Julia (high-order) — Fantastic Fractals

High-order Julia set

This is the natural generalization of the classic Julia set to polynomials of higher degree. Each pixel coordinate is the seed $z_{0}$ of an iteration in the complex plane; the pixel is colored by how fast (or whether) the sequence escapes to infinity.

The iteration formula is a monic depressed polynomial of degree $n$ :

z_{k + 1} = z_{k}^{n} + C_{n - 2} z_{k}^{n - 2} + C_{n - 3} z_{k}^{n - 3} + \dots + C_{1} z_{k} + C_{0}

Two coefficients are fixed by convention rather than left to the user:

The leading coefficient is $1$ (monic). Multiplying the whole polynomial by a non-zero complex constant only rescales and rotates the dynamical plane — it does not produce a genuinely new fractal.
The coefficient of $z^{n - 1}$ is $0$ (depressed). For any polynomial of degree $n$ , the substitution $z \to w - a / n$ removes the $z^{n - 1}$ term without otherwise changing the dynamics; the visible difference in the picture is again a pure translation.

Together these two gauge choices remove the $(n - 1)$ -parameter freedom that does not affect the shape of the fractal, leaving the $(n - 1)$ "essential" complex coefficients $C_{0}, C_{1}, \dots, C_{n - 2}$ that are exposed as draggable control points.

When $n = 2$ the formula collapses to $z_{k + 1} = z_{k}^{2} + C_{0}$ , which is exactly the classic Julia set with the iconic constant $c = C_{0}$ .

Parameters

GUI Parameters

Iterations: The maximum number of iterations applied at each pixel before declaring the orbit bounded. Higher values reveal finer detail near the boundary at the cost of compute.
Degree ( $n$ ): The degree of the iteration polynomial, $2 \leq n \leq 8$ . Changing $n$ adds or removes control points: there are $n - 1$ of them.

Complex Parameters (Control Points)

$C_{0}, C_{1}, \dots, C_{n - 2}$ : The polynomial's lower coefficients, in order from the constant term ( $C_{0}$ ) up through the coefficient of $z^{n - 2}$ . Drag any of them on the complex plane to explore the family. At the default $n = 2$ only $C_{0}$ is exposed, and it plays the role of the classic Julia parameter $c$ .