Fully Leafed Induced Subtrees in Penrose P2 Tilings

Abstract

In a recent article by C. Porrier, A. Blondin Massé and A. Goupil, a first bi-infinite fully leafed induced subcaterpillar of Penrose P2 tilings is presented. In this paper, we formally construct this caterpillar for the first time. We then prove that every fully leafed induced subtree in Penrose P2 tilings is a caterpillar with at most one appendix of at most two internal tiles, and we characterize fully leafed induced subtrees that have the property of saturation. We also refute the conjecture that there is a unique bi-infinite fully leafed induced subcaterpillar by constructing a new one. Finally, we present progress on the construction of all bi-infinite fully leafed induced subcaterpillars in Penrose P2 tilings.

Figures (40)

• Figure 1: Prototiles of P2 tilings. Tiles may meet only when filled (resp. empty) vertices coincide. $\varphi = \frac{1+\sqrt{5}}{2}$ is the golden ratio, giving the ratio of long to short edges in both prototiles.
• Figure 2: The seven minimal vertex neighborhoods in P2 tilings up to isometry
• Figure 3: Inflation of P2 tiles
• Figure 4: A region of a Penrose P2 tiling. The kites are shown in grey and the darts in white.
• Figure 5: The six possible prime caterpillars, up to isometry. A prime caterpillar is only determined by its derived path. In other words, there are six different prime caterpillars, up to the choice of leaves.

Theorems & Definitions (77)

• Definition 1: blondin2018fully
• Definition 2: blondin2018fully
• Lemma 1: porrier2019leaf
• Lemma 2: porrier2023leaf
• Corollary 1
• proof
• Lemma 3: porrier2023leaf
• Corollary 2
• proof
• Lemma 4: porrier2023leaf