Periodic continued fraction representations of different quark's mass ratios

In the present work, first we give some theorems on continued fraction expansions. As shown in [Chaos, Solitons & Fractals 21 (2004) 9-19] the limit set of the Kleinian transformation acting on the E-infinity Cantorian space-time is a set of periodic continued fractions. We expand all possible quark's mass ratios as periodic continued fractions and following [Chaos, Solitons & Fractals 22 (2004) 749-752], we establish that their vibrations are like fractal polylines.