The circle packing theorem states that a circle packing exists if and only if the pattern of adjacencies forms a planar graph; it was originally proved by Paul Koebe in the 1930s, and popularized by William Thurston, who rediscovered it in the 1970s and connected it with the theory of conformal maps and conformal geometry.
[4] The first part introduces the subject visually, encouraging the reader to think about packings not just as static objects but as dynamic systems of circles that change in predictable ways when the conditions under which they are formed (their patterns of adjacency) change.
In this case, different extensions of this pattern to larger maximal planar graphs will lead to different packings, which can be mapped to each other by corresponding circles.
[4] Unsolved problems are listed throughout the book, which also includes nine appendices on related topics such as the ring lemma and Doyle spirals.
[1][3] The book presents research-level mathematics, and is aimed at professional mathematicians interested in this and related topics.