A circle covering is an arrangement of overlapping disks which cover the entire plane. For congruent disks, a lower bound for the
translative covering density is 
(Williams 1979, p. 51).
Pairwise disjoint circles of positive radius contained in a nonempty open
disk cannot cover that open disk (Moorhouse 2009).
To prove this, suppose such a covering exists for an open
disk of radius 
. Let 
be the circle through its center, and let 
be the circle through the
center of 
. Each new circle
lies strictly inside the preceding circle, since their
boundaries cannot intersect. If 
is the radius of 
, then 
and 
, because each circle passes through the preceding center.
The nested closed disks therefore have diameters
tending to zero, so compactness gives a unique common
point. This point lies strictly inside each circle, since
it lies in the next closed disk. A covering circle
through this point must lie inside each of these closed
disks, as it cannot cross any of their boundary circles. Its positive diameter
would then be smaller than every 
, a contradiction.
Consequently, the plane has no set partition into circles of positive radius. The interior of any circle in such a set partition would have to be covered by circles lying entirely inside it (Numberphile 2023).
