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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 $s$ 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

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

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

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

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 $\phi$, $\ln 2$, $e$, $\sqrt{2}$, 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
$e$ 58
$\sqrt{2}$ 58
Euler-Mascheroni Constant $\gamma$ 63
$\pi$ 90
Feigenbaum Constant $\alpha$ 92


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.

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© 1996-9 Eric W. Weisstein