Rather, with the Explorer, you can view preselected regions of the Mandelbrot set as well as significant Julia sets. You will come to understand the meaning of the Mandelbrot set as the parameter plane, the filled Julia sets as the dynamical planes, and fascinating ideas such as the rotation numbers, the Farey tree, and the Fibonacci sequence all of which are embedded within this set. We hope that you will come away with an idea of the mathematical meaning of this object as well as an appreciation of its incredible beauty. See the section entitled Background for more information on these ideas.
Necessary Background
We assume that the user is familiar with the basic arithmetic and
geometry of complex numbers. If you are not: Sorry! Pull out your
old algebra book and start reviewing. You really need to know what
complex numbers are in order to understand what is going on in the
Explorer. We also assume that the user is familiar with the basic
terminology of dynamical systems, including the ideas of orbit,
iteration, cycle, etc. We have included a brief introduction to these
ideas as well as a glossary of all technical terms used in order to
help you get over this hurdle. Click the HELP
button to access these resources.
This software is really intended for
use by readers of the series of four books in the
Toolkit of Dynamics Activities published by Key Curriculum
Press. All of the background material is explained in detail in these
books.
How to use Explorer.
The Mandelbrot Set Explorer consists of a collection of guided and
unguided tours. Each tour begins with a Question, which is
immediately answered in one or two short and perhaps at first
unintelligible
sentences. The rest of the tour is designed to illustrate and explain
what this answer means. Using a series of interactive graphics and
animations, you will see geometrically and dynamically various facets
of the answer to the tour's question. Many of the tours come with
associated questions (and answers) to help you check that you fully
understand the concepts. Also, certain of the tours come
with suggestions for further exploration. These are designed to
help you plunge deeper into the topic at hand. No answers are give
here, since we have no idea where you are going to go!
We suggest that you take at least the first few tours in order; after that you may merrily skip around and explore various facets of the Mandelbrot and Julia sets.
A list of the various tours are listed on the MAIN PAGE. Click on this button at any time to return to the departure point for the different tours. The HOME button returns you to the welcome page for the Explorer.
Most of the images included in the Explorer have been precomputed. You do not need to waste time computing various blowups of the Mandelbrot and Julia sets in order to see a particular mathematical point. Rather, you can just click and see various images, zooms, and animations very quickly. The downside of this is that you cannot necessarily go wherever you want in these sets. For that you need special software. In order to allow you to venture forward on your own, we have included several Java applets that allow you to move about these sets to a certain extent on your own. These Java applets are platform independent, so you can use them no matter what computer you are using. You should be forewarned that these applets will run quite slowly on older machines.
There are a number of animations included in various tours. These animations are in QuickTime format, so you will need a QuickTime player in order to view these animations. You can download the player for Macs or PCs from Apple Computers.
What the colors mean. In each picture of the Mandelbrot set or the filled Julia sets, the set in question is painted in black. Other colors indicate points whose orbits escape to infinity. In the Mandelbrot set, a c-value for which the orbit of 0 escapes is colored; for the filled Julia set, it is the actual seed for the given function that is colored.
The color code is always the same. It is known that if an orbit enters the region outside the circle of radius max (2, |c|) centered at the origin, then that orbit necessarily escapes to infinity. This is what is called the Escape Criterion. So, we color an escaping orbit according to the number of iterations necessary to exceed this bound. Red points escape fastest, followed in order by orange, yellow, green, blue, and violet. The important thing is: colored points lie outside the Mandelbrot or Julia sets; black points lie within.
Good Luck! The Mandelbrot Set Explorer is a work in progress. Whenever we have time we add a few more images. We sincerely hope that, while using this, you will gain an appreciation of the remarkable mathematics that lies behind this set.
Dynamical Systems and Technology Project
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