Vieta jumping, Lagrange groups, orders of irrationality, lines of symmetry

In mathematics, Vieta jumping, also known as root flipping, is a number theory proof technique. It is most often used for problems in which a relation between two positive integers is given, along with a statement to prove about its solutions. There are multiple methods of Vieta jumping, all of which involve the common theme of infinite descent by finding new solutions to an equation using Vieta's formulas.

They are all related, and mutually related to Mersenne primes. I am exploring this fascinating topic and going to learn a lot, I am sure. I am going to find more than one level of Vieta jumping, the orders of jumping should go as the Markov numbers. The idea should be the basis of new theory, and I hope the mathematical wizards of the web beat me to it. The proton will become a form of Vieta jumping, minimizing the unit circles of N dimensions by a slight turn of the screw which redistributes irrationality to the wave power series among symmetric lines of symmetry. The integers are a composite of vector combinations of the irrationals:  

v1 * ir1 + v2 * ir2 + v3 * ir3 = N

Integers are now vectors, in finite stepped space,  maintain integers by rotations of the irrational error, each having 1,2 and 3 lines of symmetry. Each rotation is a Vieta jump.

 We need to add more dimensionality to this Reimann surface. Following the links thru Wiki and the Mobius strip lead to:

In mathematics, the topological entropy of a topological dynamical system is a nonnegative real number that is a measure of the complexity of the system. Topological entropy was first introduced in 1965 by Adler, Konheim and McAndrew. Their definition was modelled after the definition of the Kolmogorov–Sinai, or metric entropy. Later, Dinaburg and Rufus Bowen gave a different, weaker definition reminiscent of the Hausdorff dimension. The second definition clarified the meaning of the topological entropy: for a system given by an iterated function, the topological entropy represents the exponential growth rate of the number of distinguishable orbits of the iterates. An important variational principle relates the notions of topological and measure-theoretic entropy.