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Polyomino Tiling


HeptominoNontiling

A polyomino tiling is a tiling of the plane by specified types of polyominoes. Tiling by polyominoes has been investigated since at least the late 1950s, particularly by S. Golomb (Wolfram 2002, p. 943).

Interestingly, the Fibonacci number F_(n+1) gives the number of ways for 2×1 dominoes to cover a 2×n checkerboard.

Each monomino, domino, triomino, tetromino, pentomino, and hexomino tiles the plane without requiring flipping. In addition, each heptomino with the exception of the four illustrated above can tile the plane, also without flipping (Schroeppel 1972).

PolyominoTilingAperiodic

Recently, sets of polyominoes that force non-periodic patterns have been found. The set illustrated at left above was announced by Roger Penrose in 1994, and the slightly smaller set illustrated at right below was found by Matthew Cook (Wolfram 2002, p. 943).

PolyominoTilingNested

Both of these sets yield nested patterns, as illustrated above for Cook's tiles (Wolfram 2002, p. 943).

Consider now those collections of all n-ominoes which form a rectangle. The polynomials of orders n=1 and n=2 form only a square and rectangle, respectively. The two polyominoes of order n=3 cannot form a rectangle, nor can the five polyominoes of order n=4 or the 35 polyominoes of order n=6 (Beeler 1972). There are several rectangles formed by the 12 polyominoes of order n=5, as summarized in the following table (Fletcher 1965, Beeler 1972).

sizesolutions
3×202
4×15368
5×121010
6×102339
two 5×62
8×8 with 2×2 hole65

See also

Anisohedral Tiling, Domino, Fibonacci Number, Isohedral Tiling, Polyhex Tiling, Polyiamond Tiling, Polyomino

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References

Beauquier, D. and Nivat, M. "On Translating One Polyomino to Tile the Plane." Disc. Comput. Geom. 6, 575-592, 1991.Beeler, M. Item 112 in Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239, pp. 48-50, Feb. 1972. http://www.inwap.com/pdp10/hbaker/hakmem/polyominos.html#item112.Fletcher, J. G. "A Program to Solve the Pentomino Problem by the Recursive Use of Macros." Comm. ACM 8, 621-623, 1965.Friedman, E. "Puzzle of the Month (February 1999)." https://erich-friedman.github.io/mathmagic/0299.html.Gardner, M. "Tiling with Polyominoes, Polyiamonds, and Polyhexes." Ch. 14 in Time Travel and Other Mathematical Bewilderments. New York: W. H. Freeman, pp. 177-187, 1988.Martin, G. Polyominoes: A Guide to Puzzles and Problems in Tiling. Washington, DC: Math. Assoc. Amer., 1991.Myers, J. "Polyomino Tiling." http://www.srcf.ucam.org/~jsm28/tiling/.Rawsthorne, D. A. "Tiling Complexity of Small n-Ominoes (n<10)." Disc. Math. 70, 71-75, 1988.Schroeppel, R. Item 109 in Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239, p. 48, Feb. 1972. http://www.inwap.com/pdp10/hbaker/hakmem/polyominos.html#item109.Vichera, M. "Polyominoes." http://www.vicher.cz/puzzle/polyform/minio/polynom.htm.Weisstein, E. W. "Books about Polyominoes." http://www.ericweisstein.com/encyclopedias/books/Polyominoes.html.Wijshoff, H. A. G. and van Leeuwen, J. "Arbitrary Versus Periodic Storage Schemes and Tessellations of the Plane Using One Type of Polyomino." Inform. and Control 62, 1-25, 1984.Wolfram, S. A New Kind of Science. Champaign, IL: Wolfram Media, p. 943, 2002.

Referenced on Wolfram|Alpha

Polyomino Tiling

Cite this as:

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

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