Mandelbrot (high-order) — Fantastic Fractals

High-order Mandelbrot set

This is a Mandelbrot-style family for monic polynomials of arbitrary degree $n$ . As in the classic Mandelbrot set, each pixel's complex coordinate $c$ enters the iteration formula as a parameter (not the starting point: the iteration always begins at $z_{0} = 0$ ). The pixel is colored by how fast the orbit escapes to infinity.

The iteration polynomial is monic of degree $n$ , with every lower coefficient $P_{0}, P_{1}, \dots, P_{n - 1}$ free in principle:

z_{i + 1} = z_{i}^{n} + P_{n - 1} z_{i}^{n - 1} + P_{n - 2} z_{i}^{n - 2} + \dots + P_{1} z_{i} + P_{0}

What makes this a Mandelbrot family rather than a Julia family is the role of the pixel coordinate $c$ : exactly one of the lower coefficients is replaced by $c$ , so the picture sweeps over a slice of the polynomial's parameter space. The slot that holds $c$ is chosen by the index $k$ (between $0$ and $n - 1$ ); the remaining $n - 1$ coefficients are draggable control points.

This generalizes the classic case in two independent directions:

Degree $n$ . With all other coefficients zero, $z_{i + 1} = z_{i}^{n} + c$ is the multibrot of order $n$ . At $n = 2, k = 0$ this is the classic Mandelbrot.
Position of $c$ . Letting $c$ live at $P_{k}$ rather than $P_{0}$ rotates the role $c$ plays in the polynomial. With $k = n - 1$ , for example, $c$ multiplies $z^{n - 1}$ ; the parameter slice is then through a different cross-section of the family.

Unlike the high-order Julia shader, no coefficient is fixed to zero here — the "depressed" gauge would remove the $P_{n - 1}$ slot, which is exactly one of the slots a user may want $c$ to occupy.

Parameters

GUI Parameters

Iterations: Maximum number of iterations applied at each pixel before declaring the orbit bounded.
Degree ( $n$ ): Degree of the iteration polynomial, $2 \leq n \leq 8$ . Changing $n$ adjusts the number of control points to $n - 1$ .
$c$ at $P_{k}$ ( $k$ ): Which polynomial slot the pixel coordinate $c$ occupies. Range is $0 \leq k \leq n - 1$ ; the slider's visible range tracks the current $n$ . The default $k = 0$ at $n = 2$ is the classic Mandelbrot configuration.

Complex Parameters (Control Points)

$P_{j}$ for $j \neq = k$ : The $n - 1$ remaining coefficients, in ascending order of $j$ . Drag any of them on the complex plane to explore the family. At the default $n = 2, k = 0$ only $P_{1}$ is exposed; with $P_{1} = 0$ , $z_{i + 1} = z_{i}^{2} + c$ is exactly the Mandelbrot set.

Initial values: newly-exposed slots default to $0$ , except $P_{0}$ , which defaults to $0.5$ whenever it is a control point (i.e. when $k \neq = 0$ ). The iteration starts from $z_{0} = 0$ , and $p (0) = P_{0}$ , so a zero $P_{0}$ would trap the orbit at $0$ and render the entire view as the bounded "in-set" color. The nonzero $P_{0}$ default ensures the picture starts out interesting after a $k$ change.