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-- Blaise Pascal

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

Even though to my knowledge these curves have not been well characterized, there are several things of which we can be certain. For example, they are bounded by y = +- 1 . Also, since acos(1) = 0 and cos(0) = 1, the infinite number of C-K curves contain the point (1,1). This means that all curves must meet at the upper right hand of the figure. It's almost as if some geometrical god has come down and placed a pin at (1,1) to tie the majestic, unruly curves together.

The curves intersect the line y = 1 whenever any of the following conditions are met: n = 0 , x = 0 , or acos(x) = 0. This accounts for the tip of the bell-shaped curves in Gaussian Land. The bells are centered at the origin at x = 0. Zero crossings satisfy pi/2 = n*acos(x).

More information on the Carotid-Kundalini Universe appears in Keys to Infinity.

Cliff would like to hear from those of you who explore these curves in greater detail or make higher resolution plots.

Mathematical explorers are tourists in the Carotid-Kundalini Universe. Some favorite Carotid-Kundalini web pages are listed here:

- MathWorld
- Bourke
- Carotid-Kundalini Fractal Explorer
- Chaotic behaviour in the Carotid–Kundalini map function
- Wolfram Demonstrations Project
- New Julia sets for complex Carotid–Kundalini function
- Carotid-Kundalini function
- Relative Superior Julia Sets for Complex Carotid-Kundalini Function

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