Learn6 min read

How Fractal Colors Work

Every pixel's color is a mathematical answer: how long before this point escapes to infinity?

Open any fractal renderer and you'll see vivid rings of color radiating outward from a black center. The colors aren't artistic decoration — they're a direct visualization of a mathematical measurement: how fast does each point escape to infinity?

The Escape Time Algorithm

For each pixel, you pick the corresponding complex number $c$ and iterate the formula $z_{n+1} = z_n^2 + c$ starting from $z_0 = 0$ . You keep iterating until either:

$|z_n| > 2$ — the point has escaped. Record the iteration count $n$ .
You reach a maximum iteration count without escaping — the point is probably inside the Mandelbrot set. Color it black.

The escape iteration count $n$ for each pixel determines its color. Points that escape after 5 iterations get one color; points that escape after 50 iterations get another; points that escape after 500 get another still.

Each ring corresponds to a band of iteration counts before escape.

The Problem: Harsh Banding

The raw escape time produces integer values: 1, 2, 3, 4… The color transitions between bands are sharp steps, creating ugly concentric rings. Real fractal images use a smarter approach.

Smooth Coloring

The key insight is that even after escaping, you can extract more information about how far the orbit has traveled. The most common trick is to use the normalized iteration count:

smooth_n = n − log₂(log₂(|z|))

This formula uses the magnitude of $z$ at the escape step to interpolate between integer iteration counts. The result is a continuous real number, which produces a smooth gradient instead of a staircase.

Smooth coloring eliminates the harsh steps between iteration bands.

Palette Cycling

Once you have a smooth fractional iteration count, you map it onto a color palette. This is simply a 1D array of colors. The smooth count picks a position in the array; nearby positions get similar colors, producing the gradients you see.

You can cycle the palette — repeating the color array multiple times as the iteration count increases. This creates the characteristic concentric rings: each full cycle of the palette maps to one “band” of the fractal. Choosing a slower cycle rate gives smoother transitions; a faster cycle rate produces tighter, more vivid rings.

Other Coloring Methods

FractalSet supports several alternative coloring modes beyond the standard escape time:

Stripes / rings — adds the position of $z$ along a spiral to the coloring, creating sharper ring outlines.
Angle shading — colors based on the final angle of $z$ at escape, creating directional light effects.
Curvature shading — detects how much the orbit bends, giving embossed 3D-style relief.
Orbit trap — colors by how close the orbit comes to a geometric shape (a line, circle, or cross), creating entirely different aesthetics.

Try it in the viewer

Open any location in the FractalSet viewer and click the shader selector. Switch between “smooth”, “rings”, “neon”, and “angle” to see how the same underlying math produces dramatically different visuals.

Open the Viewer Browse palette examples What is the Mandelbrot Set?

The Escape Time Algorithm

For each pixel, you pick the corresponding complex number

c

and iterate the formula

z_{n+1} = z_n^2 + c

starting from

z_0 = 0

. You keep iterating until either:

|z_n| > 2

— the point has escaped. Record the iteration count

n

You reach a maximum iteration count without escaping — the point is probably inside the Mandelbrot set. Color it black.

The escape iteration count

n

for each pixel determines its color. Points that escape after 5 iterations get one color; points that escape after 50 iterations get another; points that escape after 500 get another still.

Each ring corresponds to a band of iteration counts before escape.

Smooth Coloring

The key insight is that even after escaping, you can extract more information about how far the orbit has traveled. The most common trick is to use the normalized iteration count:

smooth_n = n − log₂(log₂(|z|))

This formula uses the magnitude of

z

at the escape step to interpolate between integer iteration counts. The result is a continuous real number, which produces a smooth gradient instead of a staircase.

Smooth coloring eliminates the harsh steps between iteration bands.

Palette Cycling

Other Coloring Methods

FractalSet supports several alternative coloring modes beyond the standard escape time:

Stripes / rings — adds the position of

z

along a spiral to the coloring, creating sharper ring outlines.

Angle shading — colors based on the final angle of

z

at escape, creating directional light effects.

Curvature shading — detects how much the orbit bends, giving embossed 3D-style relief.

Orbit trap — colors by how close the orbit comes to a geometric shape (a line, circle, or cross), creating entirely different aesthetics.

Try it in the viewer