Editing M. C. Escher (section)

== Mathematically inspired work ==

{{further|Mathematics and art}}

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.<ref name=Locher13/><ref name="Scotsman 2015"/> 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]] 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.<ref name=Locher13>{{harvnb|Locher|1971|pp=13–14}}</ref>

Escher is not the first artist to explore mathematical themes: J. L. Locher, a previous director of the [[Kunstmuseum Den Haag|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.<ref name="Locher11">{{harvnb|Locher|1971|pp=11–12}}</ref><ref name=NGA/> Another early artistic forerunner is [[Giovanni Battista Piranesi]] (1720–1778), whose dark "fantastical"<ref name=Carnegie/> prints such as ''The Drawbridge'' in his [[Imaginary Prisons|''Carceri'' ("Prisons")]] sequence depict perspectives of complex architecture with many stairs and ramps, peopled by walking figures.<ref name=Carnegie>{{cite web |last=Altdorfer |first=John |title=Inside A Fantastical Mind |url=http://www.carnegiemuseums.org/cmag/article.php?id=123 |publisher=Carnegie Museums |access-date=7 November 2015 |archive-url=https://web.archive.org/web/20100706192452/http://www.carnegiemuseums.org/cmag/article.php?id=123 |archive-date=6 July 2010 |url-status=dead }}</ref><ref>{{cite magazine |last=McStay |first=Chantal |title=Oneiric Architecture and Opium |url=http://www.theparisreview.org/blog/tag/giovanni-battista-piranesi/ |magazine=[[The Paris Review]] |access-date=7 November 2015 |date=15 August 2014}}</ref> Escher greatly admired Piranesi and had several of Piranesi's prints hanging in his studio.<ref>{{cite web |title=Giovanni Battista Piranesi |url=https://www.escherinhetpaleis.nl/escher-today/giovanni-battista-piranesi/?lang=en |website=Escher in het Paleis |access-date=6 August 2022 |date=14 November 2020}}</ref><ref>{{cite book |last=Hazeu |first=Wim |title=M.C. Escher, Een biografie |language=Dutch |publisher=Meulenhoff |year=1998 |page=175}}</ref>

Only with 20th century movements such as [[Cubism]], [[De Stijl]], [[Dada]]ism, and [[Surrealism]] did mainstream art start to explore Escher-like ways of looking at the world with multiple simultaneous viewpoints.<ref name=Locher13 /> However, although Escher had much in common with, for example, [[Magritte]]'s surrealism and [[Op art]], he did not make contact with any of these movements.<ref name="Scotsman 2015">{{cite news |last=Mansfield |first=Susan |title=Escher, the master of impossible art |url=http://www.scotsman.com/lifestyle/arts/visual-arts/escher-the-master-of-impossible-art-1-3815073#axzz3qnqdWYGr |newspaper=[[The Scotsman]] |access-date=7 November 2015 |date=28 June 2015| archive-url=https://web.archive.org/web/20150701214226/http://www.scotsman.com/lifestyle/arts/visual-arts/escher-the-master-of-impossible-art-1-3815073#axzz3qnqdWYGr | archive-date=July 1, 2015}}</ref><ref name="Marcus 2022">{{cite news |last=Marcus |first=J. S. |author-link=J. S. Marcus |title=M.C. Escher's illusionist art has long been ignored by the establishment due to its mass appeal. A Houston show hopes to correct that |url=https://www.theartnewspaper.com/2022/03/11/mc-escher-poster-boy-of-illusionist-art |access-date=7 August 2022 |work=[[The Art Newspaper]] |date=11 March 2022 |quote=the art world proper has [been] inclined to regard Escher, whose finished prints share formal qualities with Surrealism and Op art, as somewhat derivative or merely decorative.}}</ref>

<gallery class=center mode=nolines widths=200px heights=200px caption="Forerunners of Escher identified by J. L. Locher">
File:Parmigianino Selfportrait.jpg|Forerunner of Escher's [[Curvilinear perspective|curved perspectives]], geometries, and reflections: [[Parmigianino]]'s ''[[Self-portrait in a Convex Mirror]]'', 1524<ref name="Locher11"/>
File:William Hogarth - Absurd perspectives.png|Forerunner of Escher's impossible perspectives: [[William Hogarth]]'s ''[[Satire on False Perspective]]'', 1753<ref name="Locher11"/>
File:Giovanni Battista Piranesi - The Drawbridge - Google Art Project.jpg|Forerunner of Escher's fantastic endless stairs: [[Giovanni Battista Piranesi|Piranesi]]'s ''[[Imaginary Prisons|Carceri]]'' Plate VII&nbsp;– The Drawbridge, 1745, reworked 1761<ref name="Locher11"/>
</gallery>

