Random Fibonacci Tilings

Abstract

The Fibonacci spiral is formed by a square-joining rule: begin with a unit square and attach squares whose side lengths follow the Fibonacci sequence; in the standard construction, choices of direction produce a spiral and a tiling of the plane. We study a randomization of this construction, in which the attachment axis alternates deterministically, while fair coin flips determine direction. For a lattice point we ask when the growing rectangle first covers that point and how the expected covering time depends on its location. Tracking the rectangle's boundary reduces this question to random subset sums of even- and odd-indexed Fibonacci numbers. We extend the construction to d dimensions, where coordinate subsequences are super-increasing, allowing us to derive exact covering probabilities and formulas for the expectation and variance. For $d\geq2$, the expected covering time grows logarithmically with distance from the initial block, while the variance remains uniformly bounded. We obtain exponential tail bounds for the delay beyond the earliest feasible covering stage. Finally, we extend the Fibonacci spiral to the randomized tiling through a square-by-square construction using circular arcs and quintic connectors. This construction uses one-step lookahead, admits a continuously differentiable parameterization, and recovers the Fibonacci spiral for the standard tiling.