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I realize this question isn't strictly mathematical, and if it doesn't fit with the content on this site then feel free (moderators/high-rep users) to close it. But when I thought up the question it seemed to me that the users on this site would be best equipped to answer it.

I've always been intrigued by the paintings of M.C. Escher comprising infinitely repeating patterns. For those of you not familiar with his work, here's an example, titled Angels & Devils:

Angels & Devils by M.C. Escher

I recently wrote a blog post about the above picture and then started thinking: how did Escher do that? It's kind of like a fractal, right? (Or is that an extremely ignorant thing to say?)

Maybe if I were to attempt such a painting myself, today, I could find a computer program to generate random fractal-like patterns over and over until I found one I felt I could work with; then I could simply "fill in the space" with whatever image I chose. But surely Escher didn't have any such tool available to him, right?.

So: how might Escher have designed such patterns? Does anyone have any mathematical insight into what process might have been used to accomplish this? Alternatively, does anyone possibly have some historical knowledge of how Escher actually did do this?

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  • $\begingroup$ Escher did that particular tessellation in both hyperbolic and elliptical geometric—the Poincare disk in "Circle Limit IV" shown in the question, and a spherical wooden sculpture (reproduction shown here: mcescher.com/product/sculpture-sphere-angels-devils ). $\endgroup$
    – Buzz
    Commented Sep 17, 2021 at 1:25

7 Answers 7

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'Around 1956, Escher explored the concept of representing infinity on a two-dimensional plane. Discussions with Canadian mathematician H.S.M. Coxeter inspired Escher's interest in hyperbolic tessellations, which are regular tilings of the hyperbolic plane. Escher's works Circle Limit I–IV demonstrate this concept. In 1995, Coxeter verified that Escher had achieved mathematical perfection in his etchings in a published paper. Coxeter wrote, "Escher got it absolutely right to the millimeter."'

http://en.wikipedia.org/wiki/M._C._Escher

If Angels and Devils is a hyperbolic tessellation then it might have been inspired by Coxeter.

The construction itself was done using techniques like these:

http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.133.8746&rep=rep1&type=pdf

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  • $\begingroup$ Awesome. I am reading the PDF now; this is definitely what I was looking for—thanks! $\endgroup$
    – Dan Tao
    Commented Jan 1, 2011 at 18:54
  • $\begingroup$ Unfortunately the last link is broken. $\endgroup$
    – Jean Marie Becker
    Commented Jan 18, 2023 at 17:43
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The June/July 2010 issue of the AMS Notices here has a further article by Doris Schattschneider (a graduate school classmate of mine) on Coxeter and Escher. Doris has written extensively about Escher's work from a mathematical viewpoint, so it's worth checking out her point of view.

P.S. Don't overlook in this Notices issue Bill Casselman's column About the Cover.

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    $\begingroup$ See also her books, "M.C. Escher: Visions of Symmetry" and "Visions of Symmetry: Notebooks, Periodic Drawings, and Related Work of M.C. Escher." $\endgroup$
    – Joseph O'Rourke
    Commented Jan 1, 2011 at 18:48
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    $\begingroup$ Nice, good stuff! I'd like to upvote you but my rep is too pitiful. Maybe later, if I ever get a bit more ;) $\endgroup$
    – Dan Tao
    Commented Jan 1, 2011 at 18:56
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I think he knew about hyperbolic geometry you can read more here:http://www.math.cornell.edu/~mec/Winter2009/Mihai/index.html

There is also a beautiful article in the notices of the AMS by H. Lenstra and B. de Smit, (http://www.ams.org/notices/200304/fea-escher.pdf), in which they explain how he painted the "Print gallery" (it involves complex geometry). They also show how to fill the hole in the center of the painting, you can actually see it at dedicate web site (run by Bart de Smit) http://escherdroste.math.leidenuniv.nl/

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  • $\begingroup$ Very cool—thanks for the link! I want to give you an upvote but my rep is too low :( $\endgroup$
    – Dan Tao
    Commented Jan 1, 2011 at 18:55
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Adding to Valerio Talamanca's answer, there was a lecture by Hendrik Lenstra on Escher's "Print gallery" at Leiden University (December 12, 2012) which can be viewed here.

