# MMM LXXI

## Image Details

This is a nebulabrot rendering of the function $z_{n+1} = z_0\left(\cos(z_n)+\cosh(z_n)\right).$The usual cosine function and the hyperbolic cosine function are closely related—they deform the complex plane in a similar manner, but at right angles to each other. Hence there was reason to expect that they might interact nicely to create a symmetric set when used to generate a Mandelbrot-style set.

This particular image contains four layers, each of which was adjusted using the auto levels functionality in GIMP. The bottom layer contains 160 million exposures, tracking orbits that escape after between 50 and 200 iterations. This layer was inverted (light pixels were made dark and vice versa), then colored a dark purple using GIMPs “Colorify” command. The next layer contains another 160 million exposures, this time tracking orbits that escape after between 200 and 800 iterations. It was colored a deep vermilion or scarlet color, then added to the bottom layer. Layer three tracked exposures that escape after between 800 and 3200 iterations and contains 40 million exposures. It was colored the same vermilion color as the next layer down and added. Finally, the top layer consists of 10 million exposures tracking orbits that escape after between 3200 and 12800 iterations. It was darkened very slightly, then added to the layer below. After the layers were composited, a couple of filters were used to smooth the image a little.

The color scheme was chosen quite intentionally for some friends who are getting married in two weeks. Oddly enough, I took my first statistics class from the bride more than a decade ago, though I didn’t get to know her socially at the time. Several years later, when my wife started working at her current job—in the same office as the bride—I discovered that the two of them had known each other for years and were good friends. Small world, no?

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