Find the minimum size square capable of bounding equal squares arranged in any configuration. The first few cases are illustrated above (Friedman). The only packings which have been proven optimal are 2, 3, 5, 6, 7, 8, 14, 15, 24, and 35, in addition to the trivial cases of the square numbers (Friedman).
If for some , it is conjectured that the size of the minimum bounding square is for small . The smallest for which the conjecture is known to be violated is (with ).
The following table gives the smallest known side lengths for a square into which unit squares can be packed (Friedman 2005). An asterisk (*)indicates that a packing has been proven to be optimal.
| exact | approx. | exact | approx. | ||
| 1* | 1 | 1 | 16* | 4 | 4 |
| 2* | 2 | 2 | 17 | 4.6755... | |
| 3* | 2 | 2 | 18 | 4.822... | |
| 4* | 2 | 2 | 19 | 4.885... | |
| 5* | 2.707... | 20 | 5 | 5 | |
| 6* | 3 | 3 | 21 | 5 | 5 |
| 7* | 3 | 3 | 22 | 5 | 5 |
| 8* | 3 | 3 | 23* | 5 | 5 |
| 9* | 3 | 3 | 24* | 5 | 5 |
| 10* | 3.707... | 25* | 5 | 5 | |
| 11 | 3.877... | 26 | 5.6214... | ||
| 12 | 4 | 4 | 27 | 5.7072... | |
| 13 | 4 | 4 | 28 | 5.8285... | |
| 14* | 4 | 4 | 29 | 5.9465... | |
| 15* | 4 | 4 |
The best known packings of squares into a circle are illustrated above for the first few cases (Friedman).
The best known packings of squares into an equilateral triangle are illustrated above for the first few cases (Friedman).
The best packing of a square inside a pentagon, illustrated above, is 1.0673....
See also
Circle Packing, Packing, Square Dissection, Triangle Packing
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References
Erdős, P. and Graham, R. L. "On Packing Squares with Equal Squares." J. Combin. Th. Ser. A 19, 119-123, 1975.Friedman, E. "Erich's Packing Center." http://www.stetson.edu/~efriedma/packing.html.Friedman, E. "Circles in Squares." http://www.stetson.edu/~efriedma/cirinsqu/.Friedman, E. "Squares in Squares." http://www.stetson.edu/~efriedma/squinsqu/.Friedman, E. "Triangles in Squares." http://www.stetson.edu/~efriedma/triinsqu/.Friedman, E. "Packing Unit Squares in Squares." Elec. J. Combin. DS7, 1-24, Oct. 31, 2005. http://www.combinatorics.org/Surveys/ds7.html.Gardner, M. "Packing Squares." Ch. 20 in Fractal Music, Hypercards, and More Mathematical Recreations from Scientific American Magazine. New York: W. H. Freeman, pp. 289-306, 1992.Göbel, F. "Geometrical Packing and Covering Problems." In Packing and Covering in Combinatorics (Ed. A. Schrijver). Amsterdam: Tweede Boerhaavestraat, 1979.Hoffman, P. The Man Who Loved Only Numbers: The Story of Paul Erdős and the Search for Mathematical Truth. New York: Hyperion, p. 174, 1998.Roth, L. F. and Vaughan, R. C. "Inefficiency in Packing Squares with Unit Squares." J. Combin. Th. Ser. A 24, 170-186, 1978.
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Cite this as:
Weisstein, Eric W. "Square Packing." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/SquarePacking.html