If you divide a square into some fixed number of similar rectangles, what proportions can these rectangles have? We’ve been having fun thinking about this on Mathstodon, and here is a report.
If you divide a square into 3 similar rectangles, what proportions can these rectangles have? There are three options. The third is more complicated than the first two:
• We can divide the square into three rectangles that are 1/3 as long in one direction as the other, as in the first picture.
• We can divide it into three rectangles that are 2/3 as long in one direction as the other, as in the second picture.
• We can divide it into three rectangles that are x times as long in one direction as the other, as in the third picture.
What’s x? The yellow rectangle has height 1 and width x, so the blue rectangle has width 1-x and height x(1-x), while the red one has width 1-x and height (1-x)/x. The heights of the blue and red rectangles must sum to 1, so
x(1-x) + (1-x)/x = 1
x²(1-x) + (1-x) = x
x² – x³ + 1 – x = x
x³ – x² + 2x – 1 = 0
and the solution of this cubic is
x ≈ 0.56984
The reciprocal of this number is the square of a famous constant: the plastic ratio, ρ. This is like a cheap imitation of the golden ratio, with
ρ3 = ρ + 1
Why is the third option so much more complicated than the other two? As we’ll see, it’s because the first two have all the rectangles ‘pointing the same way’, while the third does not: it has a mix of rectangles ‘standing up’ and ‘lying down’.
To see the pattern, it helps to do a harder problem.
If you divide a square into 4 similar rectangles, what proportions can these rectangles have? A lot of people on Mathstodon worked on this puzzle! Here Dan Piker lists the 11 options we’ve found:
Note: there are more than 11 ways to divide a square into 4 similar rectangles, since you can rotate and reflect these pictures, and also rearrange the rectangles in some of them. But we’ve only found 11 possible proportions for the rectangles. Stefano Gogioso has sketched a proof that these are all the options, and Rahul Narain has given a computerized proof that these are all the options obtained from ‘guillotine cuts’:
• Wikipedia, Guillotine partition.
A ‘guillotine cut’ is a straight line going from one edge of an existing polygon to the opposite edge. And I believe there’s no way to dissect a square into 4 rectangles that doesn’t use guillotine cuts.
In the process of listing the options, we discovered something interesting: the rectangles have rational proportions only in options 1, 4, 7, 10 and 11 here:
And these are precisely the options where all the rectangles are pointing the same way! (They happen to all be lying down, wider than they are tall. But of course they’d all be standing up if we rotated the pictures.)
Puzzle 1. Show that if you subdivide a square into n similar rectangles that are all pointing the same way, the ratio x of their short side to their long side must be a rational number.
Puzzle 2. Show that the converse is not true.
We also discovered another pattern, too: when x is not rational, it is the solution to a cubic equation with integer coefficients.
Does that pattern persist when we subdivide a square into 5 similar rectangles? No, alas! Ian Henderson and Rahul Narain seem to have shown that exactly 51 proportions are possible when we subdivide a square into 5 similar rectangles. Henderson drew them:
while Narain listed the polynomial equations obeyed by the 51 possible proportions. Some obey a quartic equation with integer coefficients, not a cubic.
To be honest, Narain only considered ways of subdividing a square into 5 similar rectangles using guillotine cuts. But this should be okay, since I believe there’s just one way to subdivide a square into 5 rectangles that cannot be done using guillotine cuts. It’s shown here in a paper by Robert Dawson, in fact:
However, when we try to make all 5 rectangles similar, the central one shrinks to a point.
Ian Henderson also believes that when you subdivide a square into 6 similar rectangles, 245 proportions of rectangles are possible:
And for subdivisions of a square into 7 similar rectangles, he believes 1371 proportions are possible:
You can click to enlarge these last two pictures.
However, he adds, “I’m not 100% confident in these numbers.” So, someone should check his work.
Puzzle 3. Show that you can divide a square into 6 similar rectangles in a way that cannot be done via guillotine cuts.
Okay, back to something simpler: what proportions are possible when we divide a square into 4 similar rectangles? Let’s work it out!
For each option we get a number 0 < x ≤ 1 describing the proportion of the rectangles that subdivide the square. This number is the length of the short side divided by the length of the long side. The 11 options below are listed from the smallest possible value of x to the largest.
1) Option 1 is to divide the 1 × 1 square into 4 rectangles that are 1/4 as tall as they are wide. So, the number we get from this option is 1/4.
Note that if we rotated this option we’d get tall skinny rectangles, but the number would still be 1/4.
