[Paper Review] Penrose P2 Tilings: A Study of Fully Leafed Induced Subtrees

Mathieu Cloutier, Alain Goupil+1 more|arXiv (Cornell University)|Mar 10, 2026

Quasicrystal Structures and Properties0 citations

TL;DR

The paper analyzes fully leafed induced subtrees in Penrose P2 graphs, showing they are caterpillars (up to small appendices) and constructs bi-infinite examples, refuting prior uniqueness conjectures.

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.

Motivation & Objective

• Motivate the study by understanding extremal boundary structures in tiling-derived graphs with potential applications to adsorption on quasicrystal surfaces.
• Characterize the graph structure of fully leafed induced subtrees in Penrose P2 graphs.
• Develop construction methods for large and bi-infinite fully leafed subtrees via prime caterpillar graftings.
• Explore geometric embedding considerations and star-graph representations to analyze possible configurations.
• Disprove the conjecture of uniqueness for bi-infinite fully leafed caterpillars in Penrose P2 tilings.

Proposed method

• Classify fully leafed induced subtrees by their internal tile degrees (prime caterpillars) and use grafting operations at leaves to build larger structures.
• Show that all such subtrees are caterpillars up to an appendix of at most six tiles.
• Employ star-graph representations (centers of P2 stars connected by a path) to analyze graftings and resulting caterpillar structure.
• Analyze angles between grafted prime caterpillars, identifying three possible angle measures (4π/5, 6π/5, 8π/5) and their implications for uniqueness.
• Construct sea caterpillars and analyze their star-graph paths to build larger saturated caterpillars.
• Formalize bi-infinite fully leafed caterpillars and refute the prior uniqueness conjecture by presenting a new bi-infinite example containing cape 4.

(a) Kite

Experimental results

Research questions

• RQ1 What is the precise graph structure of fully leafed induced subtrees in Penrose P2 graphs?
• RQ2 Are all fully leafed induced subtrees in P2 graphs caterpillars (up to a small appendix), and how can they be constructed via grafting prime caterpillars?
• RQ3 Is there a unique bi-infinite fully leafed caterpillar in Penrose P2 tilings, or are there multiple such structures?
• RQ4 How do geometric constraints in P2 tilings (such as inflation and star-graph representations) influence the possible configurations of fully leafed subtrees?
• RQ5 Can we classify bi-infinite fully leafed caterpillars using star-graph paths and angle measures to enumerate possibilities?

Key findings

• Every fully leafed induced subtree of a P2-graph is a subcaterpillar of a prime caterpillar up to an appendix of at most two internal tiles (or at most six tiles in a broader sense).
• Prime caterpillars provide the primitive blocks for constructing larger fully leafed induced subtrees via grafting, and any such subtree can be built from graftings of prime caterpillars (with optional small appendices).
• The saturated fully leafed subtrees (maximizing leaves for a given order) are precisely caterpillars obtained from these graftings.
• There exist bi-infinite fully leafed caterpillars that include a cape 4, refuting the conjecture of uniqueness of bi-infinite fully leafed caterpillars in Penrose P2 tilings.
• Angles between grafted prime caterpillars are constrained to three values (4π/5, 6π/5, 8π/5), and the presence of an 8π/5 angle on one side can determine certain caterpillars uniquely, aiding classification.
• The authors introduce constructions such as Super cape 4 and ψ(Super cape 4) to demonstrate infinite (bi-infinite) fully leafed caterpillars.

(b) Dart