- #1
arkajad
- 1,481
- 4
I have six 2x2 complex matrices from the group SL(2,C). Each line is a matrix: first row, second row.
\left(<br /> \begin{array}{cc}<br /> \left\{\frac{1}{\sqrt{1-k^2}},\frac{k}{\sqrt{1-k^2}}\right\} & \left\{\frac{k}{\sqrt{1-k^2}},\frac{1}{\sqrt{1-k^2}}\right\} \\<br /> \left\{\frac{1}{\sqrt{1-k^2}},\frac{i k}{\sqrt{1-k^2}}\right\} & \left\{-\frac{i k}{\sqrt{1-k^2}},\frac{1}{\sqrt{1-k^2}}\right\} \\<br /> \left\{\frac{k+1}{\sqrt{1-k^2}},0\right\} & \left\{0,\frac{1-k}{\sqrt{1-k^2}}\right\} \\<br /> \left\{\frac{1-k}{\sqrt{1-k^2}},0\right\} & \left\{0,\frac{k+1}{\sqrt{1-k^2}}\right\} \\<br /> \left\{\frac{1}{\sqrt{1-k^2}},-\frac{i k}{\sqrt{1-k^2}}\right\} & \left\{\frac{i k}{\sqrt{1-k^2}},\frac{1}{\sqrt{1-k^2}}\right\} \\<br /> \left\{\frac{1}{\sqrt{1-k^2}},-\frac{k}{\sqrt{1-k^2}}\right\} & \left\{-\frac{k}{\sqrt{1-k^2}},\frac{1}{\sqrt{1-k^2}}\right\} \\<br /> \end{array}<br /> \right)
Let us take k=0.7 and act with these matrices on the complex plane via linear fractional transformations (Mobius transformations):
z\mapsto\frac{az+b}{cz+d}.
Selecting an arbitrary starting point and applying consecutively randomly selecting matrices several millions of times ("Chaos Game") we create a pattern. Here is the pattern in the square -8<x,y<8
The pattern seems to be self-similar. Here is the central square -1<x,y<1
Question: Is it possible, looking at the transformations to find a formula for the centers and the radii of all these circles? What kind of a formula would it be? A closed algebraic expression? An algorithm?
P.S. You can create this pattern yourself using Quantum Fractal Generator. Just choose Alpha=0.7, Chaos Game, Fractal Type: PlatonicOcta, Display Type Stereographic, XY LIMIT 8.0, Fractal Algorithm: Chaos Game, Color Scheme: Grayscale, 10 mln iterations.
P.S2: Your computer will crash if you try to implement the transformations as described in my post. The trick is to do it on the Riemann sphere rather than on the complex plane.
\left(<br /> \begin{array}{cc}<br /> \left\{\frac{1}{\sqrt{1-k^2}},\frac{k}{\sqrt{1-k^2}}\right\} & \left\{\frac{k}{\sqrt{1-k^2}},\frac{1}{\sqrt{1-k^2}}\right\} \\<br /> \left\{\frac{1}{\sqrt{1-k^2}},\frac{i k}{\sqrt{1-k^2}}\right\} & \left\{-\frac{i k}{\sqrt{1-k^2}},\frac{1}{\sqrt{1-k^2}}\right\} \\<br /> \left\{\frac{k+1}{\sqrt{1-k^2}},0\right\} & \left\{0,\frac{1-k}{\sqrt{1-k^2}}\right\} \\<br /> \left\{\frac{1-k}{\sqrt{1-k^2}},0\right\} & \left\{0,\frac{k+1}{\sqrt{1-k^2}}\right\} \\<br /> \left\{\frac{1}{\sqrt{1-k^2}},-\frac{i k}{\sqrt{1-k^2}}\right\} & \left\{\frac{i k}{\sqrt{1-k^2}},\frac{1}{\sqrt{1-k^2}}\right\} \\<br /> \left\{\frac{1}{\sqrt{1-k^2}},-\frac{k}{\sqrt{1-k^2}}\right\} & \left\{-\frac{k}{\sqrt{1-k^2}},\frac{1}{\sqrt{1-k^2}}\right\} \\<br /> \end{array}<br /> \right)
Let us take k=0.7 and act with these matrices on the complex plane via linear fractional transformations (Mobius transformations):
z\mapsto\frac{az+b}{cz+d}.
Selecting an arbitrary starting point and applying consecutively randomly selecting matrices several millions of times ("Chaos Game") we create a pattern. Here is the pattern in the square -8<x,y<8
The pattern seems to be self-similar. Here is the central square -1<x,y<1
Question: Is it possible, looking at the transformations to find a formula for the centers and the radii of all these circles? What kind of a formula would it be? A closed algebraic expression? An algorithm?
P.S. You can create this pattern yourself using Quantum Fractal Generator. Just choose Alpha=0.7, Chaos Game, Fractal Type: PlatonicOcta, Display Type Stereographic, XY LIMIT 8.0, Fractal Algorithm: Chaos Game, Color Scheme: Grayscale, 10 mln iterations.
P.S2: Your computer will crash if you try to implement the transformations as described in my post. The trick is to do it on the Riemann sphere rather than on the complex plane.
Last edited: