The geometry of markoff numbers

It turns out that v(0) ~ 1/V~ with equality only if 0 is a "noble number ' ' t whose continued fraction expansion ends in a string of ones. In 1879 Markoff improved this result by showing that there is a discrete set of values v i decreasing to 1/3 so that if v(0) > 1/3 then v(0) = v i for some i [8]. The numbers v i are called the M a r k o f f s p e c t r u m and the corresponding O's, M a r k o f f i rra t ional i t ies . Markoff irrationalities have cont inued fraction expansions whose tails satisfy a very special set of rules, often called the Dickson rules [4]. The tail 1,1,1 . . . . is the simplest example. What these rules are will become clear as we proceed. Markoff gave a prescription for determining all of these irrationalities starting from the solutions of a certain diophantine equation and linked his results to the minima of associated binary quadratic forms. Recently there has been a revival of interest in this topic, starting from the realisation that each v i together with its corresponding class of Markoff irrationalities is associated to a simple (non-self-intersecting) loop on the punctured torus, as shown in Figure 1. The details have been worked out most fully by A. Haas [6], based on earlier work of Cohn [2, 3], and Schmidt [10]. Lehner, Scheingorn and Beardon [7] tackle the same problem but base their analysis on a sphere with four punctures. It turns out that almost all the results follow from some rather simple observations about the way in which straight lines cut certain tessellations of the Euclidean and hyperbolic planes and it is these ideas which we want to explain here. Before understanding approximations we shall need to make a fairly lengthy digression to investigate such cutting patterns, for which I offer no apology, for the approach via the patterns is quite as fascinating as Markoff's theory itself.