Maurits Cornelis Escher (17 June 1898 – 27 March 1972) was a Dutch graphic artist who made woodcuts, lithographs, and mezzotints, many of which were inspired by mathematics. Despite wide popular interest, for most of his life Escher was 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 objects, explorations of infinity, reflection, symmetry, perspective, truncated and stellated polyhedra, hyperbolic geometry, and tessellations. Although Escher believed he had no mathematical ability, he interacted with the mathematicians George Pólya, Roger Penrose, and Donald Coxeter, and the crystallographer Friedrich Haag, and conducted his own research into tessellation.
https://en.m.wikipedia.org/wiki/M._C._Escher
Reptiles depicts a desk upon which is a two-dimensional drawing of a tessellated pattern of reptiles and hexagons, Escher's 1939 Regular Division of the Plane. The reptiles at one edge of the drawing emerge into three-dimensional reality, come to life and appear to crawl over a series of symbolic objects (a book on nature, a geometer's triangle, a dodecahedron, a pewter bowl containing a box of matches and a box of cigarettes) to eventually re-enter the drawing at its opposite edge. Other objects on the desk are a potted cactus and yucca, a ceramic flask with a cork stopper next to a small glass of liquid, a book of JOB cigarette rolling papers, and an open handwritten note book of many pages. Although only the size of small lizards, the reptiles have protruding crocodile-like fangs, and the one atop the dodecahedron has a dragon-like puff of smoke billowing from its nostrils.
Once a woman telephoned Escher and told him that she thought the image was a "striking illustration of reincarnation".
The critic Steven Poole commented that one of Escher's "enduring fascinations" was "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" when space and flatness exist side by side and are "each born from and returning to the other, the black magic of the artistic illusion made creepily manifest."
https://en.m.wikipedia.org/wiki/Reptiles_(M._C._Escher)
On 19 August 1960 he gave a lecture in Cambridge, during which he said of this print:
'On the page of an opened sketchbook a mosaic of reptiles can be seen, drawn in three colours. Now let them prove themselves to be living creatures. One of them extends his paw out over the edge of the sketchbook, frees himself fully and starts on his path of life. First he climbs onto a book, walks further up across a smooth triangle and finally reaches the summit on the horizontal plane of a dodecahedron. He has a breather, tired but satisfied, and he moves down again. Back to the surface, the ‘flat lands’, in which he resumes his position as a symmetrical figure. I was later told that this story perfectly sums up the theory of reincarnation.'
The reference to reincarnation must have brought a smile to his face, as he always laughed about other people’s interpretations. He also listened in amusement when people stated that the word ‘Job’ on the packet in the bottom left was a reference to the Book of Job in the Bible. Nothing was further from the truth. Escher had lived in Belgium for several years and Job was a popular brand of cigarette paper there.
Because he could not print a lithograph himself, he stayed at his printer Dieperink in Amsterdam for a few days. To his friend Bas Kist he wrote that he had to do ‘a lot of tinkering’ on the stone ‘before a definitive set of copies’ could be produced.
Escher himself called what the reptiles are freeing themselves from ‘a sketchbook’, but it is of course one of his own design sketchbooks. In 1939 he created Regular division drawing nr 25, featuring these reptiles. What is remarkable and interesting about this periodic drawing is the presence of three different rotation points, where three heads meet and three ‘knees’ meet. If you copy the figure onto transparent paper and put a pin through both pieces of paper, in one of these rotation points, you can turn the transparent one 120 degrees and the figures will cover the ones below completely.
https://escherinhetpaleis.nl/en/about-escher/escher-today/reptiles-in-wartime?lang=en
The Mathematical Side of M. C. Escher
While the mathematical side of Dutch graphic artist M. C. Escher (1898– 1972) is often acknowledged, few of his admirers are aware of the mathematical depth of his work. Probably not since the Renaissance has an artist engaged in mathematics to the extent that Escher did, with the sole purpose of understanding mathematical ideas in order to employ them in his art. Escher consulted mathematical publications and interacted with mathematicians. He used mathematics (especially geometry) in creating many of his drawings and prints. Several of his prints celebrate mathematical forms. Many prints provide visual metaphors for abstract mathematical concepts; in particular, Escher was obsessed with the depiction of infinity. His work has sparked investigations by scientists and mathematicians. But most surprising of all, for several years Escher carried out his own mathematical research, some of which anticipated later discoveries by mathematicians. And yet with all this, Escher steadfastly denied any ability to understand or do mathematics. His son George explains:
Father had difficulty comprehending that the working of his mind was akin to that of a mathematician. He greatly enjoyed the interest in his work by mathematicians and scientists, who readily understood him as he spoke, in his pictures, a common language. Unfortunately, the specialized language of mathematics hid from him the fact that mathematicians were struggling with the same concepts as he was.Scientists, mathematicians and M. C. Escher approach some of their work in similar fashion. They select by intuition and experience a likely-looking set of rules which defines permissible events inside an abstract world. Then they proceed to explore in detail the consequences of applying these rules. If well chosen, the rules lead to exciting discoveries, theoretical developments and much rewarding work. [18, p.4]
In Escher’s mind, mathematics was what he encountered in schoolwork—symbols, formulas, and textbook problems to solve using prescribed techniques. It didn’t occur to him that formulating his own questions and trying to answer them in his own way was doing mathematics.
https://www.ams.org/journals/notices/201006/rtx100600706p.pdf
by Matthew Everett and Jeffrey Mancuso
Rendering competition in Pat Hanrahan's CS 348b class: Image Synthesis Techniques in the Spring quarter of 2001.
https://graphics.stanford.edu/courses/cs348b-competition/cs348b-01/escher/