=== Tessellation ===

{{further|Tessellation}}

In his early years, Escher sketched landscapes and nature. He sketched insects such as ants, bees, grasshoppers, and mantises,<ref>{{harvnb|Locher|1971|pp=62–63}}</ref> which appeared frequently in his later work. His early love of [[Ancient Rome|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, "[[Crystallography|crystallographer]]s have opened the gate leading to an extensive domain".<ref name="Peterson's2012">{{cite book | last=Guy | first=R.K. | last2=Woodrow | first2=R.E. | title=The Lighter Side of Mathematics: Proceedings of the Eugene Strens Memorial Conference on Recreational Mathematics and Its History | publisher=Mathematical Association of America | series=Spectrum | year=2020 | isbn=978-1-4704-5731-0 | url=https://books.google.com/books?id=FsH2DwAAQBAJ&pg=PA92 | access-date=16 June 2024 | page=92}}</ref>

[[File:Study of Regular Division of the Plane with Reptiles.jpg|thumb|right|Hexagonal tessellation with animals: ''Study of Regular Division of the Plane with Reptiles'' (1939). Escher reused the design in his 1943 lithograph ''[[Reptiles (M. C. Escher)|Reptiles]]''.]]

After his 1936 journey to the [[Alhambra]] and to [[Mosque–Cathedral of Córdoba|La Mezquita]], [[Córdoba, Andalusia|Cordoba]], where he sketched the [[Moors|Moorish]] architecture and the tessellated mosaic decorations,<ref>{{harvnb|Locher|1971|pp=17, 70–71}}</ref> 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.<ref>{{harvnb|Locher|1971|pp=79–85}}</ref> 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 (M. C. Escher)|Reptiles]]''.<ref>{{harvnb|Locher|1971|p=18}}</ref>

His first study of mathematics began with papers by [[George Pólya]]<ref>{{cite journal |author=Pólya, G. |author-link=George Pólya |title=Über die Analogie der Kristallsymmetrie in der Ebene |journal=Zeitschrift für Kristallographie |volume=60 |year=1924 |issue=1–6 |pages=278–282 |language=de |doi=10.1524/zkri.1924.60.1.278|s2cid=102174323 }}</ref> and by the crystallographer [[Friedrich Haag]]<ref name=Haag>{{cite journal |author=Haag, Friedrich |title=Die regelmäßigen Planteilungen |language=de |journal=Zeitschrift für Kristallographie |volume=49 |year=1911 |issue=1–6 |pages=360–369 |url=https://zenodo.org/record/1448954 <!--open access--> |doi=10.1524/zkri.1911.49.1.360 |s2cid=100640309 }}</ref> on plane [[symmetry group]]s, sent to him by his brother [[Berend George Escher|Berend]], a geologist.<ref name=MathSide /> He carefully studied the 17 canonical [[wallpaper group]]s and created periodic tilings with 43 drawings of different types of symmetry.{{efn|Escher made it clear that he did not understand the abstract concept of a [[group theory|group]], but he did grasp the nature of the 17 wallpaper groups in practice.<ref name=StAndrews />}} 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 polygon]]s 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.<ref name=StAndrews /><ref>{{harvnb|Locher|1971|p=84}}</ref>

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").<ref>{{cite book |title=What's Happening in the Mathematical Sciences, Volume 4 |last=Cipra |first=Barry A. |author-link=Barry Arthur Cipra |editor=Paul Zorn |publisher=American Mathematical Society |page=103 |year=1998 |isbn=978-0-8218-0766-8}}</ref> The mathematician [[Doris Schattschneider]] unequivocally described this notebook as recording "a methodical investigation that can only be termed mathematical research."<ref name=MathSide/><ref name="Schattschneider  2010">{{cite journal |last=Schattschneider |first=Doris |author-link=Doris Schattschneider |date=June–July 2010 |title=The Mathematical Side of M. C. Escher |journal=[[Notices of the American Mathematical Society]] |volume=57 |issue=6 |pages=706–18 |url=https://www.ams.org/notices/201006/rtx100600706p.pdf}}</ref> She defined the research questions he was following as

