Remote Access Conformal Geometry and Dynamics
Green Open Access

Conformal Geometry and Dynamics

ISSN 1088-4173



Finite subdivision rules

Authors: J. W. Cannon, W. J. Floyd and W. R. Parry
Journal: Conform. Geom. Dyn. 5 (2001), 153-196
MSC (2000): Primary 20F65, 52C20; Secondary 05B45
Published electronically: December 18, 2001
MathSciNet review: 1875951
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We introduce and study finite subdivision rules. A finite subdivision rule $\mathcal{R}$ consists of a finite 2-dimensional CW complex $S_{\mathcal{R}}$, a subdivision $\mathcal{R}(S_{\mathcal{R}})$ of $S_{\mathcal{R}}$, and a continuous cellular map $\varphi_{\mathcal{R}}\colon\thinspace \mathcal{R}(S_{\mathcal{R}}) \to S_{\mathcal{R}}$ whose restriction to each open cell is a homeomorphism. If $\mathcal{R}$ is a finite subdivision rule, $X$ is a 2-dimensional CW complex, and $f\colon\thinspace X\to S_{\mathcal{R}}$ is a continuous cellular map whose restriction to each open cell is a homeomorphism, then we can recursively subdivide $X$ to obtain an infinite sequence of tilings. We wish to determine when this sequence of tilings is conformal in the sense of Cannon's combinatorial Riemann mapping theorem. In this setting, it is proved that the two axioms of conformality can be replaced by a single axiom which is implied by either of them, and that it suffices to check conformality for finitely many test annuli. Theorems are given which show how to exploit symmetry, and many examples are computed.

References [Enhancements On Off] (What's this?)

  • 1. P. L. Bowers and K. Stephenson, A ``regular'' pentagonal tiling of the plane, Conform. Geom. Dyn. 1 (1997), 58-68 (electronic). MR 99d:52016
  • 2. J. W. Cannon, The theory of negatively curved spaces and groups, Ergodic theory, symbolic dynamics, and hyperbolic spaces (Trieste, 1989), Oxford Univ. Press, New York, 1991, pp. 315-369. CMP 92:02
  • 3. J. W. Cannon, The combinatorial Riemann mapping theorem, Acta Math. 173 (1994), no. 2, 155-234. MR 95k:30046
  • 4. 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), Amer. Math. Soc., Providence, RI, 1994, pp. 133-212. MR 95g:20045
  • 5. J. W. Cannon, W. J. Floyd, and W. R. Parry, Sufficiently rich families of planar rings, Ann. Acad. Sci. Fenn. 44 (1999), 265-304. MR 2000k:20057
  • 6. J. W. Cannon and E. L. Swenson, Recognizing constant curvature discrete groups in dimension $3$, Trans. Amer. Math. Soc. 350 (1998), no. 2, 809-849. MR 98i:57023
  • 7. F. M. Dekking, Recurrent sets, Adv. in Math. 44 (1982), no. 1, 78-104. MR 84e:52023
  • 8. F. R. Gantmacher, The Theory of Matrices, Chelsea Publishing Company, New York, 1989.
  • 9. M. Gardner, Mathematical games: In which ``monster'' curves force redefinition of the word ``curve'', Scientific American 235 (1976), 124-133.
  • 10. J. Giles, Jr., Construction of replicating superfigures, J. Combinat. Theory A 26 (1979), 328-334. MR 80g:51013b
  • 11. J. Giles, Jr., Superfigures replicating with polar symmetry, J. Combinat. Theory A 26 (1979), 335-337. MR 80g:51013c
  • 12. B. Grünbaum and G. C. Shephard, Tilings and Patterns, W. H. Freeman and Co., New York, 1987. MR 88k:52018
  • 13. R. Kenyon, Self-replicating tilings, Symbolic dynamics and its applications (New Haven, CT, 1991), Amer. Math. Soc., Providence, RI, 1992, pp. 239-263. MR 94a:52043
  • 14. B. B. Mandelbrot, The Fractal Geometry of Nature, W. H. Freeman and Co., New York, 1977.
  • 15. B. Rodin and D. Sullivan, The convergence of circle packings to the Riemann mapping, J. Differential Geom. 26 (1987), no. 2, 349-360. MR 90c:30007
  • 16. K. Stephenson, CirclePack, software, available from
  • 17. W. P. Thurston, Groups, tilings and finite state automata, Summer 1989 AMS Colloquium Lectures, Geometry Center Preprint GCG 01.
  • 18. J. Weeks, SnapPea: A computer program for creating and studying hyperbolic 3-manifolds, available from

Similar Articles

Retrieve articles in Conformal Geometry and Dynamics of the American Mathematical Society with MSC (2000): 20F65, 52C20, 05B45

Retrieve articles in all journals with MSC (2000): 20F65, 52C20, 05B45

Additional Information

J. W. Cannon
Affiliation: Department of Mathematics, Brigham Young University, Provo, Utah 84602

W. J. Floyd
Affiliation: Department of Mathematics, Virginia Polytechnic Institute & State University, Blacksburg, Virginia 24061

W. R. Parry
Affiliation: Department of Mathematics, Eastern Michigan University, Ypsilanti, Michigan 48197

Keywords: Finite subdivision rule, conformality
Received by editor(s): September 20, 1999
Received by editor(s) in revised form: July 2, 2001
Published electronically: December 18, 2001
Additional Notes: This work was supported in part by NSF research grants and by The Geometry Center, University of Minnesota, an STC funded by NSF, DOE, and Minnesota Technology, Inc.
Article copyright: © Copyright 2001 American Mathematical Society