We show that the following are equivalent: (i) A rectangle of eccentricity v can be tiled using rectangles of eccentricity u. (ii) There is a rational function with rational coefficients, Q(z) , such that v = Q(u) and Q 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 that v = Q(u) and all roots of v = 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 let R(x, y) denote a rectangle with base x and height y. In 1903 Dehn [1 ] proved his famous result that R(x, y) can be tiled by squares if and only if y/x is a rational number. Dehn actually proved the following result. (See [4] for a generalization to tilings using triangles.)
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