A Simple Algebraic Equation, Diophantine Numbers and Patterns of Nature
Even an algebraic equation as simple as x2+ax+b=0 unfolds many interesting stories. The solutions of this equation for some specific coefficients are referred to the golden ratio, the silver ratio and the platinum ratio, which are further related to the 10-fold, 8-fold and 12-fold quasiperiodic structures, respectively. They make their presence in many patterns in nature, and even in physics problems such the Ising model, the hard hexagon model, the Hardy’s test of Bell’s inequality, etc. By stress engineering we achieved the growth of microscopic Fibonacci parastichous spirals, leading us to a deep understanding of phyllotaxis. Furthermore, by playing with the function , where x as parameter is the silver ratio or platinum ratio, and n the integer as variable, 8-fold and 12-fold quasiperiodical patterns, in a loose sense, can be generated, resulting in the discovery of directional scaling symmetry in square and triangular lattices—a problem that had frustrated Galileo long long ago. With the aid of Gauss and Eisenstein integers, it can be proven that there are infinitely many possibilities of scale symmetry for square lattice, of which one is also related to the golden ratio. Such discoveries are very interesting and can be arrived at only in a serendipitous fashion.