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The hat polykite as an Iterated Function System

Corey de Wit

TL;DR

The work addresses modeling the celebrated hat tiling with an overlapping sequential IFS (SIFS), extending IFS tiling theory to imperfect substitutions. By developing sequential IFSs with uniform contraction and defining tiling SIFS that manage overlaps, the paper constructs a limit attractor and a finite-type top code description for the hat tiling via explicit IFS maps. Key contributions include a concrete hat-tiling SIFS based on $H_8$/$H_7$ clusters and a $c$-parameter, a proof framework for existence of the limit SIFS, and a translation-detection method yielding the explicit maps. This approach unifies aperiodic tilings with IFS theory, enabling analysis of fractal boundaries and substitution-type tilings in a rigorous, symbolic setting with potential applications to other imperfect tilings.

Abstract

This paper describes the celebrated aperiodic hat tiling by Smith et al. [Comb. Theory 8 (2024), 6] as generated by an overlapping iterated function system. We briefly introduce and study infinite sequences of iterated function systems that converge uniformly in each component, and use this theory to model the hat tiling's associated imperfect substitution system.

The hat polykite as an Iterated Function System

TL;DR

The work addresses modeling the celebrated hat tiling with an overlapping sequential IFS (SIFS), extending IFS tiling theory to imperfect substitutions. By developing sequential IFSs with uniform contraction and defining tiling SIFS that manage overlaps, the paper constructs a limit attractor and a finite-type top code description for the hat tiling via explicit IFS maps. Key contributions include a concrete hat-tiling SIFS based on / clusters and a -parameter, a proof framework for existence of the limit SIFS, and a translation-detection method yielding the explicit maps. This approach unifies aperiodic tilings with IFS theory, enabling analysis of fractal boundaries and substitution-type tilings in a rigorous, symbolic setting with potential applications to other imperfect tilings.

Abstract

This paper describes the celebrated aperiodic hat tiling by Smith et al. [Comb. Theory 8 (2024), 6] as generated by an overlapping iterated function system. We briefly introduce and study infinite sequences of iterated function systems that converge uniformly in each component, and use this theory to model the hat tiling's associated imperfect substitution system.

Paper Structure

This paper contains 9 sections, 4 theorems, 35 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Iterated Function Systems
  3. Sequential Iterated Function Systems
  4. SIFS systems
  5. Tiling SIFSs
  6. General remarks
  7. The Hat Tiling as a tiling SIFS
  8. Proof of Theorem \ref{['thm:sifs']}
  9. Acknowledgements

Key Result

Proposition 1

Let $(F_n)_{n\in\mathbb{N}}$ be an SIFS. Then $A_n \to A$ with respect to $d$.

Figures (5)

  • Figure 1: First four supertiles corresponding to the SIFS \ref{['eq:hexsifs']}, with the supertiles corresponding to the first and second indexed functions outlined. The limiting boundary of the tiling construction is the (hexagonal) convex hull of the supertiles above, which is also the attractor of the limit IFS.
  • Figure 2: $H_8$ cluster (left), $H_7$ cluster (right); $c = 1/(1+\sqrt{3})$
  • Figure 3: $T_c$ (top left), $S_1$ (top right), $S_2$ (bottom left), $S_3$ (bottom right); $c = 1/(1+\sqrt{3})$. Each color represents a different indexed function.
  • Figure 4: An example of $p_3$ and $q_3$ on $S_3$, as well as a visualisation of the proof methods for Claims \ref{['claim1']} and \ref{['claim2']}; $c = 1/(1+\sqrt{3})$
  • Figure 5: The attractor of the limit IFS for the SIFS given in Theorem \ref{['thm:sifs']}; $c = 1/(1+\sqrt{3})$.

Theorems & Definitions (14)

  • Proposition 1
  • proof
  • Example 2
  • Proposition 3
  • proof
  • Remark 4
  • Theorem 5
  • Theorem 6
  • Remark 7
  • Remark 8
  • ...and 4 more