Dividing a square into similar rectangles
Dividing a square into similar rectangles (or, equivalently, tiling a square with similar rectangles) is a problem in mathematics.
Three rectangles
There is only one way (up to rotation and reflection) to divide a square into two similar rectangles.
However, there are three distinct ways of partitioning a square into three similar rectangles:[1][2]
- The trivial solution given by three congruent rectangles with aspect ratio 3:1.
- The solution in which two of the three rectangles are congruent and the third one has twice the side length of the other two, where the rectangles have aspect ratio 3:2.
- The solution in which the three rectangles are all of different sizes and where they have aspect ratio ρ2, where ρ is the plastic ratio.
The fact that a rectangle of aspect ratio ρ2 can be used for dissections of a square into similar rectangles is equivalent to an algebraic property of the number ρ2 related to the Routh–Hurwitz theorem: all of its conjugates have positive real part.[3][4]
Generalization to n rectangles
In 2022, the mathematician John Baez brought the problem of generalizing this problem to n rectangles to the attention of the Mathstodon online mathematics community.[5][6]
The problem has two parts: what aspect ratios are possible, and how many different solutions are there for a given n.[7] Frieling and Rinne had previously published a result in 1994 that states that the aspect ratio of rectangles in these dissections must be an algebraic number and that each of its conjugates must have a positive real part.[3] However, their proof was not a constructive proof.
Numerous participants have attacked the problem of finding individual dissections using exhaustive computer search of possible solutions. One approach is to exhaustively enumerate possible coarse-grained placements of rectangles, then convert these to candidate topologies of connected rectangles. Given the topology of a potential solution, the determination of the rectangle's aspect ratio can then trivially be expressed as a set of simultaneous equations, thus either determining the solution exactly, or eliminating it from possibility.[8]
(As of March 2023), the following results (sequence A359146 in the OEIS) have been obtained for the number of distinct valid dissections for different values of n:[7][9][10]
| n | # of dissections |
|---|---|
| 1 | 1 |
| 2 | 1 |
| 3 | 3 |
| 4 | 11 |
| 5 | 51 |
| 6 | 245 |
| 7 | 1372 |
| 8 | 8522 |
See also
References
- ↑ Ian Stewart, A Guide to Computer Dating (Feedback), Scientific American, Vol. 275, No. 5, November 1996, p. 118
- ↑ Spinadel, Vera W. de; Redondo Buitrago, Antonia (2009), "Towards Van der Laan's Plastic Number in the Plane", Journal for Geometry and Graphics 13 (2): 163–175, http://www.heldermann-verlag.de/jgg/jgg13/j13h2spin.pdf.
- ↑ 3.0 3.1 Freiling, C.; Rinne, D. (1994), "Tiling a square with similar rectangles", Mathematical Research Letters 1 (5): 547–558, doi:10.4310/MRL.1994.v1.n5.a3
- ↑ Laczkovich, M. (1995), "Tilings of the square with similar rectangles", Discrete & Computational Geometry 13 (3–4): 569–572, doi:10.1007/BF02574063
- ↑ Baez, John (2022-12-22). "Dividing a Square into Similar Rectangles" (in en). https://golem.ph.utexas.edu/category/2022/12/dividing_a_square_into_similar.html.
- ↑ "John Carlos Baez (@johncarlosbaez@mathstodon.xyz)" (in en). 2022-12-15. https://mathstodon.xyz/@johncarlosbaez/109517010782719784.
- ↑ 7.0 7.1 Roberts, Siobhan (2023-02-07). "The Quest to Find Rectangles in a Square" (in en-US). The New York Times. ISSN 0362-4331. https://www.nytimes.com/2023/02/07/science/puzzles-rectangles-mathematics.html.
- ↑ "cutting squares into similar rectangles using a computer program". http://ianhenderson.org/similar-rectangles/.
- ↑ Baez, John Carlos (2023-03-06). "Dividing a Square into 7 Similar Rectangles" (in en). https://johncarlosbaez.wordpress.com/2023/03/06/dividing-a-square-into-7-similar-rectangles/.
- ↑ "A359146: Divide a square into n similar rectangles; a(n) is the number of different proportions that are possible.". https://oeis.org/A359146.
External links
- Python code for dissection of a square into n similar rectangles via "guillotine cuts" by Rahul Narain
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