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 ${f(r): r in (0, 1)}=mathbb{N}$?