Editing M. C. Escher (section)

=== 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 />}}