Iterated affine fractal

Parametric equations

A single point $P(x,y)$ is sent through one of four affine maps, chosen at random with set probabilities. After many iterations the points gather into a fern-shaped attractor:

b (leaf bend) controls the horizontal offset of the sub-branches. d (side branch height) controls how far the stem reaches up and how large the side leaves are.

Interaction

• Leaf bend b: change the shear of the affine map and the curve of the leaves
• Side branch height d: move the branches and change the overall outline
• Generation speed ω: build up the point cloud faster and watch the outline of the fractal appear

What to notice

• Iterating several affine maps at random pulls the point cloud toward a self-similar structure.
• Small parameter changes move the outline continuously between a tree and a fern.
• The boundary comes from the density of many orbit points, not from a single curve.

Related

• Affine transformation pattern
• Sierpinski triangle

Further reading

• Iterated function system (Wikipedia)