Fractal Visualizations:

Iterated Function Systems

Iterated function systems are methods of generating fractals that are self-similar via transformation and scaling of each element through each iteration. Here I am showing nine popular examples of iterated function system fractals. Each fractal has a slider bar that controls the iteration level. The visualizations were made using D3.js/JavaScript. The iteration levels were calculated using recursion and good old-fashioned linear algebra.

1. Koch Curve

Begin with a single line segment. Each iteration transforms the middle third of each line segment into two equal-length lines that are rotated by 60°.


2. Koch Snowflake

Same rules as for the Koch Curve, but starting with a triangle. Produces a snowflake-like shape.


3. Cesàro Fractal

Similar to the Koch Curve, except the angle of rotation is greater than 60° and less than 90°. Here, the angle of rotation is 85°.


4. Lévy Dragon

Begin with a single line segment. Each iteration transforms each line segment into two lines with length scaled by 1/√2 and rotated by 45° in a symmetric fashion.


5. Heighway Dragon

Similar to the Lévy Dragon, but each line segment has asymmetrical rotation compared to the segment before it.


6. Sierpinski Gasket

Begin with a triangle. For each triangle in each iteration, subtract a triangle formed by connecting the midpoint of each line segment.


7. Sierpinski Carpet

Begin with a square. The first iteration subtracts a square in the middle third of the original square. Each further iteration divides the remaining area into squares and subtracts the middle third of each square.


8. Binary Tree

Begin with a vertical line segment. Each iteration produces symmetrical branching by angle θ and length scaled by r. Here, θ is 30° and r is 0.7.


9. Cantor Set

Begin with a line segment. For each iteration, create and translate two line segments that represent the first and last third of each existing line segment.