstein and Nikola Tesla, have contributed articles to it. In print since 1845, it ...
''. Escher's works have appeared on many album covers including The Scaffold's 1969 ''L the P'' with ''Ascending and Descending'';
Mott the Hoople Mott the Hoople were an English rock band formed in Herefordshire. Originally known as the Doc Thomas Group, the group changed their name after signing with Island Records in 1969. The band released albums throughout the early 1970s but fai ...
's eponymous 1969 record with ''Reptiles'',
Beaver & Krause Beaver & Krause were an American musical duo comprising Paul Beaver and Bernie Krause. Their 1967 album ''The Nonesuch Guide to Electronic Music'' was a pioneering work in the electronic music genre. The pair were Robert Moog's sales represent ...
's 1970 ''In A Wild Sanctuary'' with ''Three Worlds''; and Mandrake Memorial's 1970 ''Puzzle'' with ''House of Stairs'' and (inside) ''Curl Up''. His works have similarly been used on many book covers, including some editions of Edwin Abbott's ''Flatland'', which used ''Three Spheres'';
E. H. Gombrich Sir Ernst Hans Josef Gombrich (; ; 30 March 1909 – 3 November 2001) was an Austrian-born art historian who, after settling in England in 1936, became a naturalised British citizen in 1947 and spent most of his working life in the United King ...
's ''Meditations on a Hobby Horse'' with ''Horseman''; Pamela Hall's ''Heads You Lose'' with ''Plane Filling 1''; Patrick A. Horton's ''Mastering the Power of Story'' with ''Drawing Hands''; Erich Gamma et al.'s ''Design Patterns: Elements of Reusable Object-oriented software'' with ''Swans''; and Arthur Markman's ''Knowledge Representation'' with ''Reptiles''. The "World of Escher" markets
poster A poster is a large sheet that is placed either on a public space to promote something or on a wall as decoration. Typically, posters include both textual and graphic elements, although a poster may be either wholly graphical or wholly text ...
s,
necktie A necktie, or simply a tie, is a piece of cloth worn for decorative purposes around the neck, resting under the shirt collar and knotted at the throat, and often draped down the chest. Variants include the ascot, bow, bolo, zipper tie, crav ...
s,
T-shirt A T-shirt (also spelled tee shirt), or tee, is a style of fabric shirt named after the T shape of its body and sleeves. Traditionally, it has short sleeves and a round neckline, known as a '' crew neck'', which lacks a collar. T-shirts are genera ...
s, and
jigsaw puzzle A jigsaw puzzle is a tiling puzzle that requires the assembly of often irregularly shaped interlocking and mosaiced pieces, each of which typically has a portion of a picture. When assembled, the puzzle pieces produce a complete picture. In t ...
s of Escher's artworks. Both Austria and the Netherlands have issued
postage stamp A postage stamp is a small piece of paper issued by a post office, postal administration, or other authorized vendors to customers who pay postage (the cost involved in moving, insuring, or registering mail), who then affix the stamp to the f ...
s commemorating the artist and his works.


See also

*
Victor Vasarely Victor Vasarely (; born Győző Vásárhelyi, ; 9 April 1906 – 15 March 1997) was a Hungarian-French artist, who is widely accepted as a "grandfather" and leader of the Op art movement. His work entitled ''Zebra'', created in 1937, is consid ...
* Escher sentences, named after works like ''Ascending and Descending''


Notes


References


Further reading


Books

* * * * * * * * *


Media

* Escher, M. C. ''The Fantastic World of M. C. Escher'', Video collection of examples of the development of his art, and interviews, Director, Michele Emmer. * Phoenix Films & Video
Adventures in Perception (1973)
'


External links

* * * * * — physical replicas of some of Escher's "impossible" designs * * Copyright issue regarding Escher from the Artquest Artlaw archive.

at the
National Gallery of Canada The National Gallery of Canada (french: Musée des beaux-arts du Canada), located in the capital city of Ottawa, Ontario, is Canada's national art museum. The museum's building takes up , with of space used for exhibiting art. It is one of the ...
, Ottawa, Ontario. {{DEFAULTSORT:Escher, M. C. 1898 births 1972 deaths 20th-century Dutch artists Burials in Utrecht (province) Delft University of Technology alumni Dutch draughtsmen Dutch engravers Dutch illustrators Dutch male artists Dutch printmakers Dutch stamp designers Dutch wood engravers Mathematical artists Modern printmakers Officers of the Order of Orange-Nassau People from Leeuwarden People from Château-d'Œx 20th-century engravers