Chaos Game Fractals

Order from Randomness: The Beauty of Probabilistic Fractals

Sierpinski
Custom Polygon
Restricted
IFS

The Sierpinski Triangle

The classic chaos game: start at a random point, pick a random vertex from 3 corners, jump halfway there, plot the point, and repeat. Despite pure randomness in vertex selection, a perfect fractal triangle emerges!

  • Algorithm: 3 vertices, jump ratio = 0.5
  • Magic: No restrictions needed - the pattern just appears
  • Fractal Dimension: log(3)/log(2) = 1.585
  • Self-Similar: Each triangle contains 3 smaller copies of itself
0
Points Rendered
3
Vertices
0.50
Jump Ratio
1.585
Fractal Dim

Custom Polygon Chaos Game

Experiment with different numbers of vertices and jump ratios. Not all combinations create fractals - some create filled shapes, others create surprising patterns!

  • Square (4): With r=0.5, creates a solid square (boring!)
  • Pentagon (5): r=0.618 (golden ratio) creates beautiful patterns
  • Hexagon (6): r=0.5 creates nested hexagons
  • Higher N: Approach circular symmetry
0
Points Rendered
4
Vertices
0.50
Jump Ratio
-
Pattern Type

Restricted Chaos Game

Add rules like "can't pick the same vertex twice" or "can't pick adjacent vertices" to create wildly different patterns from the same polygon!

  • Square + No Repeat: Creates a Sierpinski carpet-like pattern
  • Pentagon + No Adjacent: Beautiful 5-fold symmetry
  • Hexagon + Two Back: Cannot pick vertex two positions back
  • Custom Rules: The possibilities are endless
0
Points Rendered
4
Vertices
0.50
Jump Ratio
No Repeat
Restriction

Iterated Function Systems

Instead of random vertices, apply one of several affine transformations chosen by probability. This creates famous fractals like the Barnsley Fern!

  • Barnsley Fern: 4 transformations create a realistic fern
  • Tree: Branching structure from simple rules
  • Spiral: Logarithmic spiral from rotation + scaling
  • Dragon Curve: Space-filling fractal dragon
0
Points Rendered
4
Transforms
Barnsley Fern
Fractal Type
High
Complexity