Английская Википедия: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]
Шаблон:As of, the following results Шаблон: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
External links
- Python code for dissection of a square into n similar rectangles via "guillotine cuts" by Rahul Narain
- ↑ Ian Stewart, A Guide to Computer Dating (Feedback), Scientific American, Vol. 275, No. 5, November 1996, p. 118
- ↑ Шаблон:Citation.
- ↑ 3,0 3,1 Шаблон:Citation
- ↑ Шаблон:Citation
- ↑ Шаблон:Cite web
- ↑ Шаблон:Cite web
- ↑ 7,0 7,1 Шаблон:Cite news
- ↑ Шаблон:Cite web
- ↑ Шаблон:Cite web
- ↑ Шаблон:Cite web
