Have anyone ever heard any reference about CONTINUED FRACTIONS OF THE THIRD DEGREE? Well, this kind of 'FRACTAL' comes from the so simple 'Rational Process' I showed in my previous messages for solving the duplication of the cube. Oh yes!, this 'FRACTAL' comes from such a trivial method!!!. There are so many ways for applying the 'Rational Mean' in order to solve any algebraic equation. There are so many paths which rules the sequence of approximations to the solution. I say 'Rational Mean' instead of 'Mediant' because the last is only related to the generation of reduced fractions as the Farey fractions and the convergents in the SIMPLE continued fractions. I hope someone could take the time to continue with the above fraction. Indeed, I hope you have fun by computing its convergents!!!. In fact, you can try to generate a similar continued fraction of the fourth, fifth degree, etc... And... why not? try to compute the convergents of such 'FRACTALS'!!!.
In another way, it is important to note the relation between the initial fractions (see my previous messages on 'the duplication of the cube ') : 4 5 6 - - - 3 4 5 and the two mean proportionals (a / x , x / y , y/2a) between two given lines, showed by Hippocrates of Chios as solution for the duplication of the cube (David Eugene Smith, History of Mathematics, Vol. II, page 313). All this compels me to say it over and over again : i can hardly believe this so simple method for solving algebraic equations hasn't been used before the sway of analytic geometry!!!. (That's why I'm here, at the math-HISTORY-list!!!)
Next message I will continue with some brief comments on the Rational Process, the Rational Mean and my paper : "New Elements for the Irrational Numbers" published by the Journal Of Transfigural Mathematics.