Occurrences of The Dragon Curve in Solution Sets
Having obtained the set of solutions to a 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 produce new points in a set, though not all IFS are linear functions. In a conceptually similar vein, a set of Littlewood polynomials can be arrived at by such iterated applications of a set of functions.
A Littlewood polynomial is a polynomial : where all .
For some |w| < 1 we define the following functions:
Starting at z=0 we can generate all Littlewood polynomials of degree d using these functions iteratively d+1 times. For instance:
It can be seen that for w = (1+i)/2, 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 w = (1+i)/2. 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.
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