Curlicue Fractal

$\begin{figure}\begin{center}\BoxedEPSF{CurlicuePhi.epsf scaled 400}\quad\BoxedEP... ...f scaled 400} \quad\BoxedEPSF{CurlicueE.epsf scaled 400}\end{center}\end{figure}$

$\begin{figure}\begin{center}\BoxedEPSF{CurlicueSqrt2.epsf scaled 400}\quad\Boxed... ... 400}\quad\BoxedEPSF{CurlicueFeigenbaum.epsf scaled 400}\end{center}\end{figure}$

The curlicue fractal is a figure obtained by the following procedure. Let be an Irrational Number. Begin with a line segment of unit length, which makes an Angle $\phi_0\equiv 0$ to the horizontal. Then define $\theta_n$ iteratively by

$\begin{displaymath} \theta_{n+1}=(\theta_n+2\pi s){\rm\ (mod\ 2\pi)}, \end{displaymath}$

with $\theta_0=0$ . To the end of the previous line segment, draw a line segment of unit length which makes an angle

$\begin{displaymath} \phi_{n+1}=\theta_n+\phi_n {\rm\ (mod\ 2\pi)}, \end{displaymath}$

to the horizontal (Pickover 1995). The result is a Fractal, and the above figures correspond to the curlicue fractals with 10,000 points for the Golden Ratio $\phi$ , $\ln 2$ ,

, $\sqrt{2}$ , the Euler-Mascheroni Constant $\gamma$ , $\pi$ , and Feigenbaum Constant $\delta$ .

The Temperature of these curves is given in the following table.

Constant	Temperature
Golden Ratio $\phi$	46
$\ln 2$	51
	58
$\sqrt{2}$	58
Euler-Mascheroni Constant $\gamma$	63
$\pi$	90
Feigenbaum Constant $\alpha$	92

References

Berry, M. and Goldberg, J. ``Renormalization of Curlicues.'' Nonlinearity 1, 1-26, 1988.

Moore, R. and van der Poorten, A. ``On the Thermodynamics of Curves and Other Curlicues.'' McQuarie Univ. Math. Rep. 89-0031, April 1989.

Pickover, C. A. ``The Fractal Golden Curlicue is Cool.'' Ch. 21 in Keys to Infinity. New York: W. H. Freeman, pp. 163-167, 1995.

Pickover, C. A. Mazes for the Mind: Computers and the Unexpected. New York: St. Martin's Press, 1993.

Sedgewick, R. Algorithms in C, 3rd ed. Reading, MA: Addison-Wesley, 1998.

Stewart, I. Another Fine Math You've Got Me Into.... New York: W. H. Freeman, 1992.