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They are often constructed that way. Aperiodic lattices in the plane arise from projection of the periodic grid in R^5 into an inclined plane (with irrational

They are often constructed that way. Aperiodic lattices in the plane arise from projection of the periodic grid in R^5 into an inclined plane (with irrational coefficients). Now in R^5 you have the famous Poisson summation formula, that says that the Fourier transform of a dirac comb on the grid is a dirac comb. By projecting this Poisson formula in the spatial and frequential domains, you obtain an aperiodic Poisson formula in dimension 2. Thus the Fourier transform of an aperiodic lattice is another aperiodic lattice.

westurner 3y ago
Reciprocal lattice: https://en.wikipedia.org/wiki/Reciprocal_lattice https://en.wikipedia.org/wiki/Reciprocal_lattice : > In physics, the reciprocal lattice represents the Fourier transform of another lattice. The direct lattice or real lattice is a periodic function in physical space, such as a crystal system (usually a Bravais lattice). The reciprocal lattice exists in the mathematical space of spatial frequencies, known as reciprocal space or k space, where `k` refers to the wavevector. > [...] The reciprocal lattice plays a fundamental role in most analytic studies of periodic structures, particularly in the theory of diffraction. In neutron, helium and X-ray diffraction, due to the Laue conditions, the momentum difference between incoming and diffracted X-rays of a crystal is a reciprocal lattice vector. The diffraction pattern of a crystal can be used to determine the reciprocal vectors of the lattice. Using this process, one can infer the atomic arrangement of a crystal. Was wondering about biomedical NIRS applications and crystallography molecular identification and just found this. Cool. How are aperiodic and reciprocal lattices different from any molecule (or subatomic particles in a field of superfluidic vortices in a fluid with viscosity<=1)? Is this potentially part of how to make a medical tricorder (and win the Tricorder XPRIZE)?

westurner 3y ago

Reciprocal lattice: https://en.wikipedia.org/wiki/Reciprocal_lattice https://en.wikipedia.org/wiki/Reciprocal_lattice : > In physics, the reciprocal lattice represents the Fourier transform of another lattice. The direct lattice or real lattice is a periodic function in physical space, such as a crystal system (usually a Bravais lattice). The reciprocal lattice exists in the mathematical space of spatial frequencies, known as reciprocal space or k space, where `k` refers to the wavevector. > [...] The reciprocal lattice plays a fundamental role in most analytic studies of periodic structures, particularly in the theory of diffraction. In neutron, helium and X-ray diffraction, due to the Laue conditions, the momentum difference between incoming and diffracted X-rays of a crystal is a reciprocal lattice vector. The diffraction pattern of a crystal can be used to determine the reciprocal vectors of the lattice. Using this process, one can infer the atomic arrangement of a crystal. Was wondering about biomedical NIRS applications and crystallography molecular identification and just found this. Cool. How are aperiodic and reciprocal lattices different from any molecule (or subatomic particles in a field of superfluidic vortices in a fluid with viscosity<=1)? Is this potentially part of how to make a medical tricorder (and win the Tricorder XPRIZE)?