What are tessellations?

Explore connections in mathematics and nature with this article on Tessellations.
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Geometry formally defines a tessellation as an arrangement of repeating shapes which leaves no spaces or overlaps between its pieces. There are usually no gaps or overlaps in patterns of octagons and squares; they "fit" perfectly together, much like pieces of a jigsaw puzzle. Not all shapes, however, can fit snugly together. Circles, for instance, would not create a tessellation by themselves, because any arrangement of circles would leave gaps or overlaps.

Despite the limitations on the types of shapes that can form this intriguing pattern, there are many varieties of tessellations. Patterns using only one regular polygon to completely cover a surface are called regular tessellations. You can find examples of these on chess- or checkerboards. Semi-regular tessellations, on the other hand, use a combination of different regular polygons, such as the pattern above, and you can typically see examples of these patterns in the tilework of bathroom and kitchen floors. Tessellations made from regular polygons (equilateral triangles, squares, and hexagons) are usually referred to as tilings; however, tessellations can be made from many irregular shapes as well. For example, the "Fish n' Chicks" animation below shows how you can alter a square to create an irregular shape that tessellates a surface.

Tessellation patterns are very old, and are found in many cultures around the world. Sumerian wall decorations, an early form of mosaic dating from about 4000 B.C., contain examples of tessellations. In fact, the nature of mosaic art naturally gives rise to some tessellating patterns. In 1619, Johannes Kepler published the first formal study of tessellations. In the present day, Oxford mathematician Sir Roger Penrose has devoted much time to the study of recreational mathematics and tessellations. He worked on the problem of creating a set of shapes that would tile a surface without a repeating pattern, called quasi-symmetry.

Tessellating patterns cut across many different disciplines. A branch of science known as x-ray crystallography studies the repeating arrangements of identical objects in nature, sort of a natural form of tessellation. In the fine arts, artists such as M.C. Escher have used the intriguing optical effect of tessellations to create a surreal mood. In fact, in working with tessellating shapes and incorporating their patterns into his work, M.C. Escher made many discoveries similar to those made in x-ray crystallography.

You can find tessellations in many different forms of art and graphic design. They are often applied as grid patterns in the design of oriental rugs. Quilting technique involves a thorough understanding of tessellations, and quilters work hard to come up with their own tessellating designs. Many quilt patterns, however, date all the way back to patterns found in Roman floor mosaics. What other examples of tessellations can you think of?

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