Today I released the interactive demonstration of iterated function system know as Chaos game. You can read more about Chaos game on the Wikipedia, but long story short, we fix a plane polygon $\Pi = \{V_1, \dots V_m\}$ and choose an arbitrary plane point $P_0$. Then we iteratively generate orbit $\mathcal O = \{P_0, P_1, P_2, …\}$ by using formula $$ \vec P_{i+1} = \frac12\left(\vec P_i+\vec V\right),$$ where $V$ is at each iteration chosen randomly from $\{V_1, \dots V_m\}$.
Choosing equilateral triangle for $\Pi$ we get a Sierpiński triangle
My program treats vertices as complex numbers, and uses formula $$ \vec P_{i+1} = \lambda\left(\vec P_i+\vec V\right),$$ where $\lambda$ is arbitrary (but fixed) complex parameter. This enables user to create many interesting orbits which are not possible in the “original” description of the system. As far as I know, for this reason this program is unique on the web.
Choosing equilateral triangle for $\Pi$ and $\lambda=-1$, we get an 'inverted Sierpiński triangle'
Interesting examples
- $m=2$: Flag, Julia-like set
- $m=3$: Sierpiński, Inverted Sierpiński, Tiles
- $m=4$: Something?, Full square
- $m=8$: Sierpiński carpet
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