Editing Dragon curve (section)

== Occurrences of the dragon curve in solution sets ==
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. In other words, new solutions are obtained by applying a function to the set of existing solutions. This is similar to how an iterated function system produces new points in a set, though not all IFS are linear functions.
In a conceptually similar vein, a set of [[Littlewood polynomial]]s can be arrived at by such iterated applications of a set of functions.

A Littlewood polynomial is a polynomial: <math> p(x) = \sum_{i=0}^n a_i x^i \, </math> where all <math>a_i = \pm 1</math>.

For some <math>|w| < 1</math> we define the following functions:
:<math> f_+(z) = 1 + wz</math>
:<math> f_-(z) = 1 - wz</math>

Starting at z=0 we can generate all Littlewood polynomials of degree d using these functions iteratively d+1 times.<ref name='ncafe'>{{Cite web|url=http://golem.ph.utexas.edu/category/2009/12/this_weeks_finds_in_mathematic_46.html|title = The n-Category Café}}</ref> For instance:
:<math>f_+(f_-(f_-(0))) = 1 + (1-w)w = 1 + 1w - 1w^2</math>

It can be seen that for <math>w=\tfrac{1+i}{2}</math>, 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 <math>w=\tfrac{1+i}{2}</math>.
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.<ref name='ncafe' /><ref>{{Cite web|url=http://math.ucr.edu/home/baez/week285.html|title=Week285}}</ref><ref>{{Cite web|url=http://johncarlosbaez.wordpress.com/2011/12/11/the-beauty-of-roots/|title = The Beauty of Roots|date = 11 December 2011}}</ref>