Journey to Infinity

and what you might find along the way

 

2 What is a fractal?

A fractal is a pattern which reveals reveals greater complexity when you enlarge it.

Another more detailed definition  is that it is a geometrical structure which has an irregular or fragmented appearance which shows similar characteristics at all levels of magnification.

Notice the words fragmented and similar and levels in that sentence. We are not dealing with straightforward geometry here. If you magnify a simple geometrical figure, such as a circle, all you see is a larger circle. If you magnify a complex figure, such as a set of small triangles of different shapes, you'll see more detail if you magnify it, but you will be looking at the same triangles. Here is a small, apparently complicated set of shapes.

If you enlarge into it, you can see more clearly how it is constructed from pentagons (regular 5-sided shapes), but you still see the same basic set of shapes.

It doesn't matter how much you enlarge a normal two-dimensional geometrical figure, you will still see the same figure. If it is a three-dimensional figure, such as a cube, and you can view all the sides on a computer screen, you will still see one thing: the original cube. If you magnify it, you will merely see a larger cube. OK?

Another way of describing a fractal is to say that it is an irregular shape that can be sub-divided into parts. Each part can look similar to the original shape. The smaller parts are not the same: they are self-similar. In some fractals, you can see these self-similar fragments when you zoom to a higher level of magnification. Obviously, you cannot zoom into a picture of a fractal printed on paper. It has only two dimensions. A computer, however, enables you to zoom in, to see fragments and parts, and in theory to continue that process to infinity. You are moving through another sort of dimension, perhaps.

Some fractals are relatively simple

Before we look at the Mandelbrot set, a very complex fractal, there are some simpler fractals which are worth a look.

Fractals are irregular shapes and patterns generated by algorithms, which are arithmetical and computational formula used in a computer. They are purely mathematical in origin.  The algorithms are not designed to produce pictures of objects in the real world. They produce fascinating and often weird patterns and shapes which might look like something in the world of nature, or even in the world of fantasy. Click below for some examples.

Some fractals

Who is Mandelbrot?

Dr Benoit Mandelbrot is a French mathematician. He coined the word fractal in 1975. That's why you won't find it in old dictionaries. It is also why there are different definitions of it it is a new word for a new concept.

Dr Mandelbrot based his new word on the Latin word fractus, which means broken or fragmented. The Mandelbrot set, to give it its correct name, looks something like a two-dimensional picture of a series of lakes, with decorative shore-lines. If you zoomed into a painting of a photograph of a lake and its shores, you could simply see larger pictures of the details. You don't see that when you zoom into the Mandelbrot fractal on a computer screen.

When you enlarge a section of it, or zoom into part of it, you see that the patterns around the 'shores' of the 'lakes' repeat themselves change slightly. You continue zooming in, and the patterns continue to repeat and mutate, to change slightly. You seem to be able to do this for ever, but you might be limited by the capacity of your computer.

Click here to to see What the Mandelbrot fractal looks like.

Now tackle BrainWorks before going to the next Unit.

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