dividing a square into similar rectangles
{{Short description|Mathematical problem}}
File:Plastic square partitions.svg
Dividing a square into similar rectangles (or, equivalently, tiling a square with similar rectangles) is a problem in mathematics.
Three rectangles
{{See also|Plastic ratio}}
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:Ian Stewart, A Guide to Computer Dating (Feedback), Scientific American, Vol. 275, No. 5, November 1996, p. 118
- 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.{{citation
| last1 = Freiling | first1 = C.
| last2 = Rinne | first2 = D.
| doi = 10.4310/MRL.1994.v1.n5.a3
| issue = 5
| journal = Mathematical Research Letters
| mr = 1295549
| pages = 547–558
| title = Tiling a square with similar rectangles
| volume = 1
| year = 1994| doi-access = free
| last1 = Laczkovich | first1 = M.
| last2 = Szekeres | first2 = G. | author2-link = George Szekeres
| doi = 10.1007/BF02574063
| issue = 3–4
| journal = Discrete & Computational Geometry
| mr = 1318796
| pages = 569–572
| title = Tilings of the square with similar rectangles
| volume = 13
| year = 1995| doi-access = free
}}
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.{{Cite web |last=Baez |first=John |date=2022-12-22 |title=Dividing a Square into Similar Rectangles |url=https://golem.ph.utexas.edu/category/2022/12/dividing_a_square_into_similar.html |access-date=2023-03-09 |website=golem.ph.utexas.edu |language=en}}{{Cite web |date=2022-12-15 |title=John Carlos Baez (@johncarlosbaez@mathstodon.xyz) |url=https://mathstodon.xyz/@johncarlosbaez/109517010782719784 |access-date=2023-03-09 |website=Mathstodon |language=en}}
The problem has two parts: what aspect ratios are possible, and how many different solutions are there for a given n.{{Cite news |last=Roberts |first=Siobhan |date=2023-02-07 |title=The Quest to Find Rectangles in a Square |language=en-US |work=The New York Times |url=https://www.nytimes.com/2023/02/07/science/puzzles-rectangles-mathematics.html |access-date=2023-03-09 |issn=0362-4331}} 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. 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.{{Cite web |title=cutting squares into similar rectangles using a computer program |url=http://ianhenderson.org/similar-rectangles/ |access-date=2023-03-09 |website=ianhenderson.org}}
The numbers of distinct valid dissections for different values of n, for n = 1, 2, 3, ..., are:{{Cite web |date=2023-03-06 |title=Dividing a Square into 7 Similar Rectangles |url=https://johncarlosbaez.wordpress.com/2023/03/06/dividing-a-square-into-7-similar-rectangles/ |access-date=2023-03-09 |website=Azimuth |language=en|first =John Carlos|last= Baez }}
{{bi|left=1.6|1, 1, 3, 11, 51, 245, 1372, 8522, ... {{OEIS|A359146}}.}}
See also
References
{{reflist}}
External links
- [https://gist.github.com/narain/820e5e0c10ace722caf7b8a000d9f212 Python code for dissection of a square into n similar rectangles via "guillotine cuts"] by Rahul Narain
{{Tessellation}}
Category:Rectangular subdivisions