Squaring rectangles for dumbbells

Cannon, J.; Floyd, W.; Parry, W.

doi:10.1090/S1088-4173-08-00182-3

Squaring rectangles for dumbbells
HTML articles powered by AMS MathViewer

by J. W. Cannon, W. J. Floyd and W. R. Parry

Conform. Geom. Dyn. 12 (2008), 109-132

DOI: https://doi.org/10.1090/S1088-4173-08-00182-3

Published electronically: August 14, 2008

PDF | Request permission

Abstract:

The theorem on squaring a rectangle (see Schramm [Israel J. Math. 84 (1993)] and Cannon-Floyd-Parry [Contemp. Math. 169, AMS, 1994]) gives a combinatorial version of the Riemann mapping theorem. We elucidate by example (the dumbbell) some of the limitations of rectangle-squaring as an approximation to the classical Riemann mapping.

References

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
J. W. Cannon, W. J. Floyd, and W. R. Parry, Finite subdivision rules, Conform. Geom. Dyn. 5 (2001), 153–196. MR 1875951, DOI 10.1090/S1088-4173-01-00055-8
J. W. Cannon, W. J. Floyd, and W. R. Parry, Expansion complexes for finite subdivision rules. I, Conform. Geom. Dyn. 10 (2006), 63–99. MR 2218641, DOI 10.1090/S1088-4173-06-00126-3
J. W. Cannon, W. J. Floyd, and W. R. Parry, precp.p, software, available from http://www. math.vt.edu/people/floyd.
Zheng-Xu He and Oded Schramm, On the convergence of circle packings to the Riemann map, Invent. Math. 125 (1996), no. 2, 285–305. MR 1395721, DOI 10.1007/s002220050076
Oded Schramm, Square tilings with prescribed combinatorics, Israel J. Math. 84 (1993), no. 1-2, 97–118. MR 1244661, DOI 10.1007/BF02761693
K. Stephenson, CirclePack, software, available from http://www.math.utk.edu/˜kens.