SpaceCollective: Quasicrystal Diffraction Patterns
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Sat, 29 May 2010 06:09:51 +0000http://spacecollective.org/michaelerule/5810/Quasicrystal-Diffraction-PatternsenNew comment on: Quasicrystal Diffraction Patterns
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http://spacecollective.org/michaelerule/5810/Quasicrystal-Diffraction-PatternsSat, 29 May 2010 06:09:51 +0000michaelerule5810<a href="http://spacecollective.org/gamma">gamma</a> wrote:<br /><br />The following book contains the application of diffraction patterns to estimate the fractal structure of dust. <br />
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Brian H. Kaye<br />
A Random Walk Through<br />
Fractal Dimensionshttp://spacecollective.org/michaelerule/5810/Quasicrystal-Diffraction-PatternsNew comment on: Quasicrystal Diffraction Patterns
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http://spacecollective.org/michaelerule/5810/Quasicrystal-Diffraction-PatternsTue, 06 Apr 2010 13:06:14 +0000michaelerule5810<a href="http://spacecollective.org/gamma">gamma</a> wrote:<br /><br />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... <br />
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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. <br />
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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) <br />
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The 3D case - I can't help you. http://spacecollective.org/michaelerule/5810/Quasicrystal-Diffraction-PatternsNew comment on: Quasicrystal Diffraction Patterns
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http://spacecollective.org/michaelerule/5810/Quasicrystal-Diffraction-PatternsMon, 05 Apr 2010 16:14:10 +0000michaelerule5810<a href="http://spacecollective.org/michaelerule">michaelerule</a> wrote:<br /><br />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.http://spacecollective.org/michaelerule/5810/Quasicrystal-Diffraction-PatternsNew comment on: Quasicrystal Diffraction Patterns
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http://spacecollective.org/michaelerule/5810/Quasicrystal-Diffraction-PatternsMon, 05 Apr 2010 13:59:55 +0000michaelerule5810<a href="http://spacecollective.org/gamma">gamma</a> wrote:<br /><br />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". <br />
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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. :-)http://spacecollective.org/michaelerule/5810/Quasicrystal-Diffraction-PatternsNew comment on: Quasicrystal Diffraction Patterns
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http://spacecollective.org/michaelerule/5810/Quasicrystal-Diffraction-PatternsSun, 04 Apr 2010 15:53:21 +0000michaelerule5810<a href="http://spacecollective.org/michaelerule">michaelerule</a> wrote:<br /><br />hmm, I guess to get... an actual quasicrystal you'd have to define some way of getting a set of points, not waves. http://spacecollective.org/michaelerule/5810/Quasicrystal-Diffraction-PatternsNew comment on: Quasicrystal Diffraction Patterns
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http://spacecollective.org/michaelerule/5810/Quasicrystal-Diffraction-PatternsSun, 04 Apr 2010 15:35:13 +0000michaelerule5810<a href="http://spacecollective.org/gamma">gamma</a> wrote:<br /><br />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... http://spacecollective.org/michaelerule/5810/Quasicrystal-Diffraction-PatternsNew comment on: Quasicrystal Diffraction Patterns
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http://spacecollective.org/michaelerule/5810/Quasicrystal-Diffraction-PatternsSat, 03 Apr 2010 19:58:22 +0000michaelerule5810<a href="http://spacecollective.org/shiftctrlesc">shiftctrlesc</a> wrote:<br /><br />The 5-fold image is stunning.<br />
Every time you latch onto one pattern it quickly tumbles you into another.http://spacecollective.org/michaelerule/5810/Quasicrystal-Diffraction-PatternsNew comment on: Quasicrystal Diffraction Patterns
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http://spacecollective.org/michaelerule/5810/Quasicrystal-Diffraction-PatternsSat, 03 Apr 2010 16:12:45 +0000michaelerule5810<a href="http://spacecollective.org/michaelerule">michaelerule</a> wrote:<br /><br /> public static void main(String[] args) {<br />
