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Our ancestors were artistic; they covered floors and walls with patterns and mosaics, in Roman houses and Muslim mosques (see Figure 20.1). They featured patterns in other decorative arts, too— carpets, fabrics, baskets, and even wallpaper and linoleum.

Such patterns use repeated shapes (“tiles”) to cover a surface, without gaps or overlaps. Apart from aesthetic appeal, repeated patterns can have practical applications. In manufacturing, for example, stamping components from a sheet of metal is most economical if the shapes of the components fit together without gaps—in other words, if the shapes form a tiling.

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Mathematics is about discerning patterns—whether in physical objects (as in Chapter 18), numbers (as in Chapter 19), geometrical shapes (this chapter), or other structures—and crafting arguments to explain them.

In this chapter, we ask: What shapes can form a tiling that repeats in a regular way?

What if you have just one kind of tile—all the tiles are the same size and shape? For example, could a regular polygon work? Only a few do. In investigating tilings by polygons that don’t all have equal sides, you will encounter significant contributions from one amateur mathematician—a housewife—and from another, despite a degree of autism.

Abandon polygons—let your imagination loose, and consider tiles in the shapes of horsemen, fish, or any other artistic shape you like. Which can tile? What if you let the tiles appear upside down as well as right side up?

What if you have two or more different shapes of tiles? Surprisingly, there is no way to decide if an arbitrary collection of shapes can tile. For some, we can exhibit tilings; for others, we can prove that they don’t tile. But mathematicians have proved that there is no algorithm (mechanical step-by-step process) that can decide for every conceivable set of tile shapes whether they will tile or not. (See Chapter 9 for other examples of mathematically “unattainable ideals,” in that case with regard to voting.)

Finally, what about “tiling” in three dimensions—filling space with solid shapes? There are more surprises in store, including quasicrystals, the discovery of which won a Nobel Prize, and their applications in new ultrastrong alloys and coatings for nonstick cookware.