Visual analogue of a Shepard tone

A Shepard tone is an auditory illusion that appears to indefinitely ascend or descend in pitch, without actually changing pitch at all.

 

Shepard tones work because they actually contain multiple tones, separated by octaves. As tones get higher in pitch, they fade out. New tones fade in at the lower pitches. The net effect is that it sounds like all the constituent tones are continually increasing in pitch -- and they are, but pitches fade in and out so that, on average, the pitch composition is constant.

Since 2D quasicrystals can be rendered as a sum of plane-waves, it is possible to form the analogue of a Shepard tone with these visual objects. Each plane wave is replaced with a collection of plane waves, at 2,4,8,16... etc times the spatial frequency of the original plane wave.

The relative amplitudes of the plane waves are set so that the spatial frequency stays approximately the same even as the underlying waves are scaled. The result is a quasicrystal that appears to zoom in or out indefinitely, without fundamentally changing in structure.

The infinite zoom effects creates a motion-fatigue optical illusion, which will cause illusory contraction of your visual field after staring at the GIF below:

 

 

 

More quasicrystal zoom GIFs can be found here. You can run and modify the code I used to generate these animation. Copy the following code into a file called QuasiZoom.java. Then, in a terminal, type "javac QuasiZoom.java" in the same directory, and then "java QuasiZoom". Various parameters to tune the output are noted in comments in the code. Then use Gimp to make an animated GIF.

import java.awt.Color;
import java.awt.image.BufferedImage;
import java.io.File;
import java.io.IOException;
import javax.imageio.ImageIO;
import static java.lang.Math.*;

public class QuasiZoom {

    // Defines a gaussian function. We will use this to define the
    // envelope of spatial frequencies
    public static double gaussian(double x) {
        return exp(-x*x/2)/sqrt(2*PI);
    }

    public static void main(String[] args) throws IOException {
        int k = 5;        //number of plane waves
        int stripes = 3;  //number of stripes per wave
        int N = 500;      //image size in pixels
        int divisions=40; //number of frames to divide the animation into
        int N2 = N/2;

        BufferedImage it = new BufferedImage(N, N, BufferedImage.TYPE_INT_RGB);

        //the range of different spatial frequencies
        int [] M=new int[]{1,2,4,8,16,32,64,128,256};
        
    //the main ( central ) spatial frequency
        double mean=log(16);

    //the spread of the spatial frequency envelope
        double sigma=1;

    //counts the frames 
        int ss=0;

    //iterate over spatial scales, scaling geometrically
        for (double sc=2.0; sc>1.0; sc/=pow(2,1./divisions)) 
        {    
            System.out.println("frame = "+ss);

            //adjust the  wavelengths for the current spatial scale
            double [] m=new double[M.length];
            for (int l=0; l<M.length; l++)
                m[l]=M[l]*sc;

            //modulate each wavelength by a gaussian envelop in log
            //frequency, centered around aforementioned mean with defined
            //standard deviation
            double sum=0;
            double [] W=new double[M.length];
            for (int l=0; l<M.length; l++) {
                W[l]=gaussian((log(m[l])-mean)/sigma);
                sum+=W[l];
            }
            sum*=k;

            for (int i = 0; i < N; i++) {
                for (int j = 0; j < N; j++) {

                    double x = j - N2, y = i - N2; //cartesian coordinates
                    double C = 0;                  // accumulator
 
                    // iterate over all k plane waves
                    for (double t = 0; t < PI; t += PI / k){
                        //compute the phase of the plane wave
                        double ph=(x*cos(t)+y*sin(t))*2*PI*stripes/N;
                        //take a weighted sum over the different spatial scales
                        for (int l=0; l<M.length; l++)
                            C += (cos(ph*m[l]))*W[l];
                    }
                    // convert the summed waves to a [0,1] interval
                    // and then convert to [0,255] greyscale color
                    C = min(1,max(0,(C*0.5+0.5)/sum));
                    int c = (int) (C * 255);
                    it.setRGB(i, j, c | (c << 8) | (c << 16));
                }
            }
            ImageIO.write(it, "png", new File("out"+(ss++)+".png"));
        }
        
    }
}