The Leaf Function of Graphs Associated with Penrose Tilings

In graph theory, the question of fully leafed induced subtrees has recently been investigated by Blondin Massé et al in regular tilings of the Euclidian plane and 3-dimensional space. The function LG that gives the maximum number of leaves of an induced subtree of a graph $G$ of order $n$, for any $n\in \N$, is called leaf function. This article is a first attempt at studying this problem in non-regular tilings, more specifically Penrose tilings. We rely not only on geometric properties of Penrose tilings, that allow us to find an upper bound for the leaf function in these tilings, but also on their links to the Fibonacci word, which give us a lower bound. Our approach rely on a purely discrete representation of points in the tilings, thus preventing numerical errors and improving computation efficiency. Finally, we present a procedure to dynamically generate induced subtrees without having to generate the whole patch surrounding them.

Fractal Tilings Based on Successive Adjacent Substitution Rule

A fractal tiling or f-tiling is a tiling which possesses self-similarity and the boundary of which is a fractal. f-tilings have complicated structures and strong visual appeal. However, so far, the discovered f-tilings are very limited since constructing such f-tilings needs special talent. Based on the idea of hierarchically subdividing adjacent tiles, this paper presents a general method to generate f-tilings. Penrose tilings are utilized as illustrators to show how to achieve it in detail. This method can be extended to treat a large number of tilings that can be constructed by substitution rule (such as chair and sphinx tilings and Amman tilings). Thus, the proposed method can be used to create a great many of f-tilings.