# Tessellation Essay

*how*representational tessellations are accomplished. As such, their creation (by others in the field) is shrouded in a fog of secrecy, either deliberately or unintentionally so. Whether this is caused by a desire to keep their own methods to themselves or is simply ignorance/lack of inherent ability is unclear. Of interest are Escher's own explanations as to his methods. As such, he wrote sparingly, and when he did attempt to discuss specifics, in

*Regelmatige vlakverderling*, (reproduced in

*Escher The Complete Graphic Work*, pages 155-173) the explanations he gives are woefully inadequate, of which frequent digressions into non-essential matters dilute the text. For instance, concerning bird and fishes, the most frequently occurring motif, their discussion is restricted to a most simple statement:

*My experience has taught me that the silhouettes of birds and fish are the most gratifying shapes of all for use in the game of dividing the plane. The silhouette of a flying bird has just the necessary angularity, while the bulges and indentations in the outline are neither too pronounced nor too subtle. In addition, it has a characteristic shape, from above and below, from the front and the side. A fish is almost equally suitable; its silhouette can be used when viewed from any direction but the front.*

Despite studying tessellations for thirty-six *years*, this was as detailed as an account he gave. Although the advice is indeed pertinent, such a statement is far too simplistic. The question *why *these motifs are more suitable than others is the crux of the matter, and thus cries out for detailed explanation as regards such specifics.

Undoubtedly, the most informative way of discovering how he achieved his tessellations would be to see the *preparatory *drawings. However, these are conspicuously absent from almost all publications. As such, the numbered (definitive) drawings are shown without any real clue as to their ease or difficulty of creation. Needless to say, these must have been accompanied by extensive studies, of which only tantalising fragments have been published, notably in *Visions of Symmetry*. What has happened to these studies? Did Escher dispose of the bulk of his preparatory material? This seems hardly likely, given Escher's thoroughness in record keeping and in his general approach to tessellation. Infuriatingly for people like myself who are interested in his tessellations, for his ‘mathematical spatial constructions' there is an *abundance *of such material, such as with *The Magic of M. C. Escher*. Even when Escher had an opportunity of *showing *his abilities of tessellation on film, along with commentary, he 'declined' to do so, discussing unimportant aspects, such as his schooling and his views on Italian landscapes – see the M.C. Escher website for this www.mcesher.com. As such, there seems to be just a steady drip of material being released every five years or so, possibly for marketing reasons by the company owning the rights, Cordon Arts. Essentially, there is a recurrent thread running throughout this, in that ‘the secret' is not disclosed. Was Escher so disposed as to refrain from revealing his methods in order to keep his exalted position? Has such a policy been followed by Cordon Arts? Whatever, due to the publication of *Visions of Symmetry*, all the numbered drawings at least are all in the public domain, from which at least analysis of these is thus possible.

Although on the market there are books purporting to show ‘How to do Escher-Like Tessellations', these are generally best described as barely adequate. For example, probably that of the most significance, albeit with shortcomings, is *Creating Escher-Type **Drawings *by E. R. Ranucci and J. L. Teeters. As such, although the book does indeed discuss methods, much is of a lightweight nature. The ‘*how*' is discussed too simply. For example, Escher's famous Reptile tessellation (No.25) is analysed with ‘refined' lines, taken from the finished reptile. The impression given is that each line is of a certain nature, with no trial and error involved, of which Escher must certainly have undertaken. What the book (and others) does not address is the crux of the matter, which is *why *certain motifs are easier to compose than others. Furthermore, although Ranucci is a renowned mathematician, he does not show a *single *example of his own (with the drawings undertaken by Teeters). Consequently, how can he possibly discuss such matters? Other sources have included the odd article in mathematics journals, but although such efforts are welcome (due to the paucity of such material), the issue cannot be addressed in a mere handful of pages.

What is required is a more in-depth, extensive account of the intricacies; of which the following essays sets out in detail the *how's *and *why's*. By so doing, I hope to encourage further development of the subject, having laid out solid foundations from which to proceed. Essentially, people who approach the subject have to start from scratch, in effect the equivalent of reinventing the wheel each time. As I demonstrate, one's mathematical limitations are *not *a hindrance or barrier in the subject. Not by no means has Escher exhausted the subject, as may have been thought due to inferior examples produced by his (few) successors. For instance, he missed a whole type, of what I consider my own speciality, ‘geometric outlines', whereby the tile is simply awaiting the addition of a motif. Even amongst individual creatures, quality examples exist of those that he missed, for example the rabbits of Makato Nakamura and kangaroos of Bruce Bilney. Developments since the time of Escher have involved the use of the computers in the drawing and design of tessellation. However, again, use of such technology is shown to be unnecessary. Therefore, demonstrably, the field is still open to *anyone*, whatever their background, who simply has the desire to attempt such things.

