# 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.)

## References

1. 1.

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

2. 2.

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

3. 3.

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

4. 4.

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

5. 5.

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

6. 6.

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

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.

8. 8.

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

## Author information

Authors

### Corresponding author

Correspondence to C. Freiling.

The first and third authors were partially supported by NSF.

## Rights and permissions

Reprints and Permissions

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