Is it possible to fill a rectangle with a countably infinite amount of circles? The circles do not need to be of equal size but they should not overlap. It seems to be clear that this should be possible but I could not find any proof for it. Basically, I am given a rectangle $R$ with area $A$ and am looking for a sequence of circles $(C_n)$ with areas $(A_n)$ such that $C_i \cap C_j = \emptyset\ \ \forall i, j\in \mathbb{N}, i\neq j$ and $C_n \subseteq R \ \ \forall n\in \mathbb{N}$ and $$\sum_{n=1}^{\infty} A_n = A$$ How can I show that such a sequence of circles exists?
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When you place maximal circles, there are triangular voids left between them. You can fill these voids with new maximal circles and create three new voids.
Clearly, you can establish a bijection between the circles and the fractional numbers written in base $3$.
