Dragon curve

A dragon curve is any member of a family of self-similar fractal curves, which can be approximated by recursive methods such as Lindenmayer systems.

It was described by Martin Gardner in his Scientific American column Mathematical Games in 1967.

Many of its properties were first published by Chandler Davis and Donald Knuth.

It appeared on the section title pages of the Michael Crichton novel Jurassic Park.

[1] The Heighway dragon can be constructed from a base line segment by repeatedly replacing each segment by two segments with a right angle and with a rotation of 45° alternatively to the right and to the left:[2] The Heighway dragon is also the limit set of the following iterated function system in the complex plane: with the initial set of points

Using pairs of real numbers instead, this is the same as the two functions consisting of The Heighway dragon curve can be constructed by folding a strip of paper, which is how it was originally discovered.

If the strip was opened out now, unbending each fold to become a 90-degree turn, the turn sequence would be RRL, i.e. the second iteration of the Heighway dragon.

Fold the strip in half again to the right, and the turn sequence of the unfolded strip is now RRLRRLL – the third iteration of the Heighway dragon.

Continuing folding the strip in half to the right to create further iterations of the Heighway dragon (in practice, the strip becomes too thick to fold sharply after four or five iterations).

The folding patterns of this sequence of paper strips, as sequences of right (R) and left (L) folds, are: Each iteration can be found by copying the previous iteration, then an R, then a second copy of the previous iteration in reverse order with the L and R letters swapped.

It can be also written as a Lindenmayer system – it only needs adding another section in the initial string: It is also the locus of points in the complex plane with the same integer part when written in base

[5] The terdragon can be written as a Lindenmayer system: It is the limit set of the following iterated function system: The Lévy C curve is sometimes known as the Lévy dragon.

[6] Having obtained the set of solutions to a linear differential equation, any linear combination of the solutions will, because of the superposition principle, also obey the original equation.

, the above pair of functions is equivalent to the IFS formulation of the Heighway dragon.

That is, the Heighway dragon, iterated to a certain iteration, describe the set of all Littlewood polynomials up to a certain degree, evaluated at the point

Indeed, when plotting a sufficiently high number of roots of the Littlewood polynomials, structures similar to the dragon curve appear at points close to these coordinates.