A tessellation (or tiling) is a covering of a surface with tiles so that there are no gaps or overlaps. The tile shapes can be all different, as in the chips used to produce a complex Byzantine mosaic, or they may consist of a limited number of shapes, each congruent to one or more “prototiles” which serve as templates. Since ancient times, almost every culture has produced tessellations for utilitarian or decorative purposes—on walls, ceilings, and roofs of buildings, for pavements and plazas, and for designs to be woven, painted, printed, incised, or inlaid on every variety of surface. Tessellations adorn many churches, temples, palaces, and mosques; perhaps the most celebrated geometric tessellations are found in the Alhambra, in Granada, Spain. Tessellations also occur in nature, in the designs formed by scales or packed cells on a living surface.
Although artisans have produced tessellations for thousands of years, only recently have mathematicians undertaken a methodical study of the subject. Grünbaum and Shephard’s book is the most complete reference. Intuitively, a shape “tiles” (is a prototile for a tessellation) if congruent copies of that shape can be fitted together exactly to fill a surface. Every triangle and every quadrilateral can tile the plane, and convex polygons with seven or more sides can never tile the plane. All convex hexagons that tile the plane have been determined, and 14 classes of convex pentagons that tile have been discovered, but it is not known if there are others. Tessellations by regular polygons appear frequently; those known as Archimedean tessellations are composed of regular polygons with edges matched, and have the same arrangement of tiles occurring at every vertex of the tiling (Figure 1).
An infinite variety of shapes tile the plane, many of them classified by special properties. Given an arbitrary set of shapes, many tests can attempt to answer the question: Will these tile the plane? But there is no guarantee of an answer. The question is mathematically
Figure 1. The 11 Archimedean tessellations. The three with only one prototile are called regular tessellations; the others are often called semiregular.
undecidable; that is, there is no algorithm that can deliver an answer of yes or no for every possible set of shapes.
M.C.Escher, a Dutch graphic artist (1898–1972), is the best-known creator of tessellations using whimsical figures as tiles (Figure 2). Inspired by the geometric tessellations in the Alhambra, he sought to answer the question: What shapes can tile the plane so that every tile is surrounded in the same way? As he discovered some answers, he developed a system of tile types that enabled him to create imaginative shapes that fit together in a prescribed manner. In his lifetime, he produced more than 150 finished tessellations; many have been used by scientists, particularly crystallographers, to illustrate their theories.
Almost all tessellations with repetition (including those in Figures 1 and 2) are periodic, that is, there is a smallest patch of the tiling that can be translated repeatedly by two independent vectors to fill out the whole tiling. It is natural to want repetition in a predictable manner to lay tile, to print, or to weave a pattern. Symmetry groups of periodic tilings are known as two-dimensional crystallographic groups, because crystals, by definition (until very recently), were defined by their periodic molecular structure. These groups consist of the translations, rotations, reflections, and glide-reflections that can act on the tiling in such a way that each tile moves to fit exactly onto another, leaving the tiling invariant. There are only 17 distinct symmetry groups for periodic tilings in the Euclidean plane; these are frequently used to classify tilings. Colored tilings (such as an extended checkerboard or Escher’s tilings) can be analyzed according to color symmetries, which permute colors of tiles as well as positions of tiles in the tiling.
Nonperiodic tessellations have no translation symmetry. Although a regular tiling by squares can be made
Figure 3. A Penrose tiling by kites and darts requires matching vertices of the same color.
