In 1619, Johannes Kepler published the first formal study of tessellations. In fact, the nature of mosaic art naturally gives rise to some tessellating patterns. Sumerian wall decorations, an early form of mosaic dating from about 4000 B.C., contain examples of tessellations. Tessellation patterns are very old, and are found in many cultures around the world. For example, the "Fish n' Chicks" animation below shows how you can alter a square to create an irregular shape that tessellates a surface. 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. 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. You can find examples of these on chess- or checkerboards. Patterns using only one regular polygon to completely cover a surface are called regular tessellations. Circles, for instance, would not create a tessellation by themselves, because any arrangement of circles would leave gaps or overlaps.ĭespite the limitations on the types of shapes that can form this intriguing pattern, there are many varieties of tessellations. Not all shapes, however, can fit snugly together. There are usually no gaps or overlaps in patterns of octagons and squares they "fit" perfectly together, much like pieces of a jigsaw puzzle. Hunt using an irregular pentagon (shown on the right).Geometry formally defines a tessellation as an arrangement of repeating shapes which leaves no spaces or overlaps between its pieces. Another spiral tiling was published 1985 by Michael D. The first such pattern was discovered by Heinz Voderberg in 1936 and used a concave 11-sided polygon (shown on the left). Lu, a physicist at Harvard, metal quasicrystals have "unusually high thermal and electrical resistivities due to the aperiodicity" of their atomic arrangements.Īnother set of interesting aperiodic tessellations is spirals. The geometries within five-fold symmetrical aperiodic tessellations have become important to the field of crystallography, which since the 1980s has given rise to the study of quasicrystals. According to ArchNet, an online architectural library, the exterior surfaces "are covered entirely with a brick pattern of interlacing pentagons." An early example is Gunbad-i Qabud, an 1197 tomb tower in Maragha, Iran. The patterns were used in works of art and architecture at least 500 years before they were discovered in the West. Medieval Islamic architecture is particularly rich in aperiodic tessellation. These tessellations do not have repeating patterns. Notice how each gecko is touching six others. The following "gecko" tessellation, inspired by similar Escher designs, is based on a hexagonal grid. By their very nature, they are more interested in the way the gate is opened than in the garden that lies behind it." In doing so, they have opened the gate leading to an extensive domain, but they have not entered this domain themselves. This further inspired Escher, who began exploring deeply intricate interlocking tessellations of animals, people and plants.Īccording to Escher, "Crystallographers have … ascertained which and how many ways there are of dividing a plane in a regular manner. His brother directed him to a 1924 scientific paper by George Pólya that illustrated the 17 ways a pattern can be categorized by its various symmetries. According to James Case, a book reviewer for the Society for Industrial and Applied Mathematics (SIAM), in 1937, Escher shared with his brother sketches from his fascination with 11 th- and 12 th-century Islamic artwork of the Iberian Peninsula. The most famous practitioner of this is 20 th-century artist M.C. A unique art form is enabled by modifying monohedral tessellations.
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