Gene asks: << If x is irrational, and I wish to best approximate the ratio x:1 by integers a:b, then I would use continued fractions. Suppose x and y are irrational, and I wish to best approximate the ratio x:y:1 by integers a:b:c. Is it known how to do this?
This is a very interesting question, and I don't have much useful to contribute, except that 1) The family of ratios x:y:1 is perhaps best naturally encoded as the projective space P^2 = R^3 - {0} / (x,y,z) ~ (ax,ay,az) where a runs over all nonzero reals, aka the set of lines through the origin in R^3. 2) The answer probably depends on how the goodness of is measured; the most natural way seems to me to be the smallness of the angle between two lines, or equivalently the nearness of cos(angle) to 1. 3) The points in question will then be of the form +-(K,L,1)/sqrt(K^2+L^2+1) for integers K,L. The asymmetry of this expression suggests the last refuge of the scoundrel -- changing the question slightly. Perhaps we should allow the approximation to be via (K,L,M) for any integers K,L,M, leading to all points of the form +-(K,L,M) / ||(K,L,M)||, with sign(s) chosen so that a dot product is always positive. 4) An even more radical change-of-problem, but perhaps equally interesting, might be to look at the (dense set of) rational points +-(p,q,r) on P^2 (which especially John Conway discussed vis-a-vis spheres a few months ago) as possible approximators of an arbitrary ratio (x:y:1). Clearing denominators shows this is a subset of the points +-(K,L,M) / ||(K,L,M)|| of 3). --Dan