Dividing a square into similar rectangles explained

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:^{[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.

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.^{[3] [4]}

The problem has two parts: what aspect ratios are possible, and how many different solutions are there for a given n.^[5] 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.^[6]

The numbers of distinct valid dissections for different values of n, for n = 1, 2, 3, ..., are:^{[5] [7]}

See also

• Squaring the square

External links

• Python code for dissection of a square into n similar rectangles via "guillotine cuts" by Rahul Narain

Notes and References

1. Ian Stewart, A Guide to Computer Dating (Feedback), Scientific American, Vol. 275, No. 5, November 1996, p. 118
2. .
3. Web site: Baez . John . 2022-12-22 . Dividing a Square into Similar Rectangles . 2023-03-09 . golem.ph.utexas.edu . en.
4. Web site: 2022-12-15 . John Carlos Baez (@[e-mail]) . 2023-03-09 . Mathstodon . en.
5. News: Roberts . Siobhan . 2023-02-07 . The Quest to Find Rectangles in a Square . en-US . The New York Times . 2023-03-09 . 0362-4331.
6. Web site: cutting squares into similar rectangles using a computer program . 2023-03-09 . ianhenderson.org.
7. Web site: 2023-03-06 . Dividing a Square into 7 Similar Rectangles . 2023-03-09 . Azimuth . en. John Carlos. Baez .