Abstract
Two kinds of lattices built up step by step to satisfy a certain condition on uniformity are introduced and proved to be self-similar for two series of irrational ratios of concentrations, \sqrt2+1, (\sqrt5+1)⁄2, …, (\sqrtk2+1+1)⁄k, … and \sqrt3, \sqrt2, …, \sqrt(k+2)⁄k, …. Consequently for the golden ratio two different self-similar lattices, the uniform and the Fibonacci ones, exist. The condition is shown to be equivalent to the projection method in which the edge of the strip contains the central point of the integer lattice. The self-similar structures of the uniform lattices presented are studied by the aid of the convergents to continued fraction of the ratio and mediants.
