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
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 Eric W. Weisstein