On a connection between the limit set of the Möbius–Klein transformation, periodic continued fractions, El Naschie’s topological theory of high energy particle physics and the possibility of a new axion-like particle

In the present work we first give a general representation of the derivatives of the irrational number φ, for instance , , etc., as periodic continued fractions. Any irrational number can then be expanded in an infinite continued fraction. The limit set of the Kleinian transformation acting on the E-infinity Cantorian spacetime turned out to be this set of periodic continued fractions, consequently the vacuum of the E-infinity is described by this limit set. As discussed by El Naschie, every particle can be interpreted geometrically as a scaling of another. This is done using the topology of hyperbolic Kleinian space of VAK, which is nothing but our limit set. Here we will present the ratios of the theoretical masses of certain elementary particles to that of some chosen particles in term of φ. Many of these masses are quite close to integer multiples of the mass of a chosen particle. Finally we discuss the possibility of new transfinite, axion-like particles as discussed recently by Krauss and El Naschie [Quintessence, Vintage, London, 1999].