Curlicue Fractal

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  to the horizontal. Then define  iteratively by

with . To the end of the previous line segment, draw a line segment of unit length which makes an angle

to the horizontal (Pickover 1995ab). The result is a fractal, and the above figures correspond to the curlicue fractals with  points for the golden ratio , , , , the Euler-Mascheroni constant , , and the Feigenbaum constant .

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

constant	temperature
golden ratio	46
	51
	58
	58
Euler-Mascheroni constant	63
	90
Feigenbaum constant	92

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  in the Gauss Plane." http://marvin.sn.schule.de/~inftreff/modul36/task36_e.htm.