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.

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.

© 1996-9

1999-05-25