  This article is a gentle introduction to the mathematical area known as
circle packing, the study of the kinds of patterns that can be formed by
configurations of non-overlapping circles. The first half of the article is an
exposition of the two most important facts about circle packings, (1) that
essentially whatever pattern (i.e. combinatorics) we ask for, we may always
arrange circles in that pattern, and (2) that under simple conditions on the
pattern, there is an essentially unique arrangement of circles in that pattern.
In the second half of the article, we consider related questions, but where we
allow the circles to overlap. The article is written with the idea that no
mathematical background should be required to read it.
