A single fractal pinwheel tile

Abstract

The pinwheel triangle of Conway and Radin is a standard example for tilings with self-similarity and statistical circular symmetry. Many modifications were constructed, all based on partitions of triangles or rectangles. The fractal example of Frank and Whittaker requires 13 different types of tiles. We present an example of a single tile with fractal boundary and very simple geometric structure which has the same symmetry and spectral properties as the pinwheel triangle.

Fractals: An Eclectic Survey, Part-I

“…We also refer to the recent works on fractal tilings by Bandt and Mekhontsev [32][33][34][35] for a comprehensive summary and current state of the art in the subject. The use of the inverses of integer matrices in the study of tilings is well-established, and can be found in particular in many of the references cited above.…”

Computer Geometry: Rep-Tiles with a Hole

“…They can be determined recursively, and if the map f (x) = x is not among them then we really have a rep-tile. Details are explained in [2,3] Working out a single three-dimensional example by hand may take a day, or even a week. The computer opens up new perspectives.…”

“…On the other hand, will computers change our attitudes? We discuss such issues for elementary problems of fractal geometry [2,3], using the free software package IFStile [15]. This note is concerned with self-similar tilings of three-dimensional space.…”

Lattice self-similar sets on the real line are not Minkowski measurable

“…We keep only those arrows which correspond to proper neighbors, that is φ i (F ) ∩ hφ j (F ) = ∅." [BMT17]. Using the terminology from [ Obviously, the first statement of the theorem follows from the second.…”