Abstract
Most quasicrystals can be generated by the cut-and-project method from higher dimensional parent lattices. In doing so they lose the periodic order their parent lattice possess, replaced with aperiodic order, due to the irrationality of the projection. However, perfect periodic order is discovered in the perpendicular space when gluing the cut window boundaries together to form a curved loop. In the case of a 1D quasicrystal projected from a 2D lattice, the irrationally sloped cut region is bounded by two parallel lines. When it is extrinsically curved into a cylinder, a line defect is found on the cylinder. Resolving this geometrical frustration removes the line defect to preserve helical paths on the cylinder. The degree of frustration is determined by the thickness of the cut window or the selected pitch of the helical paths. The frustration can be resolved by applying a shear strain to the cut-region before curving into a cylinder. This demonstrates that resolving the geometrical frustration of a topological change to a cut window can lead to preserved periodic order.
Highlights
With the announcement by Shechtman et al [1] in 1984 of a physical material exhibiting crystallographically forbidden symmetries in its diffraction patterns, and the following theoretical description by Levine et al [2], the study of aperiodically ordered structures moved from what was something of a niche specialty to a burgeoning field in both mathematics and material science
In curling up a dimension in P⊥, we find a new kind of geometric frustration, as well as a way to resolve it
In an attempt to resolve the geometric frustration in quasicrystals, which is a natural result of their aperiodic nature, we discovered the inherent periodicity associated to a quasicrystal when considering its dual space, the perpendicular space (P⊥ ) of the cut-andproject method
Summary
With the announcement by Shechtman et al [1] in 1984 of a physical material exhibiting crystallographically forbidden symmetries in its diffraction patterns, and the following theoretical description by Levine et al [2], the study of aperiodically ordered structures (dubbed quasicrystals) moved from what was something of a niche specialty to a burgeoning field in both mathematics and material science. Quasicrystals have long range order, usually the symmetry of some regular polyhedron, and they have sharp diffraction patterns, generally because they are built up from a finite number of primitive structural units. Unlike crystals, these structural units are not periodically repeated, so quasicrystals lack symmetry under any translation group. They both appear aperiodic if looked at individually.
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