This is a leaf convergence zoom into the Mandelbrot set. Its a deep zoom near a minibrot which creates very complex texture.
In this post we are going to look at a trick for figuring out what a whole type of fractals look like, before doing all the calculations needed to generate them. Occupy Math has posted on fractals before. The clue we will use to search for beautiful fractals is, itself, a very special fractal called the Mandelbrot set. Here is the core fact for today’s post. The Mandelbrot set lives in the complex numbers, which form a plane. Given that, the way the Mandelbrot set looks near a particular place (or number) in the plane of complex numbers is the way the Julia set based on that number looks all over. Another way to say this is that the Mandelbrot set indexes Julia sets.
This week Occupy Math presents another 32nd order Julia set. Cool, no?
The current Image of the Week is a thirty-second power Julia set — Occupy Math is experimenting with the round boundaries you get from this sort of high power.
This post is on a really easy method to solve some hard problems, including locating a pretty fractal. It is called the bisection method, but you may know it as a high-low game. One person thinks of a number and the other guesses. The person who knows the number replies “high” or “low”. A good strategy is for the guesser to move half way between their most recent high and low guesses. Bisection is fancy math talk for “move half way”. The thing is that this method can solve hard problems and do COOL things. Read on to find out.
This week, Occupy Math’s Image of the week is a three parameter Julia set located with a high-low game. This is a zoom into its center.
This is a Julia set with two complex parameters. Occuy math set on of them to 0.01i and then chose the other using iterated bisection driven by the appearance of the fractal. It came out pretty well. One of the future Thursday blogs will be about iterated bisection.
This is an evolved generalized Julia set with three parameters. Occupy Math picked it because it looks really interesting. if you would like a copy of the scientific paper about evolving this type of fractal, send a request to firstname.lastname@example.org.
This week Occupy Math presents a generalized Julia set animated by morping a parameter from 0.360 to 0.419 in increments of 0.001. Happy New Year!
This week we look deeper into the fourth-power Mandelbrot to find a star of trees. Peace on earth, good will to all!