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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 such as the [[ring lemma]] and [[Doyle spiral]]s.{{r|pokas|lord}}
==Audience and reception==
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