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Rectangling a rectangle


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.)


  1. 1.

    M. Dehn, Über die Zerlegung von Rechtecken in Rechtecke,Math. Ann. 57 (1903), 314–332.

    Article  MathSciNet  Google Scholar 

  2. 2.

    C. Freiling and D. Rinne, Tiling a square with similar rectangles,Math. Res. Lett. 1 (1994), 547–558.

    MATH  MathSciNet  Google Scholar 

  3. 3.

    D. Gale, More on squaring squares and rectangles,Math. Intelligencer 15(4) (1993), 60–61.

    MathSciNet  Google Scholar 

  4. 4.

    M. Laczkovich, Tilings of polygons with similar triangles,Combinatorica 10 (1990), 281–306.

    MATH  Article  MathSciNet  Google Scholar 

  5. 5.

    M. Laczkovich and G. Szekeres, Tilings of the square with similar rectangles,Discrete Comput. Geom. 13 (1995), 569–572.

    MATH  Article  MathSciNet  Google Scholar 

  6. 6.

    G. Strang,Linear Algebra and Its Applications, p. 109, Academic Press, New York, 1980.

    Google Scholar 

  7. 7.

    H. S. Wall,Analytic Theory of Continued Fractions, Chelsea, New York, 1967, originallyThe University Series in Higher Mathematics, Vol.1, Van Nostrand, New York, 1948.

    Google Scholar 

  8. 8.

    H. S. Wall, Polynomials whose zeros have negative real parts,Amer. Math. Monthly 52 (1945), 308–322.

    MATH  Article  MathSciNet  Google Scholar 

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Correspondence to C. Freiling.

Additional information

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).

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