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The humble sine waves that lie at the very foundation of trigonometry have a special beauty all their own. It takes just a little coddling to bring the beauty out. But who would guess, for example, that psychedelic fractal patterns lurk within the cosine operation applied to real numbers?

Consider the union of the infinite set of curves produced by Carotid-Kundalini functions defined by:

y = cos(n*x*acos(x))

where (-1 < x < 1, n = 1, 2, 3, ...), and "acos" designates the arccosine function. The set of superimposed curves is very simple to plot -- most computer hobbyists could easily program and plot them on a personal computer -- but the curves have an extremely complicated and beautiful structure. For example, for (x < 0 ) there appears to be an exotic fractal structure with gaps repeated at different size scales and with progressively increasing spacing as x becomes smaller.




You can compute the union of the first 25 curves using the following logic (a more complete program listing is provided at the end):

for (n=1; n < =25; n=n+1) {
    for (x= -1; x < = 1;  x=x+.01) {
        y = cos((float)n*x*acos(x));
        if (x == -1) MovePenTo(x,y);
            else  DrawTo(x,y);
    }
}

If you have the ability to display these curves, make a plot from -1 < x < 1 and -1 < y < 1. You could spend a lifetime exploring the infinite intricacies of the resulting superimposed patterns.

C program

/* Compute Carotid-Kundalini Curves */
#include <math.h>
#include <stdio.h>

main()
{
float x,y;
int n;

/* Superimpose 25 curves */
for (n=1; n < =25; n=n+1) {
    for (x = -1; x < = 1;  x = x+.01) {
        y=cos( (float)n * x * acos(x));
        /* Write out x,y points for plotting */
        printf("%f %f\n",x,y);
    }
}
}

BASIC program

10 REM Compute Carotid-Kundalini Curves
20 REM Superimpose 25 curves
30    FOR N=1 TO 25
40        FOR X = -1 TO 1 STEP 0.01
50              Y=COS(N * X * ACOS(X))
60              REM Write out x,y points for plotting
70              PRINT X, Y
80        NEXT X
90    NEXT N
100 END

Explore the Carotid-Kundalini Universe Forever

1. MathWorld
2. Bourke
3. Carotid-Kundalini Fractal Explorer
4. Chaotic behaviour in the Carotid–Kundalini map function
5. Wolfram Demonstrations Project
6. New Julia sets for complex Carotid–Kundalini function
7. Carotid-Kundalini function
8. Relative Superior Julia Sets for Complex Carotid-Kundalini Function