int W = 640; //width of image<br />
int H = 540; //height of image<br />
int N=8; //number of waves<br />
double f = 0.5;<br />
double step = Math.PI/N;<br />
float[][] data = new float[W][H];<br />
for (int i=0;i<W;i++) for (int j=0;j<H;j++) {<br />
int ii = i-W/2;<br />
int jj = j-H/2;<br />
double theta = Math.atan2(jj,ii);<br />
double r = Math.sqrt(ii*ii+jj*jj);<br />
double val = 0;<br />
for (int k=0; k<N; k++)<br />
val += cos(f*r*sin(theta + step*k));<br />
data[i][j]=(float)val;<br />
}<br />
final BufferedImage im = new BufferedImage(W,H,BufferedImage.TYPE_INT_ARGB);<br />
for (int i=0;i<W;i++) for (int j=0;j1) x=1;<br />
im.setRGB(i,j,0xff000000|Color.HSBtoRGB(0, 0, 1f-x));<br />
}<br />
try {<br />
File file = new File("quasiSpecial6-"+N+".png");<br />
ImageIO.write(im, "png", file);<br />
} catch (Exception ex) {<br />
System.out.println("write error");<br />
}<br />
}<br />
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... does space collective support code embedding and formatting ? this would seem like a useful addition.<br />
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}http://spacecollective.org/michaelerule/5810/Quasicrystal-Diffraction-PatternsNew comment on: Quasicrystal Diffraction Patterns
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http://spacecollective.org/michaelerule/5810/Quasicrystal-Diffraction-PatternsSat, 03 Apr 2010 16:06:20 +0000michaelerule5810<a href="http://spacecollective.org/michaelerule">michaelerule</a> wrote:<br /><br />public static void main(String[] args) {<br />
int W = 640; //width of image<br />
int H = 540; //height of image<br />
int N=8; //number of waves<br />
double f = 0.5;<br />
double step = Math.PI/N;<br />
final BufferedImage im = new BufferedImage(W,H,BufferedImage.TYPE_INT_ARGB);<br />
double min = 0, max = 0;<br />
float[][] data = new float[W][H];<br />
for (int i=0;i<W;i++) for (int j=0;j<H;j++) {<br />
int ii = i-W/2;// + 7043;<br />
int jj = j-H/2;// + 2684;<br />
double theta = Math.atan2(jj,ii);<br />
double r = Math.sqrt(ii*ii+jj*jj);<br />
double val = 0;//sin(f*i);<br />
for (int k=0; k<N; k++)<br />
{<br />
double newtheta = theta + step*k;<br />
double x = cos(f*r*sin(newtheta));<br />
val += x;<br />
}<br />
data[i][j]=(float)val;<br />
min=Math.min(min,val);<br />
max=Math.max(max,val);<br />
}<br />
for (int i=0;i<W;i++) for (int j=0;j1) x=1;<br />
im.setRGB(i,j,0xff000000|Color.HSBtoRGB(0, 0, 1f-x));<br />
}<br />
JPanel disp = new JPanel() {<br />
public void paint(Graphics g) {g.drawImage(im,0,0,null);}};<br />
JFrame fdisp=new JFrame();<br />
fdisp.setDefaultCloseOperation(JFrame.EXIT_ON_CLOSE);<br />
fdisp.add(disp);<br />
disp.setSize(new Dimension(W,H));<br />
fdisp.pack();<br />
fdisp.setVisible(true);<br />
write(im,"quasiSpecial6-"+N+".png");<br />
}http://spacecollective.org/michaelerule/5810/Quasicrystal-Diffraction-PatternsNew comment on: Quasicrystal Diffraction Patterns
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http://spacecollective.org/michaelerule/5810/Quasicrystal-Diffraction-PatternsSat, 03 Apr 2010 14:16:13 +0000michaelerule5810<a href="http://spacecollective.org/gamma">gamma</a> wrote:<br /><br /><cap>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.</cap><br />
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The description of the wave in plane sounds ambiguous. You practically need to define the algorithm for drawing the infinite crystal.<br />
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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:<br />
<a href="http://en.wikipedia.org/wiki/Coherent_diffraction_imaging" target="blank">http://en.wikipedia.org/wiki/Coherent_diffraction_imaging</a><br>
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http://spacecollective.org/michaelerule/5810/Quasicrystal-Diffraction-PatternsSat, 03 Apr 2010 14:05:56 +0000michaelerule5810<a href="http://spacecollective.org/gamma">gamma</a> wrote:<br /><br />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. http://spacecollective.org/michaelerule/5810/Quasicrystal-Diffraction-PatternsNew comment on: Quasicrystal Diffraction Patterns
