Mandelbrot and Julia - Mathematical visualization and beauty
I have been fascinated by the idea of chaos theory since I read about the butterfly effect. I read some more about it in the book ‘Chaos’ by James Gleick including about the Mandelbrot. But until I read ‘Make your own Mandelbrot’ by Tariq Rashid I did not realize just how simple the equation to draw the Mandelbrot was.
This is the innocuous looking equation that generates the awesome Mandelbrot Set.
A slight change in the input to the same equation will give the Julia set. It gives rise to a complicated boundary that reveals a recursive aspect; there are many smaller sized Mandelbrot one can see as one zoom’s in.
To learn more - look at this video - “What’s so special about the Mandelbrot Set?”
So I used the code from the book (python) to get the Mandelbrot set and the Julia set. I also got to see the 3D surface close ups (after smoothing).
MandelBrot 3D
Julia 3D
But the fun of the Mandelbrot is when one can zoom in and see the beautiful patterns interactively.
After a bit of search (and multiple tries), I was able to finalize on JavaScript code works reasonably without needing specific libraries or frameworks or threads. I also improved on the color palette, to get brightly colored areas to observe the beauty of the mathematical patterns.
Following is the core of the Mandelbrot code. This calculates the value of the function for a maximum number of iterations.
MandelBrot Code
Ref : https://rembound.com/articles/drawing-mandelbrot-fractals-with-html5-canvas-and-javascript
Julia Code
Ref : https://rosettacode.org/wiki/Julia_set
Color Palette Code
Ref : https://rosettacode.org/wiki/Julia_set
Results
Using mousedown to zoom in, ctrl to zoom out and shift to pan in the Mandelbrot and Julia results, some of the images / gifs obtained are shown below.
Links to the live demo are also mentioned at the end of the post.
Mandelbrot gif
Julia gif
Mandelbrot Image 1
Mandelbrot Image 2
Mandelbrot Image 3
Mandelbrot Image 4
Mandelbrot Image 5
Mandelbrot Image 6
Mandelbrot Image 7
Mandelbrot Image 8
Julia Image 1
Julia Image 2
Julia Image 3
Links
Python notebook with 3D images - Mandelbrot
Python notebook with 3D images - Julia
References
1. Make your own Mandelbrot - Tariq Rashid
If you don’t know programming or mathematics, and want to start from scratch to understand the concept behind Mandelbrot in clear, simple and step-by-step manner, this book is great. Short (only ~130 pages), with plenty of illustrations, easy to grasp. Its only con is that the code is in Python 2.7.