Some Structures arising from the Farey Fractal

Abstract

This work explores new arithmetic and combinatorial structures arising from the interplay between Farey-type graphs, Fibonacci expansions, and operadic constructions. We introduce Fibonadic numbers, defined as an inverse limit under the Zeckendorf shift, equipped with a metric, order, and commutative rig structure. A normalization lemma provides canonical representatives, while quotients of the non-zero Fibonadic numbers under shifts and it's fundamental domain covering the circle via phi-values. Levels with associated functions encoding the decomposition of X into arithmetic layers. This research links number theory, combinatorics, discrete dynamics, operads, fractal geometry and the golden ratio.