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Curlicue Fractal

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CurlicueFractal

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=0 to the horizontal. Then define theta_n iteratively by

theta_(n+1)=(theta_n+2pis) (mod 2pi),

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 (mod 2pi),

to the horizontal (Pickover 1995ab). The result is a fractal, and the above figures correspond to the curlicue fractals with 10000 points for the golden ratio phi, ln2, e, sqrt(2), the Euler-Mascheroni constant gamma, pi, and the Feigenbaum constant delta.

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

constanttemperature
golden ratio phi46
ln251
e58
sqrt(2)58
Euler-Mascheroni constant gamma63
pi90
Feigenbaum constant delta92

See also

Temperature

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References

Berry, M. and Goldberg, J. "Renormalization of Curlicues." Nonlinearity 1, 1-26, 1988.Mendès-France, M. "Entropie, dimension et thermodynamique des courbes planes." In Seminar on number theory, Paris 1981-82 (Paris, 1981/1982) (Ed. M.-J. Bertin). Boston, MA: Birkhäuser, pp. 153-177, 1983.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. Mazes for the Mind: Computers and the Unexpected. New York: St. Martin's Press, 1993.Pickover, C. A. "Is the Fractal Golden Curlicue Cold?" Visual Comput. 11, 309-312, 1995a.Pickover, C. A. "The Fractal Golden Curlicue is Cool." Ch. 21 in Keys to Infinity. New York: W. H. Freeman, pp. 163-167, 1995b.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.Stoschek, E. "Module 35: Curlicue Variations: Polygon Patterns in the Gauss Plane of Complex Numbers." http://marvin.sn.schule.de/~inftreff/modul35/task35_e.htm.Stoschek, E. "Module 36: The Feigenbaum-Constant delta in the Gauss Plane." http://marvin.sn.schule.de/~inftreff/modul36/task36_e.htm.

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Curlicue Fractal

Cite this as:

Weisstein, Eric W. "Curlicue Fractal." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/CurlicueFractal.html

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