michaelerule
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Michael Rule (M, 26)
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    Quasicrystal Diffraction Patterns
    Simulated patterns for 5,6,7, &11 plane waves :


    This is a diffraction pattern of a real 3D quasicrystal. For a 3D crystal, the diffraction pattern is also three dimensional. All of the rendered images here are for 2D quasicrystals with 2D diffraction patterns.


    Here is the spatial domain of a quasicrystal formed by 8 plane waves, spaced at even rotations, in a plane. This coloring was achieved by squaring the sum of all 8 waves at each point, then normalizing the whole image, then inverting the colors.


    Here is a 5-fold 2D quasicrystal spatial domain. This pattern is related to Penrose tiling, I believe.


    Here is a 7-fold 2D quasicrystal spatial domain


    Edit : comments have indicated that systems of overlapping plane waves might not exactly be a quasicrystal, since they don't exactly have well defined lattice "points". So, I thresholded 5 summed plane waves to generate the following point distribution. The Fourier transform of this lattice looks more like what one might expect for a crystal.



    &FT:

    Thu, Apr 1, 2010  Permanent link

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    gamma     Thu, Apr 1, 2010  Permanent link
    There's a club here with the two-colored laser system that generates this kind of patterns. I am dying to find out how it works, but I suspect that it won't be diffraction...
    michaelerule     Thu, Apr 1, 2010  Permanent link
    That is basically the same thing, actually. These images are Fourier transforms of 2D quasicrystal images. Laser light passing through a diffraction grating is essentially equivalent to taking the Fourier transform of the diffraction grating. As one of my friends put it "The Fourier transform is just what happens when light goes through stuff".
    gamma     Fri, Apr 2, 2010  Permanent link
    You're killing me. I have a feeling that I should have known that, but I am not sure even what you're talking about. They have a very small device mounted on the ceiling with the red and green laser. The output draws tens of dots on the ground. Usually they are in lines (rows, columns) with the big dot in the middle, and the progressively smaller dots to the sides. They are exactly like the laser beam exiting the micro-scale grating! Then again, would they really use the grating?? The crystals cannot diffract light like this as far as I know, they use X-rays to obtain the exact images that you showed.

    I think that the Fourier analysis would show a graph of the frequencies of occurrences per length or angle of knots (or nodes) in the crystal.
    gamma     Fri, Apr 2, 2010  Permanent link
    The Indra's pearl fractal has some similarities with the diffraction pattern:
     http://ioannis.virtualcomposer2000.com/optics/indra.html 
    luke-tudor     Sat, Apr 3, 2010  Permanent link
    Could you contextualise this please? I just bumped into your images and would like to understand this in context.
    michaelerule     Sat, Apr 3, 2010  Permanent link
    @gamma : I don't fully understand the relationship between x-ray diffraction through a 3D lattice, and light diffraction through a 2D grating. The rendered images are formally equivalent to laser diffraction through a 2D grating, so might not be exactly the same thing as a 3D quasicrystal.

    @luke-tudor
    these are frequency spectrum plots of images composed of overlapping 2D plane waves. I got the idea from here. For N plane waves, I arranged the angle of the waves in the plane uniformly over [0,π). Normally, the Fourier transform of N evenly rotated plane waves would just look like 2*N dots in a circle. You can see for the case of 7 that there are 7 dots arranged in a circle. To get more interesting patterns in the frequency domain, I thresholded the images of overlapping plane waves. This introduces harmonics and interactions between the various waves that add in new frequencies and makes the Fourier transforms more interesting to look at. I used ImageJ to take the Fourier transforms, and Java to render the original ( spatial domain ) images. I will add a snapshot of the spatial domain image of a quasicrystal to the post for reference.



    gamma     Sat, Apr 3, 2010  Permanent link
    Those people at the Discover magazine are nuts.

    The quasi-crystal is a geometric structure that is for example, repeating two interchanging patterns in space or plane. Some materials are quasicrystals. If you make a needle tip from any crystal and point the x-ray beam, it will project directly onto the film and produce the dots on the images above. So, the space of the dots is physical (shadow projection space) and not the Fourier space.

    The laser passing through the grating splits into a few spherical wave-fronts, which mutually interfere. The shadows and lights on the film are simple sums of waves.

    Using the Fourier transform in some computer implementation to draw these pictures is another topic.
    michaelerule     Sat, Apr 3, 2010  Permanent link
    I see... yes, I'm still trying to understand how 3D x-ray diffraction actually works. I think the computer FT is similar in some respects, maybe ? I'd hoped it was. At any rate the pictures do seem similar.
    gamma     Sat, Apr 3, 2010  Permanent link
    If you point the beam at an interesting molecule, you obtain the shadowy diffraction pattern. You could reconstruct the 3D model of the molecule just by intuition if it is simple. In practice, they take large crystal samples, even in powder form, and send the beam. Then there are tens or hundreds of dots, lines on the film. If the structure is ordered, the shadows are simpler, repetitive. If you remove some of the repetitions, you can collapse it and reconstruct the unit cell of the crystal. The reconstruction of the 3D shape is an ambiguous computational work.
    gamma     Sat, Apr 3, 2010  Permanent link
    Here is the spatial domain of a quasicrystal formed by 8 plane waves, spaced at even rotations, in a plane. This coloring was achieved by squaring the sum of all 8 waves at each point, then normalizing the whole image, then inverting the colors.

    The description of the wave in plane sounds ambiguous. You practically need to define the algorithm for drawing the infinite crystal.

