Fractals

The Mandelbrot and Julia sets come from one map, iterated:

$$z \mapsto z^2 + c.$$

Two ways to read it. Fix the starting point at $z = 0$ and let $c$ range over the plane — that is the Mandelbrot set, a picture of the parameter. Or fix $c$ and let the starting point range — that is a Julia set, a picture of the dynamics at that one parameter. Same map, two different planes.

The n-th iterate, not an escape count

Most renderings colour a pixel by how long it took to escape. These colour the iterate itself, $z_n(c)$, through the same domain colouring the rest of the site uses — hue is the argument, brightness the magnitude.

That is not a trick to reuse machinery. The escape count is a summary of the orbit; $z_n$ is the orbit's position, and it carries the argument as well as the size. The exterior blows up and reads as poles; the interior stays bounded and reads as colour. Turning $n$ up sharpens the boundary, because the boundary is exactly where the answer depends on how long you look.

#The Mandelbrot set, as the n-th iterate

Three settings cells and one drawable — the plot falls out of the last cell alone.

source · editable

<notatio-worksheet seed='["$\\mathsf{center}$ := [-0.5, 0]", "$\\mathsf{extent}$ := 3", "n := 64", "Mandelbrot(z, n)"]' screen="side" />

The first two cells are framing, in the settings namespace — the set lives around $-0.5$, not the origin. Drag the plot to pan and ⌘/Ctrl + scroll to zoom; sweep $n$ to watch the boundary resolve. At low $n$ the set looks fat and smooth — few points have had time to escape. Each doubling etches the filaments a little further out.

A Julia set is one parameter's dynamics

#A Julia set, one parameter at a time

c is complex, so it gets one slider per part.

source · editable

<notatio-worksheet seed='["c := -0.4 + 0.6i", "n := 64", "Julia(z, c, n)"]' screen="side" />

Sweep the two parts of $c$ and the Julia set changes shape continuously — and the Mandelbrot set is the map of which shapes are possible. Inside it, the Julia set is connected; outside, it shatters into dust. That is Mandelbrot's theorem, and here it is something you can drag a slider through: take $c$ out past the boundary and watch the picture come apart.

Where the zoom runs out

The GPU works in f32, so there is a depth past which zooming stops revealing anything and starts revealing the arithmetic: the picture goes blocky, because neighbouring pixels have stopped being distinguishable coordinates. A fractal is the most direct way to see that limit, since detail is what it has at every scale.

This is also what makes the pair a good test for framing that adapts to zoom — the useful iteration count is not fixed but a function of how far in you are, since the deeper you go the longer an orbit needs to declare itself.