2) In option 2, the bottom two rectangles have width 1 and height x. Thus the top two have height 1-2x.
Since the green rectangle is x times as tall as it is wide, its width must be (1-2x)/x. The yellow rectangle thus has width
1 – (1-2x)/x = 1 − x⁻¹ + 2 = 3 – x⁻¹
Its height is this divided by x, namely 3x⁻¹ – x⁻². But we know its height is 1-2x, so
1-2𝑥 = 3x⁻¹ – x⁻²
2x³ – x² + 3x – 1 = 0
x ≈ 0.34563
3) For option 3 the red rectangle has height x, so the blue one has height 1-x and width x(1-x) = x−x², so the other two have width 1-(x−x²) = 1-x+x² and height x−x²+x³. The total height of red, green and yellow is 1 so x + 2(x−x²+x³) = 1 or
2x³ – 2x² + 3x – 1 = 0
x ≈ 0.39661
4) Option 4 is nice: it has left-right symmetry, and its rectangles can be rearranged to give another option with left-right symmetry.
The red and blue rectangles have height x. The green and yellow ones thus have height 1-2x, and thus width (1-2x)/x. But by symmetry we know they must have width 1/2, so
(1-2x)/x = 1/2
1-2x = x/2
1 = 5x/2
x = 2/5 = 0.4
Not a cubic equation this time—linear! So x is rational.
Notice that options 3, 5, 6, and 7 are all topologically the same! They were analyzed by Lisanne, and her work helped me a lot.
5) In option 5 the red rectangle has height x, so the blue has height 1-x and thus width (1-x)/x. The yellow and green thus have width 1 – (1-x)/x = 2 – x⁻¹, hence height (2 – x⁻¹)/x = 2x⁻¹ – x⁻² .
The heights of yellow, green and red must sum to 1:
2(2x⁻¹ – x⁻²) + x = 1
so we get a cubic:
x³ − x² + 4x – 2= 0
and the solution is
x ≈ 0.53318
6) In option 6 the red rectangle again has height x, so the blue again has height 1-x and width (1-x)/x.
The yellow and green again have width 1 – (1-x)/x = 2 – x⁻¹, but now they’re different: the yellow has height x(2 – x⁻¹) while the blue has height (2 – x⁻¹)/x.
The heights of yellow, green and red sum to 1:
x(2 – x⁻¹) + (2 – x⁻¹)/x + x = 1
3x³ – 2x² + 2x – 1 = 0
x ≈ 0.55232
7) In option 7, like options 5 and 6, the red rectangle has height x, so the blue has height 1-x and width (1-x)/x.
Thus, the yellow and green again have width 1 – (1-x)/x = 2 – x⁻¹. But this time both have height x(2 – x⁻¹).
Yet again, the heights of yellow, green and red sum to 1:
x + 2x(2 – x⁻¹) = 1
5x = 3
x = 3/5 = 0.6
The equation was linear, so x is rational!
Options 8, 9, and 10 are also all topologically the same.
8) In option 8 the red rectangle has height x so the blue has height 1-x and width (1-x)/x. The green and yellow also have height 1-x, but width x(1-x).
The widths of blue, green and yellow sum to 1:
(1-x)/x + 2x(1-x) = 1
2x³ − 2x² + 2x −1 = 0
x ≈ 0.64780
9) In option 9 the red rectangle has height x so the blue and green have height 1-x and width (1-x)/x. The yellow also has height 1-x, but width x(1-x).
The widths of blue, green and yellow sum to 1:
2(1-x)/x + x(1-x) = 1
2x⁻¹ – 2 + x – x² = 1
x³ – x² + 3x – 2 = 0
x ≈ 0.71523
10) Option 10 is more symmetrical than 8 or 9 since all three rectangles on top are congruent.
The red rectangle has height x so the blue, green and yellow all have height 1-x and thus width (1-x)/x.
The widths of blue, green and yellow sum to 1:
3(1-x)/x = 1
3(1-x) = x
3 = 4x
x = 3/4 = 0.75
Another linear equation, with a rational solution!
11) Finally, option 11 is the second one where all four rectangles are congruent. This time they’re squares! Clearly
x = 1
So, we see in these examples that x is rational precisely when all rectangles are ‘pointing the same way’: in the way Dan drew them, they’re all at least as wide as they are tall.
I’m not quite sure where to go with this research next. But I think the connection to double categories, hinted at in this paper with the picture of the pinwheel configuration, is interesting:
• Robert Dawson, A forbidden-suborder characterization of binarily-composable diagrams in double categories, Theory and Applications of Categories 1 (1995), 146–153.
