Penrose P2 Tilings: A Study of Fully Leafed Induced Subtrees

TL;DR

This paper investigates the structure of fully leafed induced subtrees in Penrose P2 tilings, revealing they are caterpillars and refuting a conjecture about their uniqueness.

Contribution

It characterizes the graph structure of these subtrees and disproves the conjecture of a unique bi-infinite fully leafed caterpillar in Penrose P2 tilings.

Findings

Fully leafed induced subtrees are caterpillars.

There are multiple bi-infinite fully leafed caterpillars.

The conjecture of uniqueness is false.

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 Mass\'e that there is a unique bi-infinite fully leafed caterpillar in Penrose P2 tilings.