A Fractal Version of the Pinwheel Tiling

Abstract

Dedicated to the inspiration of Benoit Mandelbrot.The pinwheel tilings are a remarkable class of tilings of the plane, and our main goal in this paper is to introduce a fractal version of them. We will begin by describing how to construct the pinwheel tilings themselves and by discussing some of the properties that have generated so much interest. After that we will develop the fractal version and discuss some of its properties. Finding this fractile version was an inherently interesting problem, and the solution we found is unusual in the tiling literature.Like the well-known Penrose tilings [5], pinwheel tilings are generated by an "inflate-and-subdivide rule" (see Figure 1). Tilings generated by inflate-and-subdivide rules form a class of tilings that have a considerable amount of global structure called self-similarity. Self-similar tilings are usually nonperiodic but still exhibit a form of "long-range order" that makes their study particularly fruit- The technique for finding fractiles in both [3] and [8] is similar. One begins with an inflate-andsubdivide rule for which the edges of each inflated tile do not quite match up with the edges of the tiles that replace it (the Penrose kite and dart are an example). The edges of each tile are redrawn using the edges of the tiles that replace it. These new edges are revised iteratively by the The authors would like to thank Dirk Frettlöh for helpful discussions about the aorta and Edmund Harriss for pointing out our theorem on rotations in the aorta. The second author is partially supported by ARC grant 228-37-1021, Australia.

A single fractal pinwheel tile

“…They use triangles, and most of them have a larger number of tiles. There is also a fractal pinwheel version [12] which uses 13 different types of tiles. Figure 1 shows an unexpected single fractal tile with irrational rotations between neighbor tiles, denoted as 'fractal pinwheel'.…”

“…Choosing part 4 as basic tile A, and point M as origin of a coordinate system with N = 1 0 , V = 0 1 , we would get the expanding matrix as 2−1 1 2 . Thus the expansion map is the same as for the pinwheel triangle in [24,25,12]. The h i are quite different: in Figure 1, piece k is connected only with pieces k − 1, k + 1 by a fractal edge, while in the pinwheel triangle three pieces are pairwise connected by edges or 'half-edges'.…”

Fractal dual substitution tilings

“…Moreover, it allows for hands-on experimentation with an infinitude of choices, and for now it is not completely clear the significance of making one choice over another. In [16], we gave a somewhat ad-hoc method for making a fractal realization of the Pinwheel tiling. In this paper we show that a similar construction works for every primitive substitution tiling where tiles meet singly edge-to-edge and remove the ad-hoc nature of the construction in [16].…”

Pinwheel tiling fractal graph- a notion to edge cordial and cordial labeling

“…The aim of this work is to introduce a tiling substitution on fractal tiles that produces tiling that is locally derivable from the pinwheel tiling mutually. But first, following M. Baake, D. Frettlo¨ h, and U. Grimm, [10], Natalie and Michael [12] researchers have introduced a tiling substitution called the ''kite-domino'' pinwheel tiling. The pinwheel triangles in any pinwheel tiling meet up hypotenuse-to-hypotenuse to form either a kite or a domino.…”

“…Thus, it is a non-periodic tiling Simon Parzer [16]. Its construction and properties are discussed by many of researchers such as Natalie and Michael [12]. In this paper, our study is to enable the Cordial, total cordial, Edge cordial and total edge cordial labeling for the above mentioned tiling fractal curves.…”