{{blockquote|(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?<br />(2) Moreover, in what ways are the edges of such a tile related to each other by [[Isometry group|isometries]]?<ref name=MathSide />}}

=== Geometries ===

{{further|Perspective (geometry)|Curvilinear perspective}}

[[File:The Artist - Maurits Cornelis Escher - working at his Atelier (50385403156).jpg|thumb|Escher at work on ''Sphere Surface with Fish'' (1958) in his workshop, using a [[mahlstick]] for support, late 1950s]]
<!--blank lines are for readability when editing-->

Although Escher did not have mathematical training – his understanding of mathematics was largely visual and intuitive – his [[mathematics and art|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 [[perspective (geometry)|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 (M. C. Escher)|Relativity]]'' (1953).{{efn|See [[Relativity (M. C. Escher)]] article for image.}} ''[[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' [[Penrose stairs|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 (M. C. Escher)|Waterfall]]'' (1961).{{efn|See [[Waterfall (M. C. Escher)]] article for image.}}<ref name=Seckel2004>{{cite book |last=Seckel |first=Al |title=Masters of Deception: Escher, Dalí & the Artists of Optical Illusion |url=https://archive.org/details/mastersofdecepti00alse |url-access=registration |year=2004 |publisher=Sterling |isbn=978-1-4027-0577-9 |pages=[https://archive.org/details/mastersofdecepti00alse/page/81 81]–94, 262}} Chapter 5 is on Escher.</ref><ref>{{cite journal |last1=Penrose |first1=L.S. |last2=Penrose |first2=R. |title=Impossible objects: A special type of visual illusion |journal=[[British Journal of Psychology]] |year=1958 |volume=49 |issue=1 |pages=31–33 |doi=10.1111/j.2044-8295.1958.tb00634.x | pmid=13536303}}</ref><ref>{{cite book | last1=Kirousis | first1=Lefteris M. | last2=Papadimitriou | first2=Christos H. | title=26th Annual Symposium on Foundations of Computer Science (SFCS 1985) | chapter=The complexity of recognizing polyhedral scenes | author2-link=Christos Papadimitriou | doi=10.1109/sfcs.1985.59 | pages=175–185 | year=1985| isbn=978-0-8186-0644-1 | citeseerx=10.1.1.100.4844 }}</ref><ref>{{cite book | last=Cooper | first=Martin | title=Inequality, Polarization and Poverty | contribution=Tractability of Drawing Interpretation | doi=10.1007/978-1-84800-229-6_9 | isbn=978-1-84800-229-6 | pages=217–230 | publisher=Springer-Verlag | year=2008}}</ref>

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 [[Middle Ages|Mediaeval]] woman in a two-pointed headdress and a long gown in his lithograph ''[[Belvedere (M. C. Escher)|Belvedere]]'' in 1958; the image is, like many of his other "extraordinary invented places",<ref name=Poole>{{cite news |last1=Poole |first1=Steven |title=The impossible world of MC Escher |url=https://www.theguardian.com/artanddesign/2015/jun/20/the-impossible-world-of-mc-escher |newspaper=The Guardian |access-date=2 November 2015 |date=20 June 2015}}</ref> peopled with "[[jester]]s, [[wikt:knave|knaves]], and contemplators".<ref name=Poole /> Thus, Escher not only was interested in possible or impossible geometry but was, in his own words, a "reality enthusiast";<ref name=Poole /> he combined "formal astonishment with a vivid and idiosyncratic vision".<ref name=Poole />

Escher worked primarily in the media of [[Lithography|lithographs]] and [[woodcut]]s, although the few [[mezzotint]]s 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.<ref>{{cite web |url=http://www.mcescher.com/Biography/biography.htm |title=The Official M.C. Escher Website – Biography |access-date=7 December 2013 |archive-url=https://web.archive.org/web/20130702184317/http://www.mcescher.com/Biography/biography.htm |archive-date=2 July 2013 |url-status=dead }}</ref>