Also, my memory says that elliptic curves, rather than complex geometry, played the pivotal role in the explanation (that is why Hendrik Lenstra explained it).

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  • $\begingroup$ Alas, the link is dead. $\endgroup$
    – Anton Sherwood
    Commented Sep 14, 2016 at 21:01
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    $\begingroup$ The link died again $\endgroup$
    – მამუკა ჯიბლაძე
    Commented Sep 16, 2021 at 14:45
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    $\begingroup$ @მამუკაჯიბლაძე I updated the link (which now points to a lecture in 2012 rather than 2007). $\endgroup$
    – GH from MO
    Commented Sep 17, 2021 at 0:07
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    $\begingroup$ I think it's "Print Gallery", not "Paint Gallery". $\endgroup$
    – Gerry Myerson
    Commented Sep 17, 2021 at 0:07
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    $\begingroup$ @GerryMyerson Thank you, I fixed this. $\endgroup$
    – GH from MO
    Commented Sep 17, 2021 at 0:08
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Around 1976, I became enthralled by Eschers patterns. I purchased many books on his art, hoping to find his method, but as far as I could find, there weren't any published. I eventually developed a system on my own by which I could create patterns at will. I have since created about thirty different patterns of my own which include elephants, eagles, angels, bisons, marine reptiles, leaves, gargoyles, dragons, etc. This system I developed, I believe, is the same, or very similar to what Escher employed, because I have since seen books that show the underlying gridwork, and graduation from the basic square grid, to the final curves of the finished pattern. This obviously doesn't require a computer, just some graph paper, a pencil with eraser, and some basic understanding of patterns along with a minor amount of artistic creativity. The graph paper can be a basic square grid, or even triangular or hexagonal, or any other regular gridwork. In a nutshell, simplify your animal or plant shape to its very basic shape, using the grid to block it out, and using common dimensions as much as possible, make a pattern of that block shape. By understanding rotation point s-curves,etc, the block can be gradually transformed into curves. The entire system is too lengthy to go into here, but you may see some of my results on my Alfredman/facebook page which is open to the public. The page is Identified as comic strip and creative writings. I have included some of my patterns on that page,and if any interest is shown, I could go into greater detail on the subject.As far as the circular design works, a grid can be formed using a compass and a straightedge, and a previously designed pattern can be applied to the resulting grid. A basic example of this is also included on my page.

In addition, as to the question of whether this fits as a mathematical query, it does. Patterns are a form of math. Not all math directly concerns numerical computations, geometry is math, as is music. Math can be intuitive, and creative. Check out Eschers book called Fantasy and Symmentry, by Caroline H. Macgillavry, Abrams press. This book contains the periodic drawings of M C Escher, and is used to illustrate how crystals form, basically a crystalography textbook. It explains some various elements of pattern formation, translation, rotation, mirroring etc. In MY patterns, as in music, a form of Jazz can be incorporated, by including a bit of randomizing. By this I mean that while the shapes have a definite structure and regularity, The colors do not have to conform to a regular pattern, although they certainly can. Escher also does this by way of morphing his shapes within a pattern to cause change. Much like visual music.

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    $\begingroup$ I have to laugh, I asked a question of my own today, framed it as well as I could, gained a student badge for getting one up vote, but was then closed down for it not being deemed a question, and lost twenty reputation points. wow $\endgroup$
    – alfred myers
    Commented Jan 6, 2012 at 1:41
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When I was in high-school, my teacher suggested a very interesting method for generating some interesting Escher's like pictures. I think it is useful for you.

Get a mirror (or something like mirror) and construct some geometrical shape with this mirror, for example a cylinder, that the inner surface of this shape is mirror. Now, collect some shapes, for example shells, flowers or some simple things like triangles. If we put these objects inside the cylinder, we find very beautiful Escher-like shape on the mirror. With some simple techniques in photography, we can take pictures from inside the cylinder. We can change the shape of mirror and create some elegant symmetry.

I didn't see this method anywhere. I hope it will be useful for you.

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    $\begingroup$ That is exactly how a kaleidoscope works $\endgroup$
    – ARi
    Commented Jul 4, 2013 at 18:36
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The below article might help understand his process... http://pi.math.cornell.edu/~mec/Winter2009/Mihai/section8.html

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