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Tiling

A plane-filling arrangement of plane figures or its generalization to higher dimensions. Formally, a tiling is a collection of disjoint open sets, the closures of which cover the plane. Given a single tile, the so-called first corona is the set of all tiles that have a common boundary point with the tile (including the original tile itself).

Wang's conjecture (1961) stated that if a set of tiles tiled the plane, then they could always be arranged to do so periodically. A periodic tiling of the plane by polygons or space by polyhedra is called a tessellation. The conjecture was refuted in 1966 when R. Berger showed that an aperiodic set of tiles exists. By 1971, R. Robinson had reduced the number to six and, in 1974, R. Penrose discovered an aperiodic set (when color-matching rules are included) of two tiles: the so-called Penrose tiles. It is not known if there is a single aperiodic tile.

A spiral tiling using a single piece is illustrated on the cover of Grünbaum and Shephard (1986).

The number of tilings possible for convex irregular polygons are given in the following table.

 name known tilings 3 triangle tiling all 4 quadrilateral tiling all 5 pentagon tiling 14 6 hexagon tiling 3

There are no tilings for identical convex -gons for , although non-identical convex heptagons can tile the plane (Steinhaus 1999, p. 77; Gardner 1984, pp. 248-249).

Anisohedral Tiling, Aperiodic Tiling, Corona, Domino Tiling, Gosper Island, Harborth's Tiling, Heesch Number, Heesch's Problem, Honeycomb Conjecture, Isohedral Tiling, Koch Snowflake, Monohedral Tiling, Penrose Tiles, Polygon Tiling, Polyomino Tiling, Space-Filling Polyhedron, Square Tiling, Tessellation, Tiling Theorem, Triangle Tiling, Wallpaper Groups

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References

Eppstein, D. "Tiling." http://www.ics.uci.edu/~eppstein/junkyard/tiling.html.Gardner, M. The Sixth Book of Mathematical Games from Scientific American. Chicago, IL: University of Chicago Press, pp. 248-249, 1984.Gardner, M. "Tilings with Convex Polygons." Ch. 13 in Time Travel and Other Mathematical Bewilderments. New York: W. H. Freeman, pp. 162-176, 1988.Gardner, M. "Penrose Tiling" and "Penrose Tiling II." Chs. 1-2 in Penrose Tiles and Trapdoor Ciphers... and the Return of Dr. Matrix, reissue ed. New York: W. H. Freeman, pp. 1-29, 1989.Grünbaum, B. and Shepard, G. C. "Some Problems on Plane Tilings." In The Mathematical Gardner (Ed. D. Klarner). Boston, MA: Prindle, Weber, and Schmidt, pp. 167-196, 1981.Grünbaum, B. and Shephard, G. C. Tilings and Patterns. New York: W. H. Freeman, 1986.Pappas, T. "Mathematics & Moslem Art." The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, p. 178, 1989.Peterson, I. The Mathematical Tourist: Snapshots of Modern Mathematics. New York: W. H. Freeman, pp. 82-85, 1988.Rawles, B. Sacred Geometry Design Sourcebook: Universal Dimensional Patterns. Nevada City, CA: Elysian Pub., 1997.Schattschneider, D. "In Praise of Amateurs." In The Mathematical Gardner (Ed. D. Klarner). Boston, MA: Prindle, Weber, and Schmidt, pp. 140-166, 1981.Seyd, J. A. and Salman, A. S. Symmetries of Islamic Geometrical Patterns. River Edge, NJ: World Scientific, 1995.Stein, S. and Szabó, S. Algebra and Tiling: Homomorphisms in the Service of Geometry. Washington, DC: Math. Assoc. Amer., 1994.Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, 1999.Stevens, P. S. Handbook of Regular Patterns: An Introduction to Symmetry in Two Dimensions. Cambridge, MA: MIT Press, 1992.Weisstein, E. W. "Books about Tilings." http://www.ericweisstein.com/encyclopedias/books/Tilings.html.Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 177-179, 208, and 211, 1991.

Tiling

Cite this as:

Weisstein, Eric W. "Tiling." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Tiling.html