Squaring rectangles for dumbbells

J. W. Cannon, W. J. Floyd, and W. R. Parry
June 17, 2008


The theorem on squaring a rectangle (see Schramm (Squaring tilings with prescribed combinatorics, Israel J. Math. 84 (1993), 97--118) and Cannon-Floyd-Parry (Squaring rectangles: the finite Riemann mapping theorem)) 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 Riemamnn mapping.

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