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http://spacecollective.org/michaelerule/5810/Quasicrystal-Diffraction-PatternsSat, 03 Apr 2010 12:41:33 +0000michaelerule5810<a href="http://spacecollective.org/michaelerule">michaelerule</a> wrote:<br /><br />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.http://spacecollective.org/michaelerule/5810/Quasicrystal-Diffraction-PatternsNew comment on: Quasicrystal Diffraction Patterns
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http://spacecollective.org/michaelerule/5810/Quasicrystal-Diffraction-PatternsSat, 03 Apr 2010 10:12:15 +0000michaelerule5810<a href="http://spacecollective.org/gamma">gamma</a> wrote:<br /><br />Those people at the Discover magazine are nuts.<br />
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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.<br />
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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.<br />
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Using the Fourier transform in some computer implementation to draw these pictures is another topic.http://spacecollective.org/michaelerule/5810/Quasicrystal-Diffraction-PatternsNew comment on: Quasicrystal Diffraction Patterns
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http://spacecollective.org/michaelerule/5810/Quasicrystal-Diffraction-PatternsSat, 03 Apr 2010 07:39:41 +0000michaelerule5810<a href="http://spacecollective.org/michaelerule">michaelerule</a> wrote:<br /><br />@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. <br />
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@luke-tudor<br />
these are frequency spectrum plots of images composed of overlapping 2D plane waves. I got the idea from <a href="http://discovermagazine.com/2009/feb/14-what-is-this-psychedelic-place-mat" target="_blank">here</a>. 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 <a href="http://rsbweb.nih.gov/ij/" target="_blank">ImageJ</a> 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.<br />
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http://spacecollective.org/michaelerule/5810/Quasicrystal-Diffraction-PatternsSat, 03 Apr 2010 06:58:10 +0000michaelerule5810<a href="http://spacecollective.org/luketudor">luke-tudor</a> wrote:<br /><br />Could you contextualise this please? I just bumped into your images and would like to understand this in context.http://spacecollective.org/michaelerule/5810/Quasicrystal-Diffraction-PatternsNew comment on: Quasicrystal Diffraction Patterns
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http://spacecollective.org/michaelerule/5810/Quasicrystal-Diffraction-PatternsFri, 02 Apr 2010 13:24:29 +0000michaelerule5810<a href="http://spacecollective.org/gamma">gamma</a> wrote:<br /><br />The Indra's pearl fractal has some similarities with the diffraction pattern:<br />
<a href="http://ioannis.virtualcomposer2000.com/optics/indra.html" target="blank">http://ioannis.virtualcomposer2000.com/optics/indra.html</a> http://spacecollective.org/michaelerule/5810/Quasicrystal-Diffraction-PatternsNew comment on: Quasicrystal Diffraction Patterns
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http://spacecollective.org/michaelerule/5810/Quasicrystal-Diffraction-PatternsFri, 02 Apr 2010 13:16:08 +0000michaelerule5810<a href="http://spacecollective.org/gamma">gamma</a> wrote:<br /><br />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. <br />
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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. http://spacecollective.org/michaelerule/5810/Quasicrystal-Diffraction-PatternsNew comment on: Quasicrystal Diffraction Patterns
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http://spacecollective.org/michaelerule/5810/Quasicrystal-Diffraction-PatternsThu, 01 Apr 2010 15:50:49 +0000michaelerule5810<a href="http://spacecollective.org/michaelerule">michaelerule</a> wrote:<br /><br />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".http://spacecollective.org/michaelerule/5810/Quasicrystal-Diffraction-PatternsNew comment on: Quasicrystal Diffraction Patterns
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http://spacecollective.org/michaelerule/5810/Quasicrystal-Diffraction-PatternsThu, 01 Apr 2010 15:41:28 +0000michaelerule5810<a href="http://spacecollective.org/gamma">gamma</a> wrote:<br /><br />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...http://spacecollective.org/michaelerule/5810/Quasicrystal-Diffraction-Patterns