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Given $k$ strictly positive real numbers $l_1,\dots,l_k$, can one decide the existence of a periodic tiling of the plane whose fundamental domain is the union of $k$ squares of length $l_1,\dots,l_k$?

For $k\leq 2$, such tilings exist always.

How can one construct them in case of existence?

(Fun example: given a natural integer $n$ are there always (at most) four such lengths given by non-negative integers $a,b,c,d$ with $n=a^2+b^2+c^2+d^2$?)

Added: One can always decide the existence theoretically since there are a finite number of possible graphs (given by boundary of squares) on the resulting quotient-torus. This is however not usable in practice.

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What has the example to do with the question? – Stefan Kohl Jun 17 '13 at 12:46
Since every natural number is a sum of four squares, it is tempting to ask for the existence of tilings having fundamental domains of area $n$ and involving at most four squares with side-lengths given by natural numbers. A somewhat weaker question is: can one give an universal upper bound (independent of $n$) for the number of such squares (with side-lengths in $\mathbb N$) involved in a fundamenal tiling domain of area $n$? – Roland Bacher Jun 17 '13 at 14:24
Fo you need a domain to be connected? – Ilya Bogdanov Jun 17 '13 at 15:03
Not necessarily, but you can always choose a connected domain by suitably translating the connected components and gluing them together. – Roland Bacher Jun 18 '13 at 6:59

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