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Penrose P2 Tilings: A Study of Fully Leafed Induced Subtrees

Mathieu Cloutier, Alain Goupil, Alexandre Blondin Massé

Abstract

We present new results about fully leafed induced subtrees in Penrose P2 tilings. We first determine the graph structure of these subtrees and show that they are caterpillars, up to an appendix of at most six tiles. We then study bi-infinite fully leafed induced caterpillars in P2 tilings and their geometric properties. In particular, we refute the conjecture proposed by C. Porrier, A. Goupil and A. Blondin Massé that there is a unique bi-infinite fully leafed caterpillar in Penrose P2 tilings.

Penrose P2 Tilings: A Study of Fully Leafed Induced Subtrees

Abstract

We present new results about fully leafed induced subtrees in Penrose P2 tilings. We first determine the graph structure of these subtrees and show that they are caterpillars, up to an appendix of at most six tiles. We then study bi-infinite fully leafed induced caterpillars in P2 tilings and their geometric properties. In particular, we refute the conjecture proposed by C. Porrier, A. Goupil and A. Blondin Massé that there is a unique bi-infinite fully leafed caterpillar in Penrose P2 tilings.
Paper Structure (6 sections, 9 theorems, 8 figures)

This paper contains 6 sections, 9 theorems, 8 figures.

Table of Contents

  1. Introduction
  2. Definitions
  3. Graph structure
  4. Prime caterpillar graftings
  5. Bi-infinite fully leafed caterpillars
  6. Conclusion

Key Result

Lemma 1

Let $\mathcal{C}=A\diamond PC_k$ be a fully leafed induced subtree obtained by grafting the appendix $A$ to the prime caterpillar $PC_k$. Then $A$ has at most 2 internal tiles.

Figures (8)

  • Figure 1: Prototiles of P2 tilings. Tiles are edge-adjacent only when black (resp. white) vertices coincide. $\varphi = \frac{1+\sqrt{5}}{2}$ is the golden ratio, giving the ratio of long to short edges in both types of tile.
  • Figure 2: Two P2 patches
  • Figure 3: Inflation of P2 tiles
  • Figure 4: The six possible prime caterpillars, up to isometry and choice of leaves. Light green tiles are choices for leaves.
  • Figure 5: The only two derived paths of graftings of two prime caterpillars. Light brown tiles are possibilities for degree-3 tiles. Blue tiles have degree $2$. We add a segment that joins the centers of the stars adjacent to the prime caterpillars.
  • ...and 3 more figures

Theorems & Definitions (12)

  • Definition 1
  • Definition 2: masse2018saturated and porrier2023leaf
  • Lemma 1: cloutier2026fullyleafedinducedsubtrees
  • Theorem 1: cloutier2026fullyleafedinducedsubtrees
  • Proposition 1: cloutier2026fullyleafedinducedsubtrees
  • Proposition 2: cloutier2026fullyleafedinducedsubtrees
  • Proposition 3: cloutier2026fullyleafedinducedsubtrees
  • Lemma 2: cloutier2026fullyleafedinducedsubtrees
  • Proposition 4: cloutier2026fullyleafedinducedsubtrees
  • Proposition 5: cloutier2026fullyleafedinducedsubtrees
  • ...and 2 more