On the Fibonacci tiling and its modern ramifications

TL;DR

The paper provides a cohesive survey of the Fibonacci tiling as a tractable gateway to the broader theory of aperiodic order, connecting one-dimensional symbolic dynamics, cut-and-project constructions, and higher-dimensional tilings. It develops the embedding method via the return module $\mathbb{Z}[\tau]$ and a Euclidean CPS to realize Fibonacci sets as regular model sets, and it then analyzes variations, equidistribution, and ergodic properties to derive exact patch frequencies and pair correlations. A central thread is the demonstration of pure point diffraction and its dynamical-systems counterpart, established through renormalisation relations and the Koopman framework, with implications for understanding quasicrystals and monotile tilings like Hat and Spectre. The work further discusses shape changes, Rauzy fractals, and generalizations, highlighting how these methods yield deep insights into spectral theory, density of states, and higher-dimensional phenomena in aperiodic order. Overall, the Fibonacci framework provides a unifying lens for diffraction, dynamics, and geometry in aperiodic tilings and their modern ramifications.

Abstract

In the last 30 years, the mathematical theory of aperiodic order has developed enormously. Many new tilings and properties have been discovered, few of which are covered or anticipated by the early papers and books. Here, we start from the well-known Fibonacci chain to explain some of them, with pointers to various generalisations as well as to higher-dimensional phenomena and results. This should give some entry points to the modern literature on the subject.

Figures (10)

• Figure 1: The geometric inflation rule for the Fibonacci chain, with natural tile lengths.
• Figure 2: The Minkowski embedding of $\space\mathbb{Z}[\tau]$ as a planar lattice, where the projections are illustrated for the lattice point $(x,x^{\star})$ with $x=4+\tau$. The horizontal (vertical) axis represents the physical (internal) space. The grey strip contains all lattice points that are projected to the horizontal line for our guiding Fibonacci example.
• Figure 3: Diffraction of the visible lattice points. They have pure point (or Bragg) diffraction. The Bragg peaks are represented by disks whose area is the height (or intensity) of the peak, located at the centre of the disk; see TAO for more.
• Figure 4: The Fibonacci direct product inflation rule in the Euclidean plane. Using the lower left corners as control points, one can extend the projection description of the Fibonacci chain to this case by doubling all dimensions and taking direct products of the 1D windows as the new windows here; see BFG for details.
• Figure 5: The two windows (blue/top for control points of type $a$ and yellow/bottom for type $b$) for the tiling given by the substitution $(aab,ba)$. The inlay shows a stretched view of the marked region. We emphasise that the sets $W_a$ and $W_b$ are measure-theoretically disjoint, see for example Bernd, even though this is difficult to illustrate due to the high Hausdorff dimension of the boundaries.