Friday, January 25, 2013

Introduction to the Mandelbrot Set

In this post, I will briefly provide an introduction to the Mandelbrot set.  If you are already familiar with the Mandelbrot set but would like to learn how I generated these, you may want to come back and see my next post, where I will give an overview and more detailed instructions.  But for now, let's stay focused on the nature of the set itself.

In loosest terms, the Mandelbrot set is the set of complex numbers that remains bounded when a certain mapping function is applied iteratively to the number.  In more rigorous terms, let us call a given complex number c.  Then, let us set z0 to zero and apply the map:

If z remains bounded, then c is part of the Mandelbrot set.  When plotting the set, we can color the points that are part of the set black and color the points that diverge some other color, to get interesting results like this:

Choosing a coloring scheme can involve factors such as the magnitude or angle of c, or the number of iterations it takes to diverge.  I will talk plenty about generating the Mandelbrot and creating a color scheme in my next post.

If you wish to learn more about the basics of Mandelbrot set, there is a plethora of information already out there.  For now, take a look at the point where the main cardioid is tangent to the largest circular bulb:

A point is said to diverge and therefore not be part of the Mandelbrot set if at any time the magnitude of z becomes greater than two (|z| > 2).  The point where these bulbs meet (-0.75) will not diverge and is therefore part of the set.  However, any points directly above or below -0.75 (any points on the vertical green line excluding -0.75) do eventually diverge and are therefore not part of the set.  Let us consider an interesting property of points that are very close to -0.75.  Let us denote them as -0.75 + , for very small values of ϵ.  As the value of ϵ  becomes smaller and smaller, the number of iterations it takes for z to diverge become larger and larger.  Let us call n the number of iterations (as a function of ϵ).  An interesting property emerges out of this:

As the distance from a number just above or below -0.75 to the number -0.75 tends to zero, the number of iterations to diverge times that distance approaches π.

It turns out that using different functions in our mapping, we can create an entirely new set of fractals.  Of the more common is the class of Multibrot sets, where z is raised to larger powers:

The following Multibrot set was generated by allowing n=4:

n = 4

Much like Mandelbrot (n=2), the Multibrot sets also exhibit fractal self similarity.  One of my favorite variations on the Mandelbrot set uses the following iteration:

The outermost zoom appears mangled an disorderly:

Corner 1, corner 2: iterations
(-4.5 + 3i), (3.5 - 3i): 100

However, zooming in on the "tip of the blade" in the upper left corner exhibits an interesting property:

(0.359428 + 1.239502i), (0.383644 + 1.221340i): 225

The presence of Mandelbrot inside this "Expobrot" is an interesting property and can be found along any of the other bulbs in the set.  A similar property is exhibited in a similar mapping, raising to yet a higher power:

An interesting property of this fractal is that there are "islands" around the outer edges (the four darker regions along the top and left side).  These islands contain Multibrot sets of n=3.

(-2.545837 + 1.011490i) (-2.494110 + 0.972694i) 225

From the traditional Mandelbrot to Multibrot to "Expobrot", there are many interesting mappings that lead to beautiful fractal patters.  In the next post, we will explore some more patters and go into detail as to methods of generating these fractals and discuss coloring algorithms to give the fractals a nice color scheme.

Wednesday, January 23, 2013

Hello, World! Construction underway...

This blog is under construction!

( January 23, 2013)

Come back to see the first post, to be published by February.  There are many stylistic and organization decisions for me to make before setting this afoot.  My first topic will be on strange varieties of the Mandelbrot Set coming up here by February.

I will deliver.  In the meantime, enjoy this classic proof without words for the Pythagorean Theorem (imagine sliding the solid colored pieces into the lighter shaded regions in order to fill in the square of the hypotenuse).

"A Proof without Words"