Sierpinski triangle

Parametric equations

The Sierpinski triangle comes from a triangle by removing the middle upside-down triangle again and again. It can also be generated by repeatedly moving to the midpoint toward a randomly chosen vertex (the chaos game). Its Hausdorff dimension is $\log 3/\log 2\approx 1.585$, and it is a typical self-similar fractal.

Geometric iteration: at each step, split every solid triangle into four similar smaller triangles and remove the middle one.

Chaos game: $P_{k+1}=\frac{P_k+V_i}{2}$, where $V_i$ is one of the three vertices (chosen at random).

Interaction

• Recursive / Chaos: switch between two ways of generating it, recursive geometry or a chaos-game point cloud
• Recursion depth n: set the number of recursion levels or the number of accumulated points, and watch the detail appear
• Compare: connect it to the intuition of affine iteration (see Iterated affine fractal)

What to notice

• In recursive mode, each level scales every solid triangle into three copies at $1/2$, and the middle triangle is removed.
• In chaos mode, the points keep moving to the midpoint toward a random vertex and end up in the same invariant set.
• Compare mode shows geometric removal and the IFS attractor side by side. Both point to the same limiting shape.

Related

• Iterated affine fractal
• Julia set
• Logistic map bifurcation diagram

Further reading

• Sierpiński triangle (Wikipedia)