I have stumbled on an interesting problem. How many congruent rectangles on a plane can be arranged in such a way that they all touch each other but never overlap? I pondered this problem for a few days, but so far I'm not even sure what's the best way to formalize it. Has this problem been solved? Is there a useful hint? Can you recommend me something to read on the subject?
UPD: my best idea so far is to represent a rectangle in $\mathbb{E}^2$ by a tuple $r = (p, x, y)$, where $p$ is a point, and $x$ and $y$ are two orthogonal vectors representing two sides coming out of it. Then we define symmetries as (a) euclidean motions acting as usual, and (b) swapping of $x$ and $y$. Then we observe that the solutions of the problem, that is, $m$ touching rectangles, are mapped to some other solutions under euclidean motions and dilatations of the underlying space, as well as under $S_m$. Now we have to algebrize the problem, but I'm unsure how.