## 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 1995). The result is a Fractal, and the above figures correspond to the curlicue fractals with 10,000 points for the Golden Ratio , , , , the Euler-Mascheroni Constant , , and 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.

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.