Square packing
Square packing is a
Square packing in a square
Square packing in a square is the problem of determining the maximum number of unit squares (squares of side length one) that can be packed inside a larger square of side length . If is an integer, the answer is but the precise – or even
The smallest value of that allows the packing of unit squares is known when is a perfect square (in which case it is ), as well as for 2, 3, 5, 6, 7, 8, 10, 13, 14, 15, 24, 34, 35, 46, 47, and 48. For most of these numbers (with the exceptions only of 5 and 10), the packing is the natural one with axis-aligned squares, and is , where is the
The figure shows the optimal packings for 5 and 10 squares, the two smallest numbers of squares for which the optimal packing involves tilted squares.[4][5]The smallest unresolved case involves packing 11 unit squares into a larger square. 11 unit squares cannot be packed in a square of side length less than . By contrast, the tightest known packing of 11 squares is inside a square of side length approximately 3.877084 found by Walter Trump.[6][4]
Asymptotic results
What is the asymptotic growth rate of wasted space for square packing in a half-integer square?
For larger values of the side length , the exact number of unit squares that can pack an square remains unknown. It is always possible to pack a grid of axis-aligned unit squares, but this may leave a large area, approximately , uncovered and wasted.[4] Instead, Paul Erdős and Ronald Graham showed that for a different packing by tilted unit squares, the wasted space could be significantly reduced to (here written in
Some numbers of unit squares are never the optimal number in a packing. In particular, if a square of size allows the packing of unit squares, then it must be the case that and that a packing of unit squares is also possible.[2]
Square packing in a circle
Square packing in a circle is a related problem of packing n unit squares into a circle with radius as small as possible. For this problem, good solutions are known for n up to 35. Here are minimum solutions for n up to 12:[10]
Number of squares | Circle radius |
---|---|
1 | 0.707... |
2 | 1.118... |
3 | 1.288... |
4 | 1.414... |
5 | 1.581... |
6 | 1.688... |
7 | 1.802... |
8 | 1.978... |
9 | 2.077... |
10 | 2.121... |
11 | 2.214... |
12 | 2.236... |
See also
References
External links
- Friedman, Erich, "Squares in Squares", Github, Erich's Packing Center