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:<ref name=NGC>{{cite web |title=Möbius Strip II, February 1963 |url=https://www.gallery.ca/en/see/collections/artwork.php?mkey=21164 |website=Collections |publisher=National Gallery of Canada |access-date=2 November 2015 |archive-url=https://web.archive.org/web/20150719142225/http://www.gallery.ca/en/see/collections/artwork.php?mkey=21164 |archive-date=19 July 2015 |url-status=dead }} which cites {{cite book |last1=Escher |first1=M. C. |title=M. C. Escher, the Graphic Work |date=2001 |publisher=Taschen}}</ref>

{{blockquote|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.<ref name=NGC />}}

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 Sea|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".<ref name=StAndrews>{{cite web |last1=O'Connor |first1=J. J. |last2=Robertson |first2=E. F. |title=Maurits Cornelius Escher |url=http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Escher.html |website=Biographies |publisher=University of St Andrews |access-date=2 November 2015 |date=May 2000 |archive-url=https://web.archive.org/web/20150925235220/http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Escher.html |archive-date=25 September 2015 |url-status=dead }} which cites {{cite news |author=Strauss, S. |title=M C Escher |work=The Globe and Mail |date=9 May 1996}}</ref>

Escher's interest in [[curvilinear perspective]] was encouraged by his friend and "kindred spirit",<ref name=ErnstinEmmerSchattschneider2007>{{cite book |last1=Emmer |first1=Michele |last2=Schattschneider |first2=Doris|last3=Ernst |first3=Bruno |title=M.C. Escher's Legacy: A Centennial Celebration |url=https://books.google.com/books?id=5DDyBwAAQBAJ&pg=PA16 |year=2007 |publisher=Springer |isbn=978-3-540-28849-7 |pages=10–16}}</ref> the art historian and artist Albert Flocon, in another example of constructive mutual influence. Flocon identified Escher as a "thinking artist"<ref name=ErnstinEmmerSchattschneider2007 /> alongside [[Piero della Francesca]], [[Leonardo da Vinci]], [[Albrecht Dürer]], [[Wenzel Jamnitzer]], [[Abraham Bosse]], [[Girard Desargues]], and [[Père Nicon]].<ref name=ErnstinEmmerSchattschneider2007 /> Flocon was delighted by Escher's ''Grafiek en tekeningen'' ("Graphics and Drawings"), 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]]").<ref>{{cite book |author1=Flocon, Albert |author2=Barre, André | title=La Perspective curviligne |publisher=Flammarion |year=1968}}</ref>

=== Platonic and other solids ===
[[File:Universiteit Twente Mesa Plus Escher Object.jpg|thumb|left|Sculpture of a [[small stellated dodecahedron]], as in Escher's 1952 work ''[[Gravitation (M. C. Escher)|Gravitation]]'' ([[University of Twente]])]]

Escher often incorporated three-dimensional objects such as the [[Platonic solid]]s such as spheres, tetrahedrons, and cubes into his works, as well as mathematical objects such as [[cylinder (geometry)|cylinders]] and [[Stellation|stellated polyhedra]]. In the print [[Reptiles (M. C. Escher)|''Reptiles'']], he combined two- and three-dimensional images. In one of his papers, Escher emphasized the importance of dimensionality:

{{blockquote|The flat shape irritates me — I feel like telling my objects, you are too fictitious, lying there next to each other static and frozen: {{em|do}} something, come off the paper and show me what you are capable of!&nbsp;... So I make them come out of the plane.&nbsp;... My objects&nbsp;... may finally return to the plane and disappear into their place of origin.<ref name=OutOfPlane>{{cite book |last1=Emmer |first1=Michele |last2=Schattschneider |first2=Doris |title=M.C. Escher's Legacy: A Centennial Celebration |url=https://books.google.com/books?id=5DDyBwAAQBAJ&pg=PA183 |year=2007 |publisher=Springer |isbn=978-3-540-28849-7 |pages=182–183}}</ref>}}

Escher's artwork is especially well-liked by mathematicians such as [[Doris Schattschneider]] and scientists such as [[Roger Penrose]], who enjoy his use of [[Polyhedron|polyhedra]] and [[geometry|geometric]] distortions.<ref name=MathSide>{{cite journal |last1=Schattschneider |first1=Doris| volume=57 |issue=6 |pages=706–718 |journal=Notices of the AMS |title=The Mathematical Side of M. C. Escher |url=https://www.ams.org/notices/201006/rtx100600706p.pdf |date=2010}}</ref> For example, in ''[[Gravitation (M. C. Escher)|Gravitation]]'', animals climb around a [[stellation|stellated]] [[dodecahedron]].<ref name="Hargittai2014">{{cite book |last=Hargittai |first=István |title=Symmetry: Unifying Human Understanding |url=https://books.google.com/books?id=vXTiBQAAQBAJ&pg=PA128 |date=23 May 2014 |publisher=Elsevier Science|isbn=978-1-4831-4952-3 |page=128}}</ref>

