The tiles you see in your bathroom or on a kitchen floor are usually square, though you may also come across tiles that are shaped like hexagons or octagons.

But tiles don't have to be polygons. Many of the drawings by M.C. Escher, a Dutch artist who lived from 1898 to 1972, contain interlocking tiles in the shape of birds, fish, reptiles, butterflies, and other living things.

As a young boy, Escher was intrigued by the different ways in which you can cover areas neatly with small, identical pieces. Drawing was his favorite subject in school, and after graduating, he became an artist. At age 24, he visited Spain and discovered the intricate mosaics (designs made with tiles) in the Alhambra, a 13th-century Moorish palace in Granada. Those designs inspired him to create the amazing tile patterns that appear in his art.

Escher used a number of different strategies to draw intriguing tiling patterns. Suppose you had a stack of square tiles, some red and some black. You could lay them out in neat rows of alternating red and black tiles to create a checkerboard pattern. In this case, the repeating unit would be a block of four tiles, two red and two black. Starting with one such unit, you can generate the entire pattern by using it like a stamp to fill in the rows and columns of the design. If all the tiles were the same color, the repeating unit would be a single tile.

To make his artworks more interesting, Escher often disguised the underlying geometric pattern by using tiles of different shapes and colors. When you look at one of these patterns, you might think that each tile is shaped like, say, a fish. But if you imagined you had a stack of fish tiles and tried to lay out the pattern with them, you'd quickly find you couldn't do it. This is because the true tile, the repeating unit that can be used to lay out the entire pattern, is not a fish. Instead it might be several fish of different colors joined together. So each of Escher's tiling patterns is a puzzle; what is the basic tile?

One person who has studied Escher's art is Doris Schattschneider, a math professor at Moravian College in Bethlehem, Pennsylvania. While examining Escher's notebooks, Schattschneider found that Escher had worked out his own mathematical system for classifying the tiles. He used special symbols to describe how portions of the edges of tiles related to each other and to edges of adjacent tiles. The system allowed him to find all the different ways in which he could interlock and color various shapes of identical tiles to create pleasing patterns.

Escher's study of geometric shapes, combined with his artistry and imagination, led to all sorts of fabulous drawings.

Muse, April 2001, p. 26.

Doris Schattschneider. Visions of Symmetry: Notebooks, Periodic Drawings, and Related Works of M.C. Escher (New York: W.H. Freeman, 1990).

Take a closer look at some of Escher's designs at http://library.thinkquest.org/16661/ and http://www.mcescher.com/.

You can find activities and investigations related to tilings (tessellations) and symmetry at http://britton.disted.camosun.bc.ca/home.htm.

## MatheMUSEments

Articles for kids about math in everyday life, written by Ivars Peterson for *Muse* magazine.

## April 23, 2007

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