Pinwheel patterns and powder diffraction

Pinwheel patterns and their higher dimensional generalisations display continuous circular or spherical symmetries in spite of being perfectly ordered. The same symmetries show up in the corresponding diffraction images. Interestingly, they also arise from amorphous systems, and also from regular crystals when investigated by powder diffraction. We present first steps and results towards a general frame to investigate such systems, with emphasis on statistical properties that are helpful to understand and compare the diffraction images. We concentrate on properties that are accessible via an alternative substitution rule for the pinwheel tiling, based on two different prototiles. Due to striking similarities, we compare our results with a toy model for the powder diffraction of the square lattice. 1. Pinwheel patterns The Conway-Radin pinwheel tiling [14], a variant of which is shown in Figure 1, is a substitution tiling with tiles occurring in infinitely many orientations. Consequently, it is not of finite local complexity (FLC) with respect to translations alone, though it is FLC with respect to Euclidean motions. This property distinguishes the pinwheel tiling from the majority of substitution tilings considered in the literature. As a consequence, its diffraction differs considerably from that of other tilings, and despite a growing interest in such structures [13, 12, 1, 18], the diffraction properties have only been partially understood to date. Whereas the pinwheel tiling is the most commonly investigated example, there are other tilings with infinitely many orientations, compare [15] for an entire family of generalisations. Yet another example is shown in Figure 2. It has a single prototile, an equilateral triangle with side lengths 1, 2 and 2. Under substitution, the prototile is mapped to nine copies, some rotated by an angle θ = arccos(1/4), which is incommensurate to π (i.e., θ / ∈ πQ). Thus, the corresponding rotation Rθ is of infinite order, and the tiles occur in infinitely many orientations in the infinite tiling. Here and below, Rα denotes the rotation through the angle α about the origin. More examples of tilings with tiles in infinitely many orientations can be found in [7]. It was shown constructively in [12] that the autocorrelation γ of the pinwheel tiling has full circular symmetry, a result that was implicit in previous work [14]. As a consequence, the diffraction measure γ̂ of the pinwheel tiling shows full circular symmetry as well. To make this concrete, we now construct a Delone set from the tiling. Recall that a Delone set Λ in Euclidean space is a point set which is uniformly discrete (i.e., there is r > 0 such that each ball of radius r contains at most one point of Λ) and relatively dense (i.e., there is R > 0 such that each ball of radius R contains at least one point of Λ). Let T be the unique fixed point of the pinwheel substitution of Figure 1 that contains the triangle with vertices ( 1 2 ,− 1 2 ), (− 1 2 ,− 1 2 ), (− 1 2 , 3 2 ). This fixed point T is the same as the one considered in [12]. We now define the set of control points ΛT of T to be the set of all points u + u−v 2 + u−w 4 such that the triangle with vertices u,v,w is in T and uv is the edge of length one. This choice of control points is indicated in Figure 1 (left) and is the same as in [12]. Recall that the natural autocorrelation measure of a Delone set Λ is defined as (1) γ := lim R→∞ 1 πR2 ∑