SQUARING RECTANGLES : THE FINITE RIEMANN MAPPING THEOREM

The classical Riemann mapping theorem asserts that any topological quadrilateral in the complex plane can be mapped conformally onto a rectangle. The finite Riemann mapping theorem asserts that any topological quadrilateral tiled by finitely many 2-cells can be mapped with minimal combinatorial distortion onto a rectangle tiled by squares. We prove the finite Riemann mapping theorem, discuss its connections with the classical theories of conformal mapping and electric circuits, and develop algorithms for calculating the finite Riemann mapping. Table of contents. 0. Introduction. 1. Squared rectangles. 1.1. Scientific American. 1.2. Resistive circuits and the laws of Ohm and Kirchhoff. 1.3. The solution of the resistive circuit problem. 1.4. Planar circuits and rectangles tiled by rectangles. 1.5. Max Dehn’s contribution. 2. Optimal weight functions. 2.1. The classical Riemann mapping theorem. 2.2. The finite problem and the existence and uniqueness of its solution. 2.3. The geometry of an optimal weight function: general results. 1991 Mathematics Subject Classification. Primary 20F32, 30C62, 31A15; Secondary 30F10, 30F40, 57N10. This research was supported in part by The Geometry Center, University of Minnesota, an STC funded by NSF, DOE, and Minnesota Technology, Inc. The first author was supported in part by NSF Research Grant No. DM-8902071. Typeset by AMS-TEX 1 2 J. W. CANNON, W. J. FLOYD AND W. R. PARRY 2.4. The geometry of an optimal weight function: special results for quadrilaterals and rings. 2.4.1. Further definitions. 2.4.2. The correspondence. 2.4.3. Level curves and the relationship between skinny flows and fat cuts. 2.4.4. The relationship between fat flows and skinny cuts for tilings. 2.4.5. The relationship between the four moduli. 3. The finite Riemann mapping theorem. 3.1. Examples. 3.2. The Kirchhoff inequalities. 4. Algorithms which calculate the finite Riemann mapping. 4.1. The minimal path algorithm. 4.2. Proofs of the claims about the minimal path algorithm. 4.3. The efficiency of the minimal path algorithm. 4.4. Flow diagrams and a hybrid algorithm for the finite Riemann mapping problem. 4.5. An example of the effectiveness of the hybrid algorithm. 5. Optimal weight functions for 2-layer valence 3 tilings of quadrilaterals. 5.1. Basic definitions and properties. 5.2. Unique prime factorization theorem. 5.3. Prime tilings. 5.4. Preparations for factorization algorithms. 5.5. The left factor algorithm. 5.6. The joint ratio algorithm. 6. Approximating combinatorial moduli. 6.1. Skinny approximation. 6.2. The averaging trick. 6.3. Examples. 7. Variable negative curvature versus constant curvature groups. 0. Introduction We dedicate this paper to Wilhelm Magnus with admiration and appreciation. This paper has mild ties with Magnus through the work of his thesis advisor, Max Dehn, who proved long ago that, for any rectangle which can be tiled by finitely many squares, the ratio of height to width is rational. This paper is directed in the long term to the recognition of discrete groups of isometries in hyperbolic 3-space, and the subject of discrete groups was one of Magnus’ favorites. Riemann, in formulating his famous Riemann mapping theorem, surely relied on the physics of electrical networks and conducting metal plates for motivation. In turn, Riemann’s theorem and its many proofs can be used as starting points for a number of beautiful finite approximations which have combinatorial, geoSQUARING RECTANGLES: THE FINITE RIEMANN MAPPING THEOREM 3 metric, and physical interpretations. These finite Riemann mapping theorems are the subject of very intense interest. One of them is the subject of Cannon’s paper [8]. Another is of central importance to this paper. The paper is partly expository, especially in the beginning. However, in the later sections, we develop material important to our program of recognizing groups of hyperbolic motions combinatorially. 1. Squared rectangles This section is purely expository. Our aim is to show a correspondence between rectangles tiled by rectangles and connected planar resistive circuits with one battery. 1.1. Scientific American. We begin our story with a puzzle whose solution is recounted in Chapter 17 of The 2nd Scientific American Book of Mathematical Puzzles and Diversions by Martin Gardner [13]: Can a square be subdivided into smaller squares of which no two are alike? Gardner’s story teller is William T. Tutte, renowned geometer and combinatorialist at the University of Toronto. Tutte, as a student at Trinity College, Cambridge, during the years 1936-38 pursued an answer to this question with three other students, C. A. B. Smith, A. H. Stone, and R. L. Brooks. “Stone was intrigued by a statement in Dudeney’s Canterbury Puzzles [12] which seemed to imply that it is impossible to cut up a square into unequal smaller squares. ” Soon the four friends “were spending much time constructing, and arguing about, dissections of rectangles into squares. ” They used a mixture of geometric and algebraic methods to create a large catalog of squared rectangles. “In the next stage of the research [they] abandoned experiment in favor of theory. ” The students “tried to represent squared rectangles by diagrams of different kinds. ” Smith suggested a diagram which showed that the problem could be reinterpreted as a part of the theory of electrical networks. We shall now explain Smith’s diagram. Consider as an example the rectangle of Figure 1.1.1 which is tiled by smaller rectangles. Each maximal horizontal line segment of the diagram is to be represented by a node of a simple resistive circuit. The top and the bottom of the rectangle represent nodes of the circuit which are to be connected to the terminals of a battery. Each small rectangle is to be represented by an edge of the circuit consisting of a single resistor. This resistor joins the nodes (maximal horizontal line segments) in which the top and bottom of the corresponding rectangle lie. We must assign values to the voltage of the battery and the resistance of the resistors. We interpret height as voltage drop and width as current flow. Hence the height of the large rectangle is assigned as the voltage of the battery. We apply Ohm’s law which gives resistance as the quotient of voltage by current to determine the resistance of each resistor as the ratio of height to width in the corresponding small rectangle. We observe that a square in the tiling corresponds to a resistance of 1. 4 J. W. CANNON, W. J. FLOYD AND W. R. PARRY