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]] now known as [[Compound of three octahedra|Escher's solid]]. Escher had used this solid in his 1948 woodcut ''[[Stars (M. C. Escher)|Stars]]'', which contains all five of the [[Platonic solid]]s and various stellated solids, representing stars; the central solid is animated by [[chameleon]]s climbing through the frame as it whirls in space. Escher possessed a 6&nbsp;cm [[refracting telescope]] and was a keen-enough amateur [[astronomer]] to have recorded observations of [[binary star]]s.<ref>{{harvnb|Locher|1971|p=104}}</ref><ref name=Beech>{{cite journal |title=Escher's ''Stars'' |last=Beech |first=Martin |journal=Journal of the Royal Astronomical Society of Canada |year=1992 |volume=86 |pages=169–177|bibcode=1992JRASC..86..169B }}</ref><ref name=CoxeterReview>{{cite journal |last=Coxeter |first=H. S. M. |author-link=Harold Scott MacDonald Coxeter |doi=10.1007/BF03023010 |issue=1 | journal=The Mathematical Intelligencer |pages=59–69 |title=A special book review: M. C. Escher: His life and complete graphic work |volume=7 |year=1985|s2cid=189887063 }}</ref>

=== 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.{{efn|See [[Drawing Hands]] article for image.}} The critic Steven Poole commented that

{{blockquote|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.<ref name=Poole />}}

=== Infinity and hyperbolic geometry ===

[[File:Schattschneider Reconstruction of Escher's Coxeter Diagram.jpg|thumb|left|upright=1.4<!--size for very low image-->|[[Doris Schattschneider]]'s reconstruction of the diagram of hyperbolic tiling sent by Escher to the mathematician [[Harold Scott MacDonald Coxeter|Donald Coxeter]]<ref name=MathSide />]]

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 [[Harold Scott MacDonald Coxeter|H. S. M. Coxeter]] were deeply impressed with Escher's intuitive grasp of mathematics. Inspired by ''Relativity'', Penrose devised his [[Penrose tribar|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''.<ref name=MathSide />
In 1957 Coxeter obtained Escher's permission to use two of his drawings in his paper "Crystal symmetry and its generalizations".<ref name=MathSide /><ref>{{cite journal |last=Coxeter |first=H. S. M. |title=Crystal symmetry and its generalizations |journal=A Symposium on Symmetry, Transactions of the Royal Society of Canada |volume=51 |issue=3, section 3 |date=June 1957 |pages=1–13}}</ref> 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 [[Models of the hyperbolic plane|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.<ref name=MathSide /><ref>{{cite web |last=Malkevitch |first=Joseph |title=Mathematics and Art. 4. Mathematical artists and artist mathematicians |url=https://www.ams.org/samplings/feature-column/fcarc-art4 |publisher=American Mathematical Society |access-date=1 September 2015}}</ref>

Escher carefully studied Coxeter's figure, marking it up to analyse the successively smaller circles{{efn|Schattschneider notes that Coxeter observed in March 1964 that the white arcs in ''[[Circle Limit III]]'' "were not, as he and others had assumed, badly rendered hyperbolic lines but rather were branches of equidistant curves."<ref name=MathSide />}} 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".<ref name=MathSide /> Among the results were the series of wood engravings ''Circle Limit I–IV''.{{efn|See [[Circle Limit III]] article for image.}}<ref name=MathSide /> In 1959, Coxeter published his finding that these works were extraordinarily accurate: "Escher got it absolutely right to the millimeter".<ref name=StAndrewsCoxeter>{{MacTutor|title=Maurits Cornelius Escher|id=Escher|mode=cs1}} which cites {{cite book |author=Schattschneider, D. |contribution=Escher: A mathematician in spite of himself |title=The Lighter Side of Mathematics |editor1=Guy, R. K. |editor2=Woodrow, R. E. |publisher=The Mathematical Association of America | location=Washington |year=1994 |pages=91–100}}</ref>