We're going to look at fractals and explore the Mandelbrot set. However, this is not a course in mathematics. There are quite a few websites where you can learn more about the mathematical algorithms and computer programming behind fractals.
This will be an exploration. If you like, it's a quest. Along the way, you'll be encouraged to:
— Look at the natural world in a new way.
— Link ideas which are apparently not related to each other.
— Work through ideas and issues in what might be a new way.
— Consider apparent contradictions.
— Exercise both logic and creativity.
— Use your imagination very fully.
— Think 'fractally'.
— Think philosophically.
— Tackle questions which have no answers.
The Mandelbrot set is one of many fractals, each with its own complex repeating internal patterns. They are generated on the computer screen by means of mathematical calculations. The Mandelbrot set is named after the person who discovered and developed it. It forms the basis of a new and developing branch of mathematics known as the geometry of chaos. This can be applied to some of the irregular and otherwise unpredictable things you see in the natural world.
The leaves on a tree, for example, appear to be the same as each other, but when you study them closely you can see that they are different in several ways. Flowers germinate from apparently identical seeds, but the results are not identical. Clouds form in the sky and change continually, due to the many factors which influence their appearance, shape and size, and eventual disappearance. The geometry of chaos has the potential for scientists and mathematicians to understand and 'measure' such apparently random and unorganised shapes and movements.
One of the most unusual aspects of fractals is that their repeating and changing patterns are infinite. You can draw a fractal pattern, but you cannot explore it fully without a computer — a two-dimensional drawing cannot lead you towards inifinity. On a computer monitor, it is possible to zoom into one small area, which becomes the new pattern, and then to zoom into a small area of that pattern, infinitely (given a computer with adequate facilities).
These are questions, ideas and suggestions for you to think about and act on.
There are no single correct 'answers' to any of them. All responses are valid.
You can use the logical and the creative parts of your mind to come up with whatever you think is a suitable response.
You can have a go at just one in each group, or at more than one. It's up to you.
1 Random shapes in the natural world
2 What is a fractal?
3 Chaos and Order
4 Consider a brain cell
5 Towards infinity
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Important notes. Please read.
There are several stunning pictures of fractals in the other section on The Brain Rummager called The Amazing Mandelbrot Fractal. Use one of the Menus to have a look it.
You cannot zoom into the fractal graphics on these pages. They are simply screen-grabs, not working fractals. If you wish to do some zooming, you will need to access an actual 'live' Mandelbrot fractal. There are several versions on the Internet. Be careful when you search. Make sure the website you use is safe and does not, for instance, bombard you with pop-ups and advertisements. I found one at this address which was both efficient and safe: http://www.thorsen.priv.no/services/mandelbrot/about.html
Fractal graphics used in this section were collected from disk-based public domain software programs several years ago. As far as I know, they are not subject to copyright restrictions.