The word "tessellation" comes from the Latin "tessela," a small tile used in making mosaics. Essentially, a tessellation is the symmetrical tiling of a surface with a repetitive pattern or series of shapes. While an infinite number of irregular shapes can be used to produce tessellations, the only regular polygons that tessellate are triangles, squares and hexagons.
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A proper tessellation is formed when a patterned shape can be matched together side to side infinitely without producing any overlaps or gaps. Because, in theory, a tessallation repeats forever, it does not necessarily need to fit into a rectangular edge. Realistically, however, tessellations must end somewhere. Some tessellations, particularly those using squares, rectangles or triangles, can fit neatly into a rectangular wall or sheet of paper. Others do not, but this does not disqualify them as tessellations.
Tessellations appear in nature, most obviously in honeycombs. These are produced by a series of interlocking hexagons that leave no gap and produce no overlap. Tessellations are also usually present in any floor or wall tile design, in brick paving, and in geodesic domes. Some of the artworks of M.C. Escher represent the most famous tessellations in the world.
In basic geometry there are two major types of tessellation, regular and semiregular. In regular tessellations, a single shape is involved, it must be a regular polygon, and all the points of intersection (called vertices) are therefore identical. Semi-regular polygons involve more than one regular polygon and therefore do not necessarily have all identical vertices. In more advanced math, there are many other classifications of tessellations, such as those that are periodic versus aperiodic, symmetric versus asymmetric, or fractal. In the 1970s, mathematician Roger Penrose discovered at least three new kinds of tessellations, formed of two or more shapes with an infinite number of possible nonrepeating connections and other unique properties.
When used in education, activities related to tessellations generally appear in lessons on symmetry. Principles of geometry, such as rotation, translation, and reflection, can be exemplified with tessellations. Students are typically asked to identify these properties within tessellations, as well as to produce their own tessellations.
Though tessellation is usually associated with flat surfaces like walls and pages, three-dimensional objects can also be tessellated. Many soccer balls, in fact, are made of 32 pentagonal tiles tessellated into a sphere. Similarly, a torus (doughnut shape) can be tessellated by connected squares. The familiar honeycomb is another example of a 3-D tessellation.
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