Mathematical Imagery by Jos Leys

Hyperbolic Escher

M.C. Escher only made four 'Circle limit' drawings: tilings using hyperbolic geometry.
Here are 29 of his famous Euclidian tilings transformed into hyperbolic ones.
All M.C. Escher Works 2008 The M.C. Escher Company, the Netherlands. All rights reserved. Used by permission.


M.C. Escher's four circle limit drawings:

Circle limit 1

Circle limit 2

Circle limit 3

Circle limit 4

Here is what M.C.Escher wrote to his son Arthur in 1960 (speaking about Circle limit 3):

" Ik heb mij rot gewerkt om eindelijk die litho af te maken en vervolgens, de tanden op elkaar, vier dagen lang nog eens negen mooie afdrukken van die hoogst bewerkelijke cirkellimiet-in-kleuren gemaakt. Elke druk bestaat uit een serie van twintig maal afdrukken: vijf planken, elke plank vier keer. Dit alles met het werkwaardige gevoel dat dit werkstuk een mijlpaal in mijn ontwikkeling betekent en dat er nooit iemand zal zijn, behalve ikzelf, die dat zal inzien."

( " I worked terribly hard to finally finish that litho, and then with gritted teeth, spent another four days making beautiful prints of that extremely complex circle limit in colors. Each print is a series of twenty printings: five pieces, and each piece four times. All this with the remarkable feeling that this work is a milestone in my development, and that nobody, except myself, will ever realize this." )

(From: De magie van M.C.Escher. Taschen Uitgeverij, Keulen 2003)

He was wrong when he said that nobody would realize the importance of these works, but he was certainly right in saying that the creation of such drawings by hand is extremely difficult and tiresome, much more so than plane (Euclidian) tilings. M.C. Escher made over a hundred such tilings, and something that he most probably never knew is that some of them (not all) can be transformed to hyperbolic, "circle limit", tilings with the help of a computer.

The images below were created starting from relatively low resolution images of original Escher prints, and are therefore not perfect. In some cases the lines where the basic tile was cut can clearly be seen. Where this is the case, it is due in part to the quality of the original image file, and in part to some inaccuracy both on my part and on M.C.Escher's part.

It is an open question how M.C. Escher's work would have been influenced had computers been available in his lifetime.

All the images below were created using Ultrafractal.

E15 by M.C.Escher

E70 by M.C.Escher

E66 by M.C.Escher

E42 by M.C.Escher

E35 by M.C.Escher

E57 by M.C.Escher

E67 by M.C.Escher

E61 by M.C.Escher

E104 by M.C.Escher

E105 by M.C.Escher

E45 by M.C.Escher

E55 by M.C.Escher

E69 by M.C.Escher

E85 by M.C.Escher

E119 by M.C.Escher

E99 by M.C.Escher

E94 by M.C.Escher

E103 by M.C.Escher

E44 by M.C.Escher

E128 by M.C.Escher

E124 by M.C.Escher

E106 by M.C.Escher

E34 by M.C.Escher

E34B by M.C.Escher

E58 by M.C.Escher

E62 by M.C.Escher

E63 by M.C.Escher

E82 by M.C.Escher

E123 by M.C.Escher

Some transformations of other Escher works

Unknown title by M.C.Escher

Tetraedral planetoid by M.C.Escher


Copyright 2014 Jos Leys