By such clear showing of the *how's *and *why's*, I hope to stimulate discussion and debate. Anyone who feels so inclined to discuss such matters, whether of a minor or major nature, agreeing or disagreeing, will be welcome to so do.

*Last updated 14 November 2005 *

The value of

*is about 78.07 degrees.*

**A** The 2003 Math Awareness Month poster design is just one example of the connection between mathematics and art. Of course there are numerous other connections, including those inspired by Escher in the recent book *M.C.Escher's Legacy* [Sc2]. My article [Du3] and electronic file on the CD Rom that accompanies that book contain many examples of computer-generated hyperbolic tessellations inspired by Escher's art. For more on Escher's work, see the Official M. C. Escher Web site http://www.mcescher.com/ [Es1].

### References

[Ab1] Abas, S. Jan, Web site: http://www.bangor.ac.uk/~mas009/part.htm

[Bo1] Bool, F.H., Kist, J.R., Locher, J.L., and Wierda, F., editors, *M. C. Escher, His life and Complete Graphic Work*, Harry N. Abrahms, Inc., New York, 1982. ISBN 0-8109-0858-1

[Co1] Coxeter, H. S. M., "Crystal symmetry and its generalizations," *Royal Society of Canada*(3), **51** (1957), 1-13.

[Co2] Coxeter, H. S. M., "The non-Euclidean symmetry of Escher's Picture 'Circle Limit III'," *Leonardo*, **12** (1979), 19-25, 32.

[Co3] Coxeter, H. S. M., "The Trigonometry of Escher's Woodcut 'Circle Limit III'," *The Mathematical Intelligencer*, **18** no. 4 (1996) 42-46. Updated and corrected version appears in [Sc2] below.

[Co4] Coxeter, H. S. M., "Angels and devils," in *The Mathematical Gardner*, David A. Klarner, editor, Wadsworth International, 1981 (out of print). ISBN 0-534-98015-5

Republished as: *Mathematical Recreations: A Collection in Honor of Martin Gardner*, David A. Klarner, editor, Dover Publishers, 1998. ISBN 0-486-40089-1

[De1] Deraux, Martin, Interactive tessellation web site: http://www.math.utah.edu/~deraux/tessel/

[Du1] Dunham, D., "Hyperbolic symmetry," *Computers and Mathematics with Applications*, Part B **12** (1986), no. 1-2, 139-153.

[Du2] Dunham, D., "Transformation of Hyperbolic Escher Patterns,"*Visual Mathematics* (an electronic journal), **1**, No. 1, March, 1999.

[Du3] Dunham, D., "Families of Escher Patterns," in [Sc2] below, pp. 286-296.

[Es1] *Official M. C. Escher Web site*, published by the M.C. Escher Foundation and Cordon Art B.V. http://www.mcescher.com/

[Fe1] Ferguson, Helaman, Web site: http://www.helasculpt.com/gallery/index.html

[Go1] Goodman-Strauss, Chaim, "Compass and straightedge in the Poincaré disk," *Amer. Math. Monthly*, **108** (2001), no. 1, 38-49.

[Gr1] Greenberg, Marvin, *Euclidean and Non-Euclidean Geometries*, 3rd Edition, W. H. Freeman and Co., 1993. ISBN 0-7167-2446-4

[Ha1] Hatch, Don, Hyperbolic tessellations web site: Hyperbolic Planar Tesselations [Ha1].

[He1] Henderson, David W., and Daina Taimina, *Experiencing Geometry: In Euclidean, Spherical and Hyperbolic Spaces*, 2nd Ed., Prentice Hall, 2000. ISBN 0130309532 Web link: http://www.mathsci.appstate.edu/~sjg/class/3610/hen.html

[Jo1] Joyce, David, Hyperbolic tessellations web site: http://aleph0.clarku.edu/~djoyce/poincare/poincare.html

[Ka1] Kaplan, Craig S., "Computer generated Islamic star patterns," *Bridges 2000, Mathematical Connections in Art, Music and Science.* Winfield, Kansas, USA, 28-30 July 2000. ISBN 0-9665201-2-2 Web link: Abstract and PDF

[Ma1] Magnus, Wilhelm, *Noneuclidean Tesselations and Their Groups*, Academic Press, 1974. ISBN 0-12-465450-9

[Sc1] Schattschneider, Doris, *Visions of Symmetry: Notebooks, Periodic Drawings, and Related Work of M. C. Escher*, W. H. Freeman, New York, 1990. ISBN 0-7167-2126-0

[Sc2] Schattschneider, Doris, and Michele Emmer, editors, *M. C. Escher's Legacy: A Centennial Celebration*, Springer Verlag, 2003. ISBN 3-540-42458-X

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