Diffraction from Aperiodic Einstein Tiling

What is it about?

The research analyzed the diffraction pattern from a recently reported aperiodic 'einstein' or 'hat' monohedral tiling. The structure is the hexagonal MTA net, a kite tiling, with aperiodic vertex deletions. The diffraction pattern displays a robust sixfold periodicity in plane group p6 and a repeating, roughly triangular motif of 'diffused intensity' between the strongest Bragg peaks. The pattern is chiral and breaks mirror symmetry. As the model size increases, the diffused scattering pattern becomes more non-mirror-symmetric, and the speckles sharpen. The pattern resolves into weak discrete spots in the limit of an infinitely large model.

Featured Image

Why is it important?

The study of the diffraction pattern from the recently reported aperiodic 'einstein ', or 'hat', monohedral tiling has important implications for our understanding of quasicrystalline structures. This research shows that the vertex structure of the aperiodic, quasicrystalline hat monotiling generates a 2-periodic chiral diffraction pattern in the limit of a large model of point scatterers. This discovery highlights the unique properties of the hat tiling and its diffraction pattern, which differ from other known quasicrystalline tilings. Key Takeaways: 1. The hat tiling generates a 2-periodic chiral diffraction pattern in the limit of a large model of point scatterers. 2. The diffraction pattern's point symmetry is 6, but its 2-periodicity for point scatterers is also consistent with plane group symmetry p6, with no mirror lines. 3. As the model size increases, the diffused scattering continuously resolves into a system of ever closer-spaced, self-similar, sharp satellite peaks centered at the 2m=3 2n=3 reciprocal-lattice locations. 4. The diffraction pattern details are strikingly different from those of the quasicrystalline Penrose tiling.