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.
This week we have a generalized Julia set with the interior of the set colored using a shade of yellow and the cosine coloring algorithm. This yields a pleasant, metallic appearance.
This is a context shot on a meandering fractal river in the Mandelbrot set. It uses a new type of convergence called wedge convergences that deems a point to have diverged when they move into cones with a vertex at one of (1,1) (1,-1) (-1,1) and (-1,-1).
It’s a little subtle, but this is a Mandelbrot set view, rendered using a new convergence condition — a diamond shaped box that the fractal iterator must escape.
The frenetic little bubble animated above is a symbot, Occupy Math’s own name for a type of super-simple robot. These robots exist only in the computer; we don’t actually make physical versions. There is an interesting book about this type of robot and its more complex cousins: Vehicles: Experiments in Synthetic Psychology by Valentino Braitenberg. These robots have sensors and wheels; the input to the sensors controls how fast the wheels turn. The interesting thing is the number of different behaviors that you can get out of even really simple robots. The robot above can sense the flashing light and is trying to approach it. It lacks the ability to slow down, so it’s learned to run the light over repeatedly. Think Labrador puppy.
The week’s image is an animation of 48 different two parameter Julia sets. One of the parameters is changed by small amounts causing the fractal to morph. Like it?
This week a deep zoom into the cubic Mandelbrot set. Way too much detail!
This week is a hybrid fourth degree fifth degree Julia set with quadrant convergence with a pleasant five-fold symmetry. “The Sheriff of Fractal Town”.
This week we have a zoom into a quintic Julia set. Notice that both five and three fold symmetry appear.
This is a twisted Newton’s method fractal with four roots.