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
E. M. Andreev, Convex polyhedra in Lobačevskiĭ spaces, Mat. Sb. (N.S.) 81 (123) (1970), 445–478 (Russian). MR 0259734
E. M. Andreev, Convex polyhedra of finite volume in Lobačevskiĭ space, Mat. Sb. (N.S.) 83 (125) (1970), 256–260 (Russian). MR 0273510
Alan F. Beardon and Kenneth Stephenson, The uniformization theorem for circle packings, Indiana Univ. Math. J. 39 (1990), no. 4, 1383–1425. MR 1087197, DOI 10.1512/iumj.1990.39.39062
Alan F. Beardon and Kenneth Stephenson, The Schwarz-Pick lemma for circle packings, Illinois J. Math. 35 (1991), no. 4, 577–606. MR 1115988
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, DOI 10.1090/S0002-9947-03-03239-2
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, DOI 10.4007/annals.2006.164.231
Mario Bonk and Bruce Kleiner, Quasisymmetric parametrizations of two-dimensional metric spheres, Invent. Math. 150 (2002), no. 1, 127–183. MR 1930885, DOI 10.1007/s00222-002-0233-z
Mario Bonk and Bruce Kleiner, Conformal dimension and Gromov hyperbolic groups with 2-sphere boundary, Geom. Topol. 9 (2005), 219–246. MR 2116315, DOI 10.2140/gt.2005.9.219
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, DOI 10.1016/0040-9383(93)90044-V
Philip L. Bowers and Kenneth Stephenson, A “regular” pentagonal tiling of the plane, Conform. Geom. Dyn. 1 (1997), 58–68. MR 1479069, DOI 10.1090/S1088-4173-97-00014-3
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, DOI 10.1090/memo/0805
James W. Cannon, The combinatorial Riemann mapping theorem, Acta Math. 173 (1994), no. 2, 155–234. MR 1301392, DOI 10.1007/BF02398434
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, DOI 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).
M. Gromov, Hyperbolic groups, Essays in group theory, Math. Sci. Res. Inst. Publ., vol. 8, Springer, New York, 1987, pp. 75–263. MR 919829, DOI 10.1007/978-1-4613-9586-7_{3}
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
Richard S. Hamilton, Three-manifolds with positive Ricci curvature, J. Differential Geometry 17 (1982), no. 2, 255–306. MR 664497
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, DOI 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.
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
Oded Schramm, Square tilings with prescribed combinatorics, Israel J. Math. 84 (1993), no. 1-2, 97–118. MR 1244661, DOI 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.
- 1.
- E.M. Andre'ev, Convex polyhedra in Lobacevskii space, Math. USSR Sbornik 10 (1970), 413-440. MR 0259734
- 2.
- E.M. Andre'ev, Convex polyhedra of finite volume in Lobacevskii space, Math. USSR Sbornik 12 (1970), 255-259. MR 0273510
- 3.
- A.F. Beardon and K. Stephenson, The uniformization theorem for circle packings, Indiana Univ. Math. J. 39 (1990), 1383-1425. MR 1087197
- 4.
- A.F. Beardon and K. Stephenson, The Schwarz-Pick lemma for circle packings, Ill. J. Math. 35 (1991), 577-606. MR 1115988
- 5.
- A.I. Bobenko and B.A. Springborn, Variational principles for circle patterns and Koebe's theorem, Trans. AMS 356 (2003), 659-689. MR 2022715
- 6.
- A.I. Bobenko, T. Hoffmann, and B.A. Springborn, Minimal surfaces from circle patterns: geometry from combinatorics, Annals of Math. 164:1 (2006), 231-264. MR 2233848
- 7.
- M. Bonk and B. Kleiner, Quasisymmetric parametrizations of two-dimensional metric spheres, Invent. Math. 150 (2002), no. 1, 127-183. MR 1930885
- 8.
- M. Bonk and B. Kleiner, Conformal dimension and Gromov hyperbolic groups with
-sphere boundary, Geometry & Topology 9 (2005), 219-246. MR 2116315
- 9.
- P.L. Bowers and K. Stephenson, Circle packings in surfaces of finite type: An in situ approach with applications to moduli, Topology 32 (1993), 157-183. MR 1204413
- 10.
- P.L. Bowers and K. Stephenson, A ``regular'' pentagonal tiling of the plane, Con. Geom. and Dynamics 1 (1997), 58-86. MR 1479069
- 11.
- P.L. Bowers and K. Stephenson, Uniformizing dessins and Belyĭ maps via circle packing, Mem. Amer. Math. Soc. 170 (2004), no. 805, xii+97 pp. MR 2053391
- 12.
- J.W. Cannon, The combinatorial Riemann mapping theorem, Acta. Math. 173 (1994), 155-234. MR 1301392
- 13.
- J.W. Cannon, W.J. Floyd, and W.R. Parry, ``Squaring rectangles: the finite Riemann mapping theorem, in The Mathematical Heritage of Wilhelm Magnus--Groups, Geometry, and Special Functions'', Contemporary Mathematics, vol. 169, Amer. Math. Soc., Providence, 1994, pp. 133-212. MR 1292901
- 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'', in Essays in Group Theory, G.M. Gersten, ed., MSRI Publ. 8, 1987, pp. 75-263. MR 0919829
- 16.
- M. Gromov, ``Asymptotic invariants of infinite groups'', in Geometric Group Theory, Vol. 2 (Sussex, 1991), LMS Lecture Note Series, 182, Cambridge University Press, Cambridge, 1993, pp. 1-295 MR 1253544
- 17.
- R. Hamilton, Three-manifolds with positive Ricci curvature, J. Diff. Geom. 17 (1982), 255-306. MR 0664497
- 18.
- Z.-X. He and O. Schramm, Fixed points, Koebe uniformization and circle packings, Annals of Math. 137 (1993), 369-406. MR 1207210
- 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.
- B. Rodin and D. Sullivan, The convergence of circle packings to the Riemann mapping, J. Diff. Geom. 26 (1987), 349-360. MR 0906396
- 22.
- O. Schramm, Square tilings with prescribed combinatorics, Israel J. Math. 84 (1993), 97-118. MR 1244661
- 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
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.
