  We classify and enumerate all rational numbers with approximation constant at
least $\frac{1}{3}$ using hyperbolic geometry. Rational numbers correspond to
geodesics in the modular torus with both ends in the cusp, and the
approximation constant measures how far they stay out of the cusp neighborhood
in between. Compared to the original approach, the geometric point of view
eliminates the need to discuss the intricate symbolic dynamics of continued
fraction representations, and it clarifies the distinction between the two
types of worst approximable rationals: (1) There is a plane forest of Markov
fractions whose denominators are Markov numbers. They correspond to simple
geodesics in the modular torus with both ends in the cusp. (2) For each Markov
fraction, there are two infinite sequences of companions, which correspond to
non-simple geodesics with both ends in the cusp that do not intersect a pair of
disjoint simple geodesics, one with both ends in the cusp and one closed.
