## Variations of the Mandelbrot Set

Here’s some variations of the Mandelbrot Set I’ve been looking at. They aren’t terribly interesting from a mathematical perspective, but their pretty!

### Mandelbrot Set with 2*x

The escape time fractal of the function:

$x\to \lim _{n\to\infty} (z\to z^2 +2*x)^n(x)$

My interpretation of this is that it does to the Mandelbrot variations described by functions of the form $x\to \lim _{n\to\infty} (z\to z^2 + x + k)^n(x)$ what the Mandelbrot set does to Julia sets: for every point x, look at the xth function!

It has the same sort of local variation as the Mandelbrot set:

And has some rather visually stunning regions:

But it is also very different from the Mandelbrot Set. For example, it’s version of Seahorse Valley:

And it would appear to not be connected and in fact have an embedding of Cantor Set grouped in an odd way $4^{\aleph_0}$:

### x=Mandelbrot Mandelbrot Set

One interesting thing to do is to replace the added value with the value of another fractal forming function. In this case, we can replace the it with the Mandelbrot itself:

$x\to \lim _{n\to\infty} (z\to z^2 +M(x))^n(x)$

$M(x) = \lim _{n\to\infty} (z\to z^2 +x)^n(x)$

Notice that the fractal must be a subset of the fractal the generates the added values because if that divereged so would this.

We can apply this same idea different ways. For example, we could use constants from the 3-Multibrot and use them in the Mandelbrot set.

Or use the Mandelbrot set to generate constants for the 3-Mutlibrot!