Carotid-Kundalini Fractal

A fractal-like structure is produced for $x<0$ by superposing plots of Carotid-Kundalini Functions $CK_n$ of different orders $n$. The region $-1<x<0$ is called Fractal Land by Pickover (1995), the central region the Gaussian Mountain Range, and the region $0<x<1$ Oscillation Land. The plot above shows $n=1$ to 25. Gaps in Fractal Land occur whenever

$$x\cos^{-1} x=2\pi{p\over q}$$

for $p$ and $q$ Relatively Prime Integers. At such points $x$, the functions assume the $\left\lceil{(q+1)/2}\right\rceil$ values $\cos(2\pi r/q)$ for $r=0$, 1, ..., $\left\lfloor{q/2}\right\rfloor$, where $\left\lceil{z}\right\rceil$ is the Ceiling Function and $\left\lfloor{z}\right\rfloor$ is the Floor Function.

References

Pickover, C. A. ``Are Infinite Carotid-Kundalini Functions Fractal?'' Ch. 24 in Keys to Infinity. New York: Wiley, pp. 179-181, 1995.