    The output from the diffraction could be simulated - dishonestly :-) Just make a simple fractal algorithm for putting the dots on the screen.... Here's some simulation:
    http://en.wikipedia.org/wiki/Coherent_diffraction_imaging
    michaelerule     Sat, Apr 3, 2010  Permanent link
    public static void main(String[] args) {
    int W = 640; //width of image
    int H = 540; //height of image
    int N=8; //number of waves
    double f = 0.5;
    double step = Math.PI/N;
    final BufferedImage im = new BufferedImage(W,H,BufferedImage.TYPE_INT_ARGB);
    double min = 0, max = 0;
    float[][] data = new float[W][H];
    for (int i=0;i<W;i++) for (int j=0;j<H;j++) {
    int ii = i-W/2;// + 7043;
    int jj = j-H/2;// + 2684;
    double theta = Math.atan2(jj,ii);
    double r = Math.sqrt(ii*ii+jj*jj);
    double val = 0;//sin(f*i);
    for (int k=0; k<N; k++)
    {
    double newtheta = theta + step*k;
    double x = cos(f*r*sin(newtheta));
    val += x;
    }
    data[i][j]=(float)val;
    min=Math.min(min,val);
    max=Math.max(max,val);
    }
    for (int i=0;i<W;i++) for (int j=0;j1) x=1;
    im.setRGB(i,j,0xff000000|Color.HSBtoRGB(0, 0, 1f-x));
    }
    JPanel disp = new JPanel() {
    public void paint(Graphics g) {g.drawImage(im,0,0,null);}};
    JFrame fdisp=new JFrame();
    fdisp.setDefaultCloseOperation(JFrame.EXIT_ON_CLOSE);
    fdisp.add(disp);
    disp.setSize(new Dimension(W,H));
    fdisp.pack();
    fdisp.setVisible(true);
    write(im,"quasiSpecial6-"+N+".png");
    }
    michaelerule     Sat, Apr 3, 2010  Permanent link
    public static void main(String[] args) {
    int W = 640; //width of image
    int H = 540; //height of image
    int N=8; //number of waves
    double f = 0.5;
    double step = Math.PI/N;
    float[][] data = new float[W][H];
    for (int i=0;i<W;i++) for (int j=0;j<H;j++) {
    int ii = i-W/2;
    int jj = j-H/2;
    double theta = Math.atan2(jj,ii);
    double r = Math.sqrt(ii*ii+jj*jj);
    double val = 0;
    for (int k=0; k<N; k++)
    val += cos(f*r*sin(theta + step*k));
    data[i][j]=(float)val;
    }
    final BufferedImage im = new BufferedImage(W,H,BufferedImage.TYPE_INT_ARGB);
    for (int i=0;i<W;i++) for (int j=0;j1) x=1;
    im.setRGB(i,j,0xff000000|Color.HSBtoRGB(0, 0, 1f-x));
    }
    try {
    File file = new File("quasiSpecial6-"+N+".png");
    ImageIO.write(im, "png", file);
    } catch (Exception ex) {
    System.out.println("write error");
    }
    }




    ... does space collective support code embedding and formatting ? this would seem like a useful addition.


    }
    shiftctrlesc     Sat, Apr 3, 2010  Permanent link
    The 5-fold image is stunning.
    Every time you latch onto one pattern it quickly tumbles you into another.
    gamma     Sun, Apr 4, 2010  Permanent link
    It seems that you defined the matter with the unchanged continuous waves, and the ability of such object to pattern the entire plane. Then, you can use the "wave of matter" as the known structure that diffracts some ray into the perfect result from the theoretical standpoint. It sounds like a student exercise in the condensed matter physics, but I think I haven't seen any quite like this one...
    michaelerule     Sun, Apr 4, 2010  Permanent link
    hmm, I guess to get... an actual quasicrystal you'd have to define some way of getting a set of points, not waves.
    gamma     Mon, Apr 5, 2010  Permanent link
    Yes, I did not think that the whole ordeal behind these images was so complicated. Usually I expect to see something similar to any of the Sierpinski sets in principle. You make the "crystal" and go straight and add some more shadows and waves around it and we say "oh wow its like diffraction patterns".

    Reconstructing the molecular structure from the diff. patterns is a big problem. Its not deterministic, its about the supercomputing, catalogs of structures, experts... In concept, you possess the neatly defined structure and then you make the diff. pattern using some ... wrong formula I am sure. Not really. :-)
    michaelerule     Mon, Apr 5, 2010  Permanent link
    I'm fairly certain that for the 2D crystals the FT is equivalent to the diffraction pattern, and that these systems are equivalent to laser light passing through a diffraction grating. The 3D case is where the confusion starts.
    gamma     Tue, Apr 6, 2010  Permanent link
    The laser and the X-ray diffraction are the same. The particle diffraction too. The difference starts when the beam damages the matter for example, or when the gamma ray excites the nucleus...

    When you say FT, it sounds as though you input a signal and obtain the infinite sum of the component cosine waves, with phases and amplitudes ("powers"). Its confusing to me, because I forgot about the waves of matter (e.g. the periodic potentials). There is some formula for the reconstruction of the grating from the diff. patterns.

    If I recall... the laser at wavelength lambda interferes with the grating of size d. At the distance x to the projection screen, there are several dots at the same distance D to each other. All these variables are in one formula... The FT of the dots on the screen as a horizontal waveform is a single frequency equivalent to the distance D per angle on the horizontal plane. The single "frequency" is a dot in 2D space, or a vertical bar that represents the actual element of the grating. (poof)

    The 3D case - I can't help you.
    gamma     Sat, May 29, 2010  Permanent link
    The following book contains the application of diffraction patterns to estimate the fractal structure of dust.

    Brian H. Kaye
    A Random Walk Through
    Fractal Dimensions
     
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