Marjorie Senechal is a mathematician at Smith College who studies patterns, especially tilings. Tilings are patterns made of shapes, which we call tiles, fitted together to cover a surface such as the infinite plane or a curved one, such as a sphere. Tilings suggest many fascinating (but difficult!) mathematical problems. For example, the clay or linoleum tiles with which we cover walls and floors are usually either squares or hexagons. What other shapes could we have used instead? You can explore these questions with cardboard and scissors, or with software such as Geometer's Sketchpad. There are also some good programs for exploring tilings, such as Kali.You can also learn about tiles and tilings in Marjorie Senechal's chapter in On The Shoulders of Giants.
Tilings span all dimensions, and stretch our imaginations. Suppose we require that all the tiles in a tiling must be arranged in parallel position. In two dimensions, only quadrilaterals and hexagons can be arranged in this way. What about three dimensional shapes? The answer to this question was found over one hundred years ago by the Russian crystallographer E. S. Fedorov. But we enter onto research ground when we ask the same question about four dimensional space, or five, or six ....
Once we have constructed a tiling, with whatever number of tile shapes, parallel or not, in any dimension, there are still many questions to answer. How can we determine the tiling's properties? For example, does it repeat, like wallpaper, or is it one of those intriguing patterns that continues forever but never repeats? (And how can we prove that it never repeats?)
The most famous non-repeating tilings are those discovered (or invented, depending on your philosophy) by the mathematician and physicist Sir Roger Penrose. The Penrose tilings use two shapes that must be fitted together according to certain rules. There are two versions of the tiles, known as "kites and darts" and "thick and thin rhombs"; both types of tilings have the same remarkable properties. A portion of a Penrose tiling by rhombs is shown at the right. For more information about the kite and dart version, see Wolfram's site or the Java applet of Shuxiang Zheng.
The sculpture, "Aperiodic Penrose Alpha" was designed by the sculptor and mathematician Helaman Ferguson in collaboration with Marjorie Senechal. It stands in the lobby of Burton Hall at Smith College, where Marjorie Senechal has taught for many years. The base of the sculpture is a portion of a Penrose tiling with rhomb tiles. The lines winding around the torus at the top encode many of the properties of the tiling, including a proof of its nonperiodicity. A larger version of this sculpture is available .
The mathematical theory of tilings draws on many branches of mathematics, such as geometry, algebra, and number theory. Tiling theory also draws on sources outside mathematics for inspiration, such as ornamental patterns, textiles, and crystals. Conversely tiling theory helps us to design interesting patterns and to understand many natural phenomena such as crystal structures.
For nearly a century, scientists believed that the molecules in crystals were arranged in three-dimensional repeating patterns, and that this repetition was the very definition of the crystalline state. This belief was shattered in 1984 with the discovery, by materials scientist Dany Shechtman, of some crystals that couled not have a repeating molecular structure. This discovery provoked a torrent of research, by mathematicians, physicists, and materials scientists, on the Penrose tilngs and other nonrepeating tilings. One of the most important techniques used to study these tilings was actually developed before the discovery of the strange new crystals. The Penrose tilings, and others, are closely related to slices cut through perfectly "normal" repeating patterns in some higher dimensional space! By studying these nonrepeating tilings in a higher dimensional context, we can create them to order. You can explore this yourself using the software Quasitiler, which Marjorie Senechal helped develop at the Geometry Center several years ago. And you can read more about it in her book, Quasicrystals and Geometry.
Here is a collection of other sites dealing with the mathematics of two- and three-dimensional patterns, including wallpaper design, quilts, origami, and fractals (objects that have fractional dimensions between two and three).