Fibonacci sequences and the space of compact sets

Abstract

The Fibonacci numbers appear in many surprising situations. We show that Fibonacci-type sequences arise naturally in the geometry of $\mathcal{\mathscr{H}}\left({ℝ}^2\right)$, the space of all nonempty compact subsets of ${ℝ}^2$ under the Hausdorff metric, as the number of elements at each location between finite sets. The results provide an interesting interplay between number theory, geometry, and topology.