## Mandelbrot set sketch in Scratch

Despite my personal disbelieve in and dislike of the colored blocks dragging simulator 3, I nevertheless wanted to extract functionality other than the hardcoded cat mascot path tracing from the aforementioned software; one of the most efficient visual result to build effort ratio yields a simple plot of the Mandelbrot set, formally known as

$M:=\{z\in\mathbb{C}:|\lim_{n\to\infty}\mathrm{itr}^n(z)|<\infty\}$

where the iterator is defined as

$\mathrm{itr}^n(z) := \mathrm{itr}^{n-1}(z)^2+z, \\ \mathrm{itr}^0(z) := 0.$

The render resolution is kept at a recognizable minimum as not to overburden the machine tasked with creating it. Source: mandelbrot-set.sb3

## Christmas MMXVII

Barren trees, white snow.
Cold and lasting winter nights.
Quiet fire crackling.

## Mandelbrot Set III

I wrote my first ever Mandelbrot Set renderer back in 2015 and used Python to slowly create fractal images. Over a year later, I revisited the project with a Java version which — due to its code being actually compiled — ran much faster, yet had the same clunky interface; a rectangle the user had to draw and a key they had to press to the view change to the selected region.
In this post, over half a year later, I present my newest Mandelbrot Set fractal renderer (download the .jar), written in Java, which both runs fast and allows a much more intuitive and immersive walk through the complex plane by utilizing mouse dragging and scrolling.
The still time demanding task of rendering fractals — even in compiled languages — is split up into a low quality preview rendering, a normal quality display rendering and a high quality 4K (UHD-1 atÂ 3840×2160 pixels to keep a 16:9 image ratio) rendering, all running in seperate threads.

The color schemes where also updated, apart from the usual black-and-white look there are multiple rainbow color schemes which rely on the HSB color space, zebra color schemes which use the iterations taken modulo some constant to define the color and a prime color scheme which tests if the number of iterations taken is prime.

Apart from the mouse and keyboard control, there is also a menu bar (implemented using Java’s JMenuBar) which allows for more conventional user input through a proper GUI.

#### Controls

• Left mouse dragging: pan view
• Left mouse double click: set cursor’s complex number to image center
• Mouse scrolling: zoom view
• Mouse scrolling +CTLR: pan view
• ‘p’: render high definition fractal
• ‘r’: reset view to default
• ‘w’, ‘s’: zoom frame
• Arrow keys: pan view
• Arrow keys +CTRL: zoom view
• Fractal: extra info about current fractal rendering
• Color Scheme: change color scheme and maximum iteration depth
• HD: controls for high definition rendering
The high definition renderings are saved as .png files and named with their four defining constants. Zim and Zre define the image’s complex center, Zom defines the complex length above the image’s center. Clr defines the number of maximum iterations.
A 4K fractal at 3840×2160 pixels with an iteration depth of 256 would in the worst case scenario (no complex numbers actually escape) require $3840 \cdot 2160 \cdot 256 \cdot 4 = 8493465600$ double multiplications. If you had a super-optimized CPU which could do one double multiplication a clock tick (which current CPUs definitely cannot) and ran at 4.00 GHz, it would still take that massively overpowered machine $\frac{8493465600}{4 \cdot 10^9} = 2.123$ seconds. Larger images and higher maximum iterations would only increase the generated overhead.
// Java Code; Jonathan Frech; 22nd, 23rd, 24th, 25th, 26th, 27th of July 2017