Maybe we should seriously study the double category where 2-cells are rectangles that are subdivided into smaller rectangles by guillotine cuts!
But I also keep hoping there’s some interesting number-theoretic significance to the proportions that come up when we divide a square into the similar rectangles. Can anyone see interesting patterns in Rahul Narain’s table of the polynomials obeyed by these ratios when we divide a square into 5 similar rectangles? His u is my 1/x:
u - 5 u^3 - 4*u^2 + u - 3 u^3 - 4*u^2 + 2*u - 4 2*u - 7 u^3 - 4*u^2 + 3*u - 3 2*u^3 - 6*u^2 + u - 2 u^3 - 3*u^2 + 2*u - 5 u^3 - 3*u^2 + 2*u - 4 2*u^3 - 6*u^2 + 2*u - 2 u^3 - 3*u^2 + u - 1 3*u - 8 u^3 - 3*u^2 + 3*u - 5 2*u^3 - 5*u^2 + u - 2 u^5 - 3*u^4 + 3*u^3 - 4*u^2 + u - 1 -2*u^2 + 6*u - 3 2*u^3 - 5*u^2 + 2*u - 3 3*u - 7 u^4 - 3*u^3 + 3*u^2 - 4*u + 2 2*u^3 - 5*u^2 + 3*u - 3 u^3 - 3*u^2 + 4*u - 4 -u^2 + 4*u - 4 4*u - 8 3*u^3 - 6*u^2 + u - 1 2*u^3 - 4*u^2 + 2*u - 3 u^3 - 2*u^2 + 3*u - 5 3*u^3 - 6*u^2 + 2*u - 2 u^5 - 2*u^4 + 3*u^3 - 5*u^2 + u - 1 2*u^3 - 4*u^2 + 3*u - 4 4*u - 7 u^3 - 2*u^2 + 4*u - 6 -u^3 + 3*u^2 - 4*u + 3 2*u^3 - 5*u^2 + 4*u - 2 2*u^3 - 4*u^2 + 3*u - 3 3*u^3 - 5*u^2 + 2*u - 3 5*u - 8 u^5 - 2*u^4 + 3*u^3 - 4*u^2 + u - 1 -3*u^2 + 6*u - 2 2*u^4 - 4*u^3 + 3*u^2 - 3*u + 1 3*u^3 - 5*u^2 + 2*u - 2 4*u^3 - 6*u^2 + u - 1 -2*u^3 + 4*u^2 - 3*u + 2 2*u^3 - 3*u^2 + 3*u - 4 u^5 - 2*u^4 + 3*u^3 - 4*u^2 + 2*u - 1 5*u - 7 u^3 - 2*u^2 + 3*u - 3 2*u^3 - 3*u^2 + 2*u - 2 -u^3 + 2*u^2 - 4*u + 4 3*u^3 - 5*u^2 + 3*u - 2 3*u^3 - 4*u^2 + u - 1 4*u^3 - 6*u^2 + 2*u - 1 4*u - 5 2*u^3 - 3*u^2 + 4*u - 4 -5*u + 6 6*u - 7
I think the number theory side of things interacts in some nice way with guillotine partitions. Say we want to dissect a 1 × 1 square into similar rectangles using guillotine cuts. We can try to describe this process recursively: each guillotine cut slices an existing rectangle into two smaller one either horizontally or vertically. I believe there should be a way to keep track of this using some sort of a tree, but I haven’t figured it out yet. The total number of ways to do this slicing (just keeping track of the topology, not the position of the slices) is the nth Schröder number.
For example, the 4th Schröder number is 22, and here are the 22 topologies of guillotine partitions of a square into 4 rectangles:
The black dots are just an aid to thinking about these.
When the slicing is completed we need to say whether each of the final pieces is a ‘vertical’ or ‘horizontal’ copy of our similar rectangles (except perhaps in the case where they’re all squares). From this data we can get a polynomial equation that determines the number I was calling x above, which determines the proportions of our similar rectangles.
So, I’d like to systematize this and figure out how the polynomial obeyed by the number x is related to the combinatorial data we use to describe a guillotine partition and the labeling of each of the smallest rectangles by ‘vertical’ or ‘horizontal’.
“Okay, back to something simpler: what proportions are possible when we divide a square into 4 congruent rectangles? Let’s work it out!”
Judging from your answer, I think you mean “similar” rather than “congruent”. Same for many other occurrences of “congruent” in the text.
