Recently I've dived into studying complex analysis on my own and it led me to find an interesting way of displaying the Mandelbrot set:

Consider $$f(f(f(f(...x))))$$ Then, $$ \frac{d}{dx} (f(f(f(f(...x))))) = f'(f(f(f(...x))))\cdot f'(f(f(...x)))...$$ Assuming that this function converges for $x$, $$\frac{d}{dx}(f(f(...x))) = \prod_{p=1}^{\infty} f'(f^{p}(x))$$ Since all non-edge-points in a Julia set always diverge or converge to the same cycle, this gave me an idea. Even though this product is always equal to $0$ or $\infty$, I wondered if it was possible to still compare the derivatives by how quickly they converge to $0$. I thought about the best way to do this for a while and came up with the following expression to color the points in the Mandelbrot set: $$ 1-4\sqrt[p]{ \prod_{i=1}^{p} d(v_{i}) }$$ where $p$ is the length of the cycle the point converges to, $v_i$ is the $i$th point of the cycle, and $d(x)$ is a function which returns the squared distance of $x$ from the origin. The radical normalizes the value of the product based on the length of the cycle, in effect determining the "average" value of $f'(f^p(x))$. Finding the cycle and its values can be done efficiently using pre-existing algorithms. When I made a program to implement this algorithm, I got the following really cool image:

The Mandelbrot set image I rendered

If this has been done before, what is this method of displaying the set called?


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.