Mandelbrot Art

Zachary Russ

The Project

I wanted to create a program that would allow me to create beautiful images using data from the Mandelbrot set, a fascinating piece of mathematics.

Although I started out using my own complex number class, I discovered that doing the complex number math explicitly with two doubles runs a bit faster.

Once I had things working, I began experimenting with different drawing, coloring, and reconstruction techniques.

Drawing Techniques

Early on while testing my program, it became clear that I needed to significantly speed up draw time to keep my program interactive while zoomed in close to the fractal's border.

Instead of doing the complex number calculations for every pixel in the image (top), I could instead do every other pixel. This would cut the number of calculations while leaving gaps in the image.

These gaps could then be filled in two different ways:

I will talk more about filling in the gaps later.

With the ability to skip \(n\) pixels, I can pixelate the image into \(n + 2\) by \(n + 2\) chunks (middle). This reduces the number of calculations by a factor of up to \((n+1)^2\). I justify this with the following scary calculus maths, where \(x^2\) is the total number of pixels and \({\lceil \frac{x}{n+1} \rceil}^2\) is the total number of pixels when skipping every \(n\) pixels: \[\lim_{x \to \infty} \frac {x^2}{{\lceil {x \over {n+1}} \rceil}^2} = \lim_{x \to \infty} \frac {x^2}{{ ({x \over {n+1}} + \delta) }^2} = \lim_{x \to \infty} \frac {x^2}{{ ({{x+\delta(n+1)} \over {n+1}}) }^2}\] \[= \lim_{x \to \infty} \frac {x^2(n+1)^2}{{({x+\delta(n+1)}) }^2} = (n+1)^2 \cdot \lim_{x \to \infty} \frac {x^2}{{({x+\delta(n+1)}) }^2} \] \[= (n+1)^2 \cdot \lim_{x \to \infty} \frac {x^2}{{({x \cdot (1+\frac{\delta(n+1)}{x}})) }^2} \] \[= (n+1)^2 \cdot \lim_{x \to \infty} \frac {1}{{ (1+\frac{\delta(n+1)}{x}) }^2} = (n+1)^2 \cdot \frac{1}{1^2}\] \[= (n+1)^2\]

For example, by skipping every other pixel, I reduce the number of calculations by a factor of \((1+1)^2 = 2^2 = 4\) as the image size approaches infinity.

In an attempt to speed things up without sacrificing quality, I took my new ability to pixelate the image into \(n + 2\) by \(n + 2\) chunks and added functionality to recursively subdivide chunks near regions of high complexity (bottom). This ended up making things slower, but it looks cool, especially when color is used.

Coloring Techniques

My simplest coloring technique is monochromatic. For each pixel, or chunk of pixels, I calculate a complexity value. This value is the number of iterations it took for the point to escape the set divided by the maximum number of iterations. Thus, this complexity value ranges from 0 to 1. For points that do not escape after the maximum number of iterations, the complexity is defined to be 1. The complexity value does not increase linearly as you approach the boundary of the set, so it is useful for aesthetic reasons to use the square root of the complexity value. I multiply this value by 255 and type cast it to an unsigned char (top).

For my second coloring technique, I take the previously described square root of the complexity value and rescale it to range from 0 to 360. I pretend that this new value is the hue and value of an HSV color with a saturation of 1. Finally, I convert this constructed HSV color to an RGB color (middle).

My third coloring technique, the one shown in the nine videos at the start, is a subtle variation of my second coloring technique. After I construct the HSV color and convert it to an RGB color (middle), I then pretend that the RGB color is an LAB color and convert it to an RGB color (bottom).

This third coloring technique produces both smooth gradients between some colors, and sharp bands between others. The reason it produces sharp bands is caused by negative values produced in the final LAB to RGB being type casted to unsigned chars (I think).

This could be seen as a bug, but because I found the problem and chose to keep it for aesthetic reasons, I consider it a surprise feature.

Reconstruction Techniques

If I skip 0 pixels, no reconstruction is applied to the image. There are no gaps to fill, so reconstruction is not necessary (top).

If I skip \(n \gt 0\) pixels, there will be gaps in the image that need to be filled (second).

I can fill these gaps two different ways:

With nearest neighbor, I fill in an \((n + 2)\) by \((n + 2)\) chunk with the color of the top left pixel in the chunk. This fills in the gaps between pixels with a uniform color, giving the image a blocky appearance (third).

Bilinear interpolation is interesting. The way I implement it is fairly straightforward. Given the following:

I can calculate an interpolated color between the known surrounding four pixel colors this way, where \(c_{f}\) is the interpolated color: \[c_{f} = v \cdot (c_{bl} \cdot (1 - u) + c_{br} \cdot u)\] \[+ (1 - v) \cdot (c_{tl} \cdot (1 - u) + c_{tr} \cdot u)\]

This color \(c_{f}\) is used to color the pixel \(\vec{p}\) (bottom).

When the image is reconstructed using square chunks of different sizes, my images using bilinear interpolation look odd. This is because color values are changing at different rates for each chunk size.

I will showcase an image like this in The Gallery section.

Building the Program

All of these images come from a single program I wrote in C++ called mandelbrot. There are several make targets that build the program in specific ways:

Using the Program

Depending on how the program was built, it can be used in different ways:

Known Problems

Some of these are genuine problems. Others are problems that I have chosen to keep because they are entertaining.

The Future

I am unhappy with how slow my program is. It runs on a single thread, on a single core. This was something I knew, and something I did experiment with, but I was not yet able to get multi-threading functional.

In the beginning, I planned to implement cubic interpolation, as well, but the project was already getting really big, especially when I added dynamic pixel skipping.

Dynamic pixel skipping was a huge time sink. My first idea was to allow non-square chunks, creating a kind of voronoi pattern. This ended up being an absolute nightmare, and I scrapped the idea of non-square chunks in a couple of days. I finally got square chunks to subdivide properly, but it took a lot longer than I thought it would. This was the most difficult part of the project for me.

The Gallery