Yes, in all but two instances when I wrote “congruent” I meant “similar”. I’ve now tried to fix that.
This paper is a nice starting point for the ‘topological’ classification/counting of the rectangulations.
Bryan Dawei He,
A simple optimal binary representation of mosaic floorplans and Baxter permutations
Thanks very much! That paper is available here.
(Elsevier may be offering it to me for free because they’re spying on me and know I’m at U. C. Riverside, even though I deleted all their cookies a while back. I read a great blog article about this topic recently but now I can’t find it.)
Puzzle 3 is solved by multiple pictures of Ian Henderson already! look at Row 10, Col 3 for example.
A remark about the polynomials: The degree is always at most the number of rectangles and there are always configurations obtaining this bound. (Except for N = 2)
I can type out a proof for this later if someone wants to see it. In fact the polynomial is the determinant of a matrix with only +-1,+-X,0 in it.
I’d love to see the proof! And I’d love to see that matrix. I’m working on this stuff myself, slowly.
Start with a generic rectangulation into N rectangles. So no 4 rectangles are allowed to meet at a single corner. (This disallows for example 4 smaller squares inside the big square, which all meet at the center of the big square.)
Let be the long side of rectangle and the short side, so the ratio is . The initial square will have side length s.
Now for each vertical line segment (always take the full segment) in the rectangulation we get a relation
For the outer line segments (the sides of the square) do something similar with
And similarly for the horizontal segments.
Using some induction argument you can see that this always gives you relations. (This is easy for guillotine cuts a little bit harder in general). Here you need the generic property.
However the relation on the right side is implied by all the other vertical relations and the same is true for the bottom side and the horizontal relations. Removing these we end up with relations.
Some intense staring will show that a solution to these equations is not just necessary but also sufficient to get a similar-rectangulation.
Now we will have a closer look at the relations. We have variables ( and ). So we can fit them into an by matrix and we need to solve the equation . Now this has a (non-zero) solution if and only if and this is just a polynomial in . Since the column corresponding to the variable has only in it this polynomial can have degree at most .
This proof also suggests an efficient way to compute and . Write down the matrix, solve the determinant polynomial. Substitute this back in and solve a linear equation. You should be carefull that even if there is a solution some of the might be negative. Geometrically this corresponds to two rectangles overlapping, but the overlap also having the same ratios!
We can also estimate the coefficients of . There is a naive bound coming from the Leibniz formula saying that the sum of all the coefficients is at most Can we do better?
Thanks. I’m trying to understand this. Does mean “maybe we include a factor of maybe not?”
Yes it does!
How about the analogous problem of dividing a (n-)cube into similar rectangular (n-)cuboids? (One can’t “cube the cube” per https://en.wikipedia.org/wiki/Squaring_the_square#Cubing_the_cube — but finding similar rectangular (n-)cuboids in an (n-)cube is definitely a different problem than that — and there exist at least some n>2 solutions that aren’t totally trivial, e.g. a 3-cube divided into 5 rectangular 3-cuboids that are each 2/3 as long in one direction as in the other two directions [i.e. one “big one” and 4 other same-sized “small” rectangular 3-cuboids that have all dimensions reduced by half relative to the big one — a solution that is analogous to the second one of the 3 divisions of the square into 3 similar rectangles at the top of your post.) Do there exist some such solutions in more than 2 dimensions with irrational but algebraic ratios, perhaps somewhat analogous to the plastic ratio-related third of the 3 divisions of the square into 3 similar rectangles? (I’m a non-mathematical physicist, so I need to ask, rather than being able to answer it myself! :) My apologies if this very-closely-related question has already been discussed on Mathstodon!
People have raised this issue on Mathstodon but not really worked on it.
I think there’s a huge amount of interesting structure in the case of dividing a square into similar rectangles: see my blog article
• Guillotine partitions and the Hipparchus operad.
and also the comments on the original problem starting here—especially the comments by David Speyer and Jeffery Opoku.
Unfortunately I haven’t had time to work on these things in the last month.
There is now a continuation of my post “Dividing a square into similar rectangles”.
Some people on Mathstodon put a lot of work into this and made some nice progress. But there’s been a surprising new twist! I’m not talking about how the answer to this puzzle is now listed on the listed on the Online Encyclopedia of Integer Sequences as sequence A359146. I’m not even talking about the fact that the New York Times ran an article about this puzzle:
• Siobhan Roberts, The quest to find rectangles in a square, New York Times, February 7, 2023. Open-access version here.