Polygonal corona limit on multigrid dual tilings

TL;DR

This work generalizes the Penrose corona limit phenomenon from decagons to all regular multigrid dual tilings by introducing the characteristic polygon $\\chi$ determined by multigrid directions. The authors first analyze corona growth on the multigrid, using dominant lines to derive $\\chi$ and show that coronas along these lines grow with distance factors $\\chi_i$, then extend the result to the dual tiling via an almost-linear dual map $\\mathcal{F}$, proving the corona limit of the tiling is $\\widetilde{\\chi}=\\mathcal{F}(\\chi)$. A key outcome is that the corona limit depends only on the multigrid directions, not the offsets, unifying the behavior across Penrose and other regular multigrid tilings. The framework clarifies why highly regular polygonal limits emerge from aperiodic substrates and paves the way for exploring neighborhood-based growth and percolation in these tilings. These results have potential implications for understanding diffusion-like processes in quasicrystal-inspired tilings and their crystallographic counterparts.

Abstract

The growth pattern of an invasive cell-to-cell propagation (called the successive coronas) on the square grid is a tilted square. On the triangular and hexagonal grids, it is an hexagon. It is remarkable that, on the aperiodic structure of Penrose tilings, this cell-to-cell diffusion process tends to a regular decagon (at the limit). In this article we generalize this result to any regular multigrid dual tiling, by defining the characteristic polygon of a multigrid and its dual tiling. Exploiting this elegant duality allows to fully understand why such surprising phenomena, of seeing highly regular polygonal shapes emerge from aperiodic underlying structures, happen.

Figures (13)

• Figure 1: Examples of multigrids.
• Figure 2: Example of a regular pentagrid and its dual tiling. Some elements of the multigrid and their dual in the tiling have been colored: in red and yellow two cells of the multigrid and the corresponding vertices in the tiling, and in blue and green two lines of the multigrid and the corresponding ribbons of tiles (each intersection of two grid lines corresponds to one tile).
• Figure 3: A patch of a Penrose tiling in dark grey and its 4 first coronas in greyscale.
• Figure 4: Successive coronas from an initial single tile in a Penrose tiling. The 40-th corona $P_{40}$ is already very close to a regular decagon which is the corona limit of Penrose tilings.
• Figure 5: A simple tiling by $1\times1$ and $2\times 2$ square tiles where the corona limit does not exist.

Theorems & Definitions (46)

• Definition 1: Grid and multigrid
• Definition 2: Dualization and multigrid dual tiling
• Lemma 1: Regular multigrid dual tiling db81db86
• Definition 3: Types of crossings and tiles
• Definition 4: Corona
• Definition 5: Corona limit
• Lemma 2: Corona limit of single tiles
• proof
• Remark 1
• Definition 6: Characteristic polygon $\chi$