Dragon Curve
Nonintersecting curves which can be iterated to yield more and more sinuosity. They can be constructed by taking a patharound a set of dots, representing a left turn by 1 and a right turn by 0. The first-order curve is then denoted 1. Forhigher order curves, add a 1 to the end, then copy the string of digits preceding it to the end but switching its centerdigit. For example, the second-order curve is generated as follows: (1)1 
(1)1(0) 110, and the third as: (110)1 (110)1(100) 1101100. Continuing gives 110110011100100... (Sloane's A014577). The Octal representationsequence is 1, 6, 154, 66344, ...(Sloane's A003460). The dragon curves of orders 1 to 9 are illustrated below.
This procedure is equivalent to drawing a Right Angle and subsequently replacing each Right Angle withanother smaller Right Angle (Gardner 1978). In fact, the dragon curve can be written as a Lindenmayer Systemwith initial string "FX", String Rewriting rules "X" -> "X+YF+", "Y" -> "-FX-Y", and angle 90°.
See also Lindenmayer System, Peano Curve
References
Dickau, R. M. ``Two-Dimensional L-Systems.''http://forum.swarthmore.edu/advanced/robertd/lsys2d.html. Dixon, R. Mathographics. New York: Dover, pp. 180-181, 1991. Dubrovsky, V. ``Nesting Puzzles, Part I: Moving Oriental Towers.'' Quantum 6, 53-57 (Jan.) and 49-51 (Feb.), 1996. Dubrovsky, V. ``Nesting Puzzles, Part II: Chinese Rings Produce a Chinese Monster.'' Quantum 6, 61-65 (Mar.) and 58-59 (Apr.), 1996. Gardner, M. Mathematical Magic Show: More Puzzles, Games, Diversions, Illusions and Other Mathematical Sleight-of-Mind from Scientific American. New York: Vintage, pp. 207-209 and 215-220, 1978. Lauwerier, H. Fractals: Endlessly Repeated Geometric Figures. Princeton, NJ: Princeton University Press, pp. 48-53, 1991. Peitgen, H.-O. and Saupe, D. (Eds.). The Science of Fractal Images. New York: Springer-Verlag, p. 284, 1988. Sloane, N. J. A. SequencesA014577 andA003460/M4300in ``An On-Line Version of the Encyclopedia of Integer Sequences.''http://www.research.att.com/~njas/sequences/eisonline.html and Sloane, N. J. A. and Plouffe, S.The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995. Vasilyev, N. and Gutenmacher, V. ``Dragon Curves.'' Quantum 6, 5-10, 1995.
- Unsolved Problem
- Unstable Improper Node
- Unstable Node
- Unstable Spiral Point
- Unstable Star
- Untouchable Number
- U-Number
- Upper Bound
- Upper Integral
- Upper Limit
- Upper Sum
- Upper-Trimmed Subsequence
- Upward Drawing
- Urchin
- Utility Graph
- Utility Problem
- Valence
- Valency
- Valuation
- Valuation Group
- Valuation Ring
- Valuation Theory
- Value
- Value (Game)
- Vampire Number