[HN Gopher] Experiments with plane-filling curves and Fourier tr...
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       Experiments with plane-filling curves and Fourier transform
        
       Author : pyinstallwoes
       Score  : 89 points
       Date   : 2023-07-08 10:50 UTC (12 hours ago)
        
 (HTM) web link (github.com)
 (TXT) w3m dump (github.com)
        
       | munro wrote:
       | wait so.... what??? haven't groked the code, but seems very
       | interesting and may have cryptographic implications
       | 
       | shot at what i think may be happening: using fft to estimate
       | these various tilings
        
       | downvotetruth wrote:
       | Fork of https://news.ycombinator.com/item?id=35590429 159 points
       | 82 days ago 19 comments
        
       | infogulch wrote:
       | Wow this is a cool idea. I wonder if it would also be interesting
       | to analyze the aperiodic tilings, including monotiles, that have
       | popped up recently.
        
         | [deleted]
        
         | enriquto wrote:
         | 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 wrote:
           | 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)?
        
         | QuadmasterXLII wrote:
         | I bashed together the linked fft visualization code and my hat
         | tiling code, the result looks pretty cool!
         | 
         | https://colab.research.google.com/drive/1cUTNiln5fMupkrlPwGK...
         | 
         | (explanation of the tiling code at
         | https://www.hgreer.com/HatTile/ )
        
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