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The book is divided into four parts, in progressive levels of difficulty.{{r|mumford}} 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. The second part concerns the proof of the circle packing theorem itself, and of the associated [[Rigidity (mathematics)|rigidity theorem]]: every [[maximal planar graph]] can be associated with a circle packing that is unique up to [[Möbius transformation]]s of the plane.{{r|pokas|lord}} More generally the same result holds for any triangulated [[manifold]], with a circle packing on a topologically equivalent [[Riemann surface]] that is unique up to conformal equivalence.{{r|cfp}}
The third part of the book concerns the degrees of freedom that arise when the pattern of adjacencies is not fully triangulated (it is a planar graph, but not a maximal planar graph). 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. The book explores the connection between these mappings, which it calls discrete analytic functions, and the [[analytic function]]s of classical [[mathematical analysis]]. The final part of the book concerns a conjecture of William Thurston, proved by [[Burton Rodin]] and [[Dennis Sullivan]], that makes this analogy concrete: conformal mappings from any topological disk to a circle can be approximated by filling the disk by a hexagonal packing of unit circles, finding a circle packing that adds to that pattern of adjacencies a single outer circle, and constructing the resulting discrete analytic function. This part also includes applications to number theory and the visualization of brain structure.{{r|pokas|lord}}
Stephenson has implemented algorithms for circle packing and used them to construct the many illustrations of the book,{{r|cfp}} giving to much of this work the flavor of [[experimental mathematics]], although it is also mathematically rigorous.{{r|mumford}} Unsolved problems are listed throughout the book, which also includes nine appendices on related topics.{{r|pokas|lord}}
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