Three partitions of 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]

  1. The trivial solution given by three congruent rectangles with aspect ratio 3:1.
  2. 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.
  3. 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

  1. ↑ Ian Stewart, A Guide to Computer Dating (Feedback), Scientific American, Vol. 275, No. 5, November 1996, p. 118
  2. ↑ Spinadel, Vera W. de; Redondo Buitrago, Antonia (2009), "Towards Van der Laan's Plastic Number in the Plane" (PDF), Journal for Geometry and Graphics, 13 (2): 163–175.
  3. 1 2 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, MR 1295549
  4. ↑ Laczkovich, M.; Szekeres, G. (1995), "Tilings of the square with similar rectangles", Discrete & Computational Geometry, 13 (3–4): 569–572, doi:10.1007/BF02574063, MR 1318796
  5. ↑ Baez, John (2022-12-22). "Dividing a Square into Similar Rectangles". golem.ph.utexas.edu. Retrieved 2023-03-09.
  6. ↑ "John Carlos Baez (@johncarlosbaez@mathstodon.xyz)". Mathstodon. 2022-12-15. Retrieved 2023-03-09.
  7. 1 2 Roberts, Siobhan (2023-02-07). "The Quest to Find Rectangles in a Square". The New York Times. ISSN 0362-4331. Retrieved 2023-03-09.
  8. ↑ "cutting squares into similar rectangles using a computer program". ianhenderson.org. Retrieved 2023-03-09.
  9. ↑ Baez, John Carlos (2023-03-06). "Dividing a Square into 7 Similar Rectangles". Azimuth. Retrieved 2023-03-09.
  10. ↑ "A359146: Divide a square into n similar rectangles; a(n) is the number of different proportions that are possible". On-Line Encyclopedia of Integer Sequences. Retrieved 2023-03-09.
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