Abstract 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 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.)