# co.combinatorics – Packing equal-size disks in a unit disk

Inspired by the delicious buns and Siu Mai in bamboo steamers I saw tonight in a food show about Cantonese Dim Sum, here is a natural question. It probably has been well studied in the literature, but I cannot find related reference.

Given a real number $$r le 1$$, let $$f(r)$$ be the maximum number of radius-$$r$$ disks that can be packed into a unit disk. For example, $$f(1)=1$$ for $$r in (1/2, 1)$$, $$f(r)=2$$ for $$r in (2sqrt{3}-3, 1/2)$$, etc.

Question: Is it true that $${f(r): r in (0, 1)}=mathbb{N}$$?