If you divide a square into 4 similar rectangles, what proportions can these rectangles have? A lot of people here have been working on this!
Here @Danpiker lists the 11 options we've found. Find more, or prove these are all!
Note: there are > 11 ways to divide a square into 4 similar rectangles: you can rotate and reflect these pictures, and also rearrange some. But we've only found 11 possible proportions for the rectangles.
Let me go through them here, in order.
(0/n)
@johncarlosbaez @Danpiker
Do different ways of dividing the square always give different proportions for the rectangles?
More formally: say two ways to divide the square are equivalent, if I can get from one to the other by taking some subset of the similar rectangles, that itself forms a rectangle (not necessarily of the same proportion), and reflecting it vertically or horizontally. Two ways of dividing the square are “different” if they are in separate equivalence classes.
Are there two different divisions of the square, each into the same number of subrectangles, such that all the subrectangles in both squares are similar?
