The Fibonacci Tiling The Fibonacci Tiling

Let PR be the set of patches of radius R, modulo translation. The tiling has finite local complexity (FLC), if and only if PR is a finite set for all R. In particular R → PR is locally constant and non-decreasing. Thus there is a sequence R0 = 0 < R1 < · · · < Rn < · · · with Rn→ ∞ such that PR = Pn for Rn ≤ R < Rn+1. Inverse Limit There is a restriction map pi: Pn+1 → Pn. Then the transversal is defined by the inverse limit Ξ = lim←pi Pn Inverse Limit For The Fibonacci and Octagonal Tilings, as for all cut-and-project tilings, the transversal coincides with the window provided the window is endowed with a topology that makes all acceptance domains closed and open Rooted Tree Since all the Pn’s are finite set, Ξ is a Cantor set. A point of Ξ is an infinite sequence ξ = (pn)∞n=0 of compatible patches, so it defines a unique tiling. This inverse limit can be represented by a rooted tre