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Square Tiling


There are a number of interesting results related to the tiling of squares. For example, M. Laczkovich has shown that there are exactly three shapes of non-right triangles that tile the square with similar copies, corresponding to angles (pi/8,pi/4,5pi/8), (pi/4,pi/3,5pi/12), and (pi/12,pi/4,2pi/3) (Stein and Szabó 1994). In particular, given triangles of shape 1-2-sqrt(5) with no two the same size, tile the square. The best known solution has 8 triangles (Berlekamp 1999).

The total number of squares contained in a grid of n×n unit square is the square pyramidal number

 N(n)=sum_(k=1)^nk^2=1/6n(n+1)(2n+1).

See also

Square Grid, Tiling

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References

Berlekamp, E. and Rodgers, T. (Eds.). The Mathemagician and the Pied Puzzler: A Collection in Tribute to Martin Gardner. Boston, MA: A K Peters, 1999.Flannery, S. and Flannery, D. In Code: A Mathematical Journey. London: Profile Books, p. 134, 2000.Laczkovich, M. "Tilings of Polygons with Similar Triangles." Combinatorica 10, 281-306, 1990.Schattschneider, D. "Unilateral and Equitransitive Tilings by Squares." Disc. Comput. Geom. 24, 519-525, 2000.Stein, S. and Szabó, S. Algebra and Tiling: Homomorphisms in the Service of Geometry. Washington, DC: Math. Assoc. Amer., 1994.

Referenced on Wolfram|Alpha

Square Tiling

Cite this as:

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

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