TITLE: Can all circles of radius $1/n$ be packed in a unit disk, excluding the circle of radius $1/1$?
QUESTION [39 upvotes]: This problem occurred to me when I came across a similar problem where the radii were taken over only the primes. That question was unanswered, but it seems to me infinitely many circles of radius $1/2, 1/3, 1/4...$ can fit into a unit disk. The area of all those circles would be $\pi \sum_2^\infty 1/n^2 = \pi^3/6 -\pi$, which is less than the area of the unit disk $\pi$. But can the circles actually be packed with no overlaps?

REPLY [13 votes]: Just to make geometry more attractive:

(on "pingpong ball, a pea and sand in a bottle"... see comment of @Patrick Da Silva :)


$4\times$ scaled left-top corner:


$4\times$ scaled left-top corner of $2$nd image:


One can see fractal property of these images.