nonperiodic by shifting a few rows a small distance, the most interesting nonperiodic tessellations are produced by an aperiodic set of prototiles—every tiling formed by such a set is nonperiodic. It is not known if there is a single aperiodic tile, but there are several known aperiodic sets of two or more prototiles. The most well-known pairs were discovered by Roger Penrose in the 1970s: a kite and a dart (or a thick and a thin rhombus). In each Penrose tiling formed by these pairs (Figure 3), every patch of the tiling repeats infinitely often, but never by a translation that leaves the tiling invariant. Some properties of Penrose tilings are similar to those exhibited by unusual alloys discovered in the 1980s. These were named quasicrystals because their X-ray diffraction patterns displayed a crystal-like orderly repetition of bright spots, but also exhibited rotation symmetry forbidden in a periodic structure. Many unusual properties of Penrose and other aperiodic tilings have been discovered, and various techniques have been developed to study their structure (Senechal, 1995). Yet the study of aperiodic tilings and of quasicrystals is in its infancy, and it remains to be seen if the connections between them are more than superficial.
A Voronoï (or Dirichlet) tessellation is determined by a given discrete set S of points in a surface. Each point x of S is surrounded by the region of points that are closer to x than to any others in S. The boundaries of these regions consist of those points that lie equidistant between at least two points of S. These regions with their boundaries (called Voronoï polygons or Dirichlet domains) are the tiles of the Voronoï tessellation determined by S. Such tessellations arise naturally in a wide variety of applications in physics (Wigner-Seitz regions); crystallography (Wirkungsbereich); physiology (capillary domains); urban planning (regions of service for schools or fire stations); biology (modeling cell arrangement); statistical spatial data analysis; and many other areas. Such tilings are almost always nonperiodic with many differently shaped tiles. Mathematical properties of such tilings and various algorithms to construct them are vigorous areas of research (Okabe et al., 1992).
Tessellations on surfaces such as spheres or the hyperbolic plane and tessellations of space of three or higher dimensions are equally important. Which shapes can fill space, and in what manner, is of great interest to scientists (and manufacturers). Little is known in this area; there is not even a complete list of space-filling tetrahedra. Other topics of investigation are tessellations with special tiles (e.g., polyominoes, or fractal tiles), relationships between local and global properties, classification, and construction of tessellations with special properties.
Bezdek, K. 2000. Space filling. In Handbook of Discrete and Combinatorial Mathematics, edited by K.H.Rosen, Boca Raton: CRC Press, pp. 824–830
Goodman, J.E. & O’Rourke, J. (editors). 2004. Handbook of Discrete and Computational Geometry, 2nd edition, Boca Raton: CRC Press (Relevant chapters: Tiling, D. Schattschneider & M.Senechal; Polyominoes, D.Klarner & S.W.Golomb; Voronoï diagrams and Delaunay triangulations, S.Fortune; Crystals and Quasicrystals, M.Senechal)
Gruber, P.M. & Wills, J.M. (editors). 1993. Handbook of Convex Geometry, Amsterdam: North-Holland (Relevant chapters: Geometric algorithms, H.Edelsbrunner, vol. A, pp. 699–735; Tilings, E.Schulte, vol. B, pp. 899–932); Geometric crystallography, P.Engel, vol. B, pp. 989–1041)
Grünbaum, B. & Shephard, G.C. 1987. Tilings and Patterns, New York: Freeman
Okabe, A., Boots, B. & Sugihara, K. 1992. Spatial Tessellations: Concepts and Applications of Voronoï Diagrams, 2nd edition, New York: Wiley (Okabe, A., Boots, B., Sugihara, K. & Chiu, S.N. 2000)
Patera, J. (editor). 1998. Quasicrystals and Discrete Geometry, Providence, RI: American Mathematical Society
Schattschneider, D. 2004. M.C. Escher: Visions of Symmetry, new edition, New York: Abrams
Schulte, E. 2002. Tilings. In Encyclopedia of Physical Science and Technology, 3rd edition, vol. 16, New York: Academic Press, pp. 763–782
Senechal, M. 1995. Quasicrystals and Geometry, Cambridge and New York: Cambridge University Press
Washburn, D.K. & Crowe, D.W. 1988. Symmetries of Culture: Symmetry and Practice of Plane Pattern Analysis, Seattle: University of Washington Press
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