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Book Review

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MathSciNet review: 2507284
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Book Information:

Author: Kenneth Stephenson
Title: Introduction to circle packing: The theory of discrete analytic functions
Additional book information: Cambridge University Press, Cambridge, 2005, xii+356 pp., ISBN 978-0-521-82356-2, £42

1. E. M. Andreev, Convex polyhedra in Lobačevskiĭ spaces, Mat. Sb. (N.S.) 81 (123) (1970), 445–478 (Russian). MR 0259734
2. E. M. Andreev, Convex polyhedra of finite volume in Lobačevskiĭ space, Mat. Sb. (N.S.) 83 (125) (1970), 256–260 (Russian). MR 0273510
3. Alan F. Beardon and Kenneth Stephenson, The uniformization theorem for circle packings, Indiana Univ. Math. J. 39 (1990), no. 4, 1383–1425. MR 1087197, https://doi.org/10.1512/iumj.1990.39.39062
4. Alan F. Beardon and Kenneth Stephenson, The Schwarz-Pick lemma for circle packings, Illinois J. Math. 35 (1991), no. 4, 577–606. MR 1115988
5. Alexander I. Bobenko and Boris A. Springborn, Variational principles for circle patterns and Koebe’s theorem, Trans. Amer. Math. Soc. 356 (2004), no. 2, 659–689. MR 2022715, https://doi.org/10.1090/S0002-9947-03-03239-2
6. Alexander I. Bobenko, Tim Hoffmann, and Boris A. Springborn, Minimal surfaces from circle patterns: geometry from combinatorics, Ann. of Math. (2) 164 (2006), no. 1, 231–264. MR 2233848, https://doi.org/10.4007/annals.2006.164.231
7. Mario Bonk and Bruce Kleiner, Quasisymmetric parametrizations of two-dimensional metric spheres, Invent. Math. 150 (2002), no. 1, 127–183. MR 1930885, https://doi.org/10.1007/s00222-002-0233-z
8. Mario Bonk and Bruce Kleiner, Conformal dimension and Gromov hyperbolic groups with 2-sphere boundary, Geom. Topol. 9 (2005), 219–246. MR 2116315, https://doi.org/10.2140/gt.2005.9.219
9. Philip L. Bowers and Kenneth Stephenson, Circle packings in surfaces of finite type: an in situ approach with applications to moduli, Topology 32 (1993), no. 1, 157–183. MR 1204413, https://doi.org/10.1016/0040-9383(93)90044-V
10. Philip L. Bowers and Kenneth Stephenson, A “regular” pentagonal tiling of the plane, Conform. Geom. Dyn. 1 (1997), 58–68. MR 1479069, https://doi.org/10.1090/S1088-4173-97-00014-3
11. Philip L. Bowers and Kenneth Stephenson, Uniformizing dessins and Belyĭ maps via circle packing, Mem. Amer. Math. Soc. 170 (2004), no. 805, xii+97. MR 2053391, https://doi.org/10.1090/memo/0805
12. James W. Cannon, The combinatorial Riemann mapping theorem, Acta Math. 173 (1994), no. 2, 155–234. MR 1301392, https://doi.org/10.1007/BF02398434
13. J. W. Cannon, W. J. Floyd, and W. R. Parry, Squaring rectangles: the finite Riemann mapping theorem, The mathematical legacy of Wilhelm Magnus: groups, geometry and special functions (Brooklyn, NY, 1992) Contemp. Math., vol. 169, Amer. Math. Soc., Providence, RI, 1994, pp. 133–212. MR 1292901, https://doi.org/10.1090/conm/169/01656
14. J.W. Cannon, W.J. Floyd, and W.R. Parry, The length-area method and discrete Riemann mappings, unpublished manuscript available from Bill Floyd that is based on a talk given by J. Cannon at the Ahlfors Celebration at Stanford University in September, 1997 (1998).
15. M. Gromov, Hyperbolic groups, Essays in group theory, Math. Sci. Res. Inst. Publ., vol. 8, Springer, New York, 1987, pp. 75–263. MR 919829, https://doi.org/10.1007/978-1-4613-9586-7_3
16. M. Gromov, Asymptotic invariants of infinite groups, Geometric group theory, Vol. 2 (Sussex, 1991) London Math. Soc. Lecture Note Ser., vol. 182, Cambridge Univ. Press, Cambridge, 1993, pp. 1–295. MR 1253544
17. Richard S. Hamilton, Three-manifolds with positive Ricci curvature, J. Differential Geometry 17 (1982), no. 2, 255–306. MR 664497
18. Zheng-Xu He and Oded Schramm, Fixed points, Koebe uniformization and circle packings, Ann. of Math. (2) 137 (1993), no. 2, 369–406. MR 1207210, https://doi.org/10.2307/2946541
19. M.K. Hurdal and K. Stephenson, Cortical cartography using the discrete conformal approach of circle packings, NeuroImage 23 (2004), Supplement 1, S119-S128.
20. P. Koebe, Kontaktprobleme der Konformen Abbildung, Ber. Sächs. Akad. Wiss. Leipzig, Math.-Phys. Kl. 88 (1936), 141-164.
21. Burt Rodin and Dennis Sullivan, The convergence of circle packings to the Riemann mapping, J. Differential Geom. 26 (1987), no. 2, 349–360. MR 906396
22. Oded Schramm, Square tilings with prescribed combinatorics, Israel J. Math. 84 (1993), no. 1-2, 97–118. MR 1244661, https://doi.org/10.1007/BF02761693
23. B. Springborn, P. Schröder, and U. Pinkall, Conformal equivalence of triangle meshes, ACM Transactions on Graphics 27:3 [Proceedings of ACM SIGGRAPH 2008], Article 77, 2008.

Review Information:

Reviewer: Philip L. Bowers
Affiliation: Department of Mathematics, The Florida State University, 1017 Academic Way, Tallahassee, Florida 32306-4510
Email: bowers@math.fsu.edu

Journal: Bull. Amer. Math. Soc. 46 (2009), 511-525
MSC (2000): Primary 52C26
DOI: https://doi.org/10.1090/S0273-0979-09-01245-2
Published electronically: February 19, 2009
Additional Notes: This review is dedicated to the memory of Oded Schramm, who worked in circle packing before his discovery of stochastic Loewner evolution and its applications to critical phenomena. This extraordinary mathematician’s untimely death on 01 September 2008 in a hiking accident was a great loss for our community.
Review copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.