Fractal Lenses

Lens

Occupy Math has already taken a shot at explaining what fractals are. He has tried to supply engaging holiday fractals. On of the least popular posts is on the incredible complexity of the Mandelbrot Set and there is a post on making family trees of fractals (and other things). What’s left? This week Occupy Math is going to turn up the weird to eleven and use fractal algorithms as lenses — a different type of lens from the one shown at the top of the post. The only thing you really need to know about fractals to get a sense of what is going on is that a fractal is based on an algorithm that moves a point around in a complex way until it is captured. The details of the algorithm and the conditions for “capture” give you the shape and then you also need a coloring algorithm. Today’s post is all about a really odd way to color fractals.

Tile the plane with a picture and used that to color the fractal.

When the point that is jumping around finished jumping, we use the point’s final position and number of steps before stopping to calculate a color. At least that’s the usual method. In this post we color points a different way. First, we tile the plane with a picture. When the jumping point stops, it is somewhere on that tiled plane. Whatever the color is right there where the point stopped is the color we use. There are a few details, like sliding and scaling the tiling to make the fractal look good, but mostly that’s it. Let’s look at some examples — right click/view image to enlarge these.

If the picture we tiled is this one,

pic01

Then the fractal comes out looking textured.

frac01

Change the picture to this one,

pic05

and the texturing comes out very different!

frac05

A while ago, Occupy Math found the mathematical formula for brown hair (this picture should tile smoothly, help yourself).

pic06

even hitting itself at right angles, this yields a cool effect…

frac06

Of course we don’t have to use just textures. If we use one of Occupy Math’s Trigonometric Flowers:

pic04

we get an okay fractal. Maybe adjust the position of the tiling more?

frac04

If we use a more representational picture:

pic02

the result might make a nice mouse pad?

frac02

You can also use a picture of a person:

pic03

Notice that some of the copies of the head are distorted. Weird, no?

frac03

The Benefit of Studying Math

This week’s post supplies an example of one of the things you get if you study math really hard. You get to be extremely weird. The fractal algorithm, in this case a variety of Julia set codes, used a tiled image as its coloring method — with the effect that the algorithm acted like a bizarre fun-house mirror of a lens on the image.

There are other ways to use images to color fractals. If we use polar coordinates on the natural convergence of a Newton’s method fractal with a marble textured image (like the first picture about but mostly red) we get this glass-like result.

glass

Once you have the fractal algorithm, and select the image, it only takes a little math — most of it available in high school — to use an image to color the fractal. One of the nice things about using fractals is that the complexity all arises from doing something simple over and over. You don’t need to be painstakingly careful and have excellent hand-eye coordination. Instead you pull art from the informational superstructure of reality. Sort of. Occupy Math hopes you have enjoyed this escapist post in his series. Got something mathy you want explained? Comment or tweet!

I hope to see you here again,
Daniel Ashlock,
University of Guelph,
Department of Mathematics and Statistics

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