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Rectangling a rectangle

Abstract

We show that the following are equivalent: (i) A rectangle of eccentricityv can be tiled using rectangles of eccentricityu. (ii) There is a rational function with rational coefficients,Q(z), such thatv =Q(u) andQ maps each of the half-planes {z ¦ Re(z) < 0} and {z ¦ Re(z) > 0 into itself, (iii) There is an odd rational function with rational coefficients,Q(z), such thatv = Q(u) and all roots ofv = Q(z) have a positive real part. All rectangles in this article have sides parallel to the coordinate axes and all tilings are finite. We letR(x, y) denote a rectangle with basex and heighty.

In 1903 Dehn [1 ] proved his famous result thatR(x, y) can be tiled by squares if and only ify/x is a rational number. Dehn actually proved the following result. (See [4] for a generalization to tilings using triangles.)

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Correspondence to C. Freiling.

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The first and third authors were partially supported by NSF.

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Freiling, C., Laczkovich, M. & Rinne, D. Rectangling a rectangle. Discrete Comput Geom 17, 217–225 (1997). https://doi.org/10.1007/BF02770874

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Keywords

  • Rational Coefficient
  • Imaginary Axis
  • Discrete Comput Geom
  • Algebraic Number
  • Companion Matrix