Dividing a Square into Similar Rectangles

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 ca...