# Learning from Escher

May 7, 2012 Leave a comment

In 1990, Doris Schattschneider[1] wrote a book about M. C. Escher ‘s work entitled ‘Visions of Symmetry.’ The book focuses on the “regular division of the plane,” one of Escher’s favorite themes (Escher, in fact, wrote a book with this title in 1958). Maurits Cornelis Escher (1898-1972) first became interested in interlocking shapes[2] when he visited Alhambra, a fourteenth century castle in Spain known for its intricate mosaics.

**Figure 1: **Tiles at Alhambra, Spain

Schattschneider reproduces many of the kaleidoscopic patterns found in his notebooks, but also includes sketches that illustrate his technique for developing complex, interlocking motifs. Escher would “divide the plane” into symmetrical shapes and let new motifs “emerge” from the changing geometry.

**Figure 2: **Dividing up the plane.

**Figure 3: **Perfecting the motif.

**Figure 4: **The final work.

His ability to envision how a simple geometric shape could evolve into a recognizable (if at times fantastic) design was overshadowed only by his ability to foresee how a multitude of them would fit together upside down and backwards to form an uninterrupted landscape. Although Escher was not a mathematician, he collaborated with mathematicians, developed his own systems to define symmetry, and ultimately his work made important contributions to crystallography. [3]

I came across Schattschneider’s book in 2001 and was mesmerized. I had seen his work, of course, including his crazy waterfalls and staircases, alligators crawling out of drawings and back in. But his notebooks (reproduced by Schattschneider) were intoxicating—the mathematics of art. I remember thinking if he had only fused two dinosaurs together, then realized he had done more than that—he had shown us how. I would try it myself, and though the result comes nowhere near the work of Escher, there is certain satisfaction to doing it yourself.

Thus, I started “dividing up the plane,” though I cheated in the sense I started with a basic sauropod shape (I wanted dinosaurs to “emerge,” not just any new shapes). In any case, it wasn’t long before I realized I could jam a few sauropods together and create an enclosed figure not unlike a pterosaur (not technically a dinosaur, but that’s splitting hairs).

**Figures 5 & 6:** Working out the motifs.

The end result is below. It’s not Escher. Even his simplest patterns have greater “interlocking” between the motifs. In my case, the pterosaur is really only bounded by three sauropods (the pterosaur only touches the sauropod above and to the left at a single point; they do not share a boundary). Nevertheless, it’s two prehistoric monsters, demonstrating how life fills up every nook and cranny, each form helping define the ones around it.

[1] Schattschneider is Professor Emerita of Mathematics at Moravian College, Bethlehem, Pennsylvania.

[2] More precisely ‘tesselations,’ two dimensional planes divided up by repeating geometric shapes that do not overlap or leave gaps.

[3] *The Mathematical Side of M.C. Escher,* Schattschneider, D., Notices of the American Mathematical Society, June/July, 2010.

Doris Schattschneider’s book **Visions of Symmetry.**