It was in fact that very artricle that led me here to this page and to join in on the discussion.

Believe it or not, and for the record, back in 2017, in connection with my independent investigations into similarity tilings of squares, I’d taken up working on this very problem; and, after a couple of months of painstaking, on and off again efforts,

I’d actually succeeded in working out by hand the case solutions, as well as the polynomials and their evaluations, for n = 4 (11 solutions) and n = 5 (51 solutions).

I meticulously recorded each of these solutions as rough sketches, along with the accompanying polynomials and their associated evaluations, on separate 3X5 index cards and filed them away for safe keeping, thinking that some time in the future I would pursue the matter further and possibly publish and share my findings; but, alas, it appears that I waited just a bit too long.

My only consolation (if that’s what it can properly be called) is that I know in my heart that, oddly enough, the idea to pursue this most intriguing and delightful geometric tiling matter in earnest had evidently occurred to me before it had occurred to anyone else.

I’d be happy to post photos of my 2017 prepared Index Card solutions for n = 4 and n = 5 cases, if anyone is interested in seeing them.

]]>here is a jupyter/sage notebook that I used to compute the number of partitions.

I will certainly add the results to OEIS once they have stewed for a while, and hopefully at least some of them have been independently verified. As has been amply demonstrated, these are tricky waters, and the OEIS editors are busy enough without having to deal with changing sequences.

Best of luck with life’s intrusions.

]]>Wow, I’m impressed that you got that far computing the sequence A359146. I hope that after you put your code on github you can add the new terms in the sequence to the OEIS and include a link to that github. If you don’t know how (I don’t), I can just send the information to Neil Sloane.

Is the rate of growth of the number of topologies similar for any positive algebraic number all whose conjugates have positive real part? Of course some start later, but do they grow slower?

Sorry, I don’t know—and right this minute, I should definitely not be thinking about that! Life intrudes. When I get more time the first thing I should think about is the math of guillotine partitions, which is very interesting and somewhat deep. But thanks so much for making this excellent progress. When you get your data onto github, please let me know here.

]]>In fact for any given ratio. If you can tile a square with rectangles of that ratio, then you can tile it with rectangles of that ratio by replacing one of the rectangles by 4 half size rectangles, by replacing on rectangle by one 2/3 size rectangle and 5 1/3 size triangle, size rectangle by replacing it by one size rectangle and size rectangles. Using these in combination one can tile a square with any rectangles for $\latex k\ge 5$. As there are lots of choices for which rectangles to subdivide, and what combinations of subdivisions to make, it is evident that the number of possible topologies grows faster than any polynomial. I am pretty sure that it grows slower than exponentially, but cannot prove it.

Is the rate of growth of the number of topologies similar for any positive algebraic number all whose conjugates have positive real part? Of course some start later, but do they grow slower?.

In not entirely unrelated news, I had free time because of the snowstorm, so I computed more terms of A359146, getting the sequence 1, 1, 3, 11, 51, 245, 1372, 8522, 58347, 433926. If someone wants, I can generate the pictures, but things seem to be getting beyond human comprehension.

I will soon clean up my sage code so that some other human might understand it and will put it on github.

I also computed the maximum number of different rectangles that can be tiled with subrectangles having aspect ratios of and and got 1, 3, 10, 38, 164, 787, 4258, 25767, 172504, 1258949. The corresponding sequence for guillotine dissections is [A338781](https://oeis.org/A338781}.

One question I have (that I may have the answer to, but haven’t looked closely at the data) is: Does there exist a rational ratio such that the smallest number of rectangles of that ratio that tile the square only tile it in a way that all the rectangles do not have the same orientation.

]]>As the number of rectangles grows, you can get arbitrarily many different topologies modulo symmetries. For example chop the unit square into N vertical strips, each a rectangle of height 1 and width 1/N. Then choose one of these vertical strips and chop it into N squares, each of height 1/N and width 1/N. Then take all of these squares and chop each of them into N vertical strips, each a rectangle of height 1/N and width 1/N^{2}. You can get a lot of different topologies this way.