Pinwheel patterns and powder diffraction

Abstract

Pinwheel patterns and their higher dimensional generalizations 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 framework 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.

Acknowledgements

It is our pleasure to thank R. V. Moody and M. Whittaker for cooperation and helpful comments. This work was supported by the German Research Council (DFG), within the CRC 701. UG gratefully acknowledges conference travel support by The Royal Society.