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The more precise mathematical definition is that there is never translational symmetry in more than n – 1 linearly independent directions, where n is the dimension of the space filled, e.g., the three-dimensional tiling displayed in a quasicrystal may have translational symmetry in two directions.
A type of quasiparticle that occurs only in two-dimensional systems, with properties much less restricted than fermions and bosons. Biexciton: A bound state of two free excitons: Bion: A bound state of solitons, named for Born–Infeld model: soliton Cooper pair: A bound pair of two electrons electron Bipolaron: A bound pair of two polarons
The book is divided into two parts. The first part covers the history of crystallography, the use of X-ray diffraction to study crystal structures through the Bragg peaks formed on their diffraction patterns, and the discovery in the early 1980s of quasicrystals, materials that form Bragg peaks in patterns with five-way symmetry, impossible for a repeating crystal structure.
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Quasicrystal: A solid in which the positions of the atoms have long-range order, but this is not in a repeating pattern. Different structural phases of polymorphic materials are considered to be different states of matter in the Landau theory. For an example, see Ice § Phases. Liquid: A mostly non-compressible fluid. Able to conform to the ...
Different phasonic modes can change the material properties of a quasicrystal. [3] In the superspace representation, aperiodic crystals can be obtained from a periodic crystal of higher dimension by projection to a lower dimensional space– this is commonly referred to as the cut-and-project method.
In this view, a 3D quasicrystal with 8-fold rotation symmetry might be described as the projection of a slab cut from a 4D lattice. The following 4D rotation matrix is the aforementioned eightfold symmetry of the hypercube (and the cross-polytope):
In mathematics, a Meyer set or almost lattice is a relatively dense set X of points in the Euclidean plane or a higher-dimensional Euclidean space such that its Minkowski difference with itself is uniformly discrete.