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The word fractal is sometimes used loosely, to describe any intricate mathematical image with artistic merit. Along with chaos theory (the infamous butterfly effect) fractals are one of the rare bits of mathematics to have a place in popular culture. This web site uses the word fractal rather inaccurately. I called the page where you can draw Spirographs and other curves "fractal.html". I'm not at all sure that any of those images are really fractals; several definitely are not.
I have a maths book with a proper definition of the type of fractals discssed in this tour. I'd put it here, but I don't know how to enter the funny Greek symbols that are needed for it... The basic idea is that a fractal is a set of points which has a similar structure on all scales. This is most easily seen in the next fractal, the Sierpiński Triangle.
Rest the mouse over the image to find out more on each page of this tour.
Take a look at the triangle on the left:
It is the Sierpiński triangle, named after the Polish mathematician who discovered it in 1915.
It is made up of three smaller triangles which are all exactly the same shape as the large triangle. They in turn are made up of three more exactly similar triangles. That is the basic idea of a fractal. However, it's not always quite so simple to spot the repeating patterns, as the larger shape may be rotated or distorted in various ways as it shrinks. In more complicated fractals there may be several different kinds of shape appearing. Click here to bring back the snowflake for a moment and see if you can see the smaller snowflakes within it.
You could start by cutting the largest triangle into four and taking away the middle one (leaving the edges). Then you do the same to the three that are left, and so on. This gets quite difficult quickly, as you have to remember to deal with more and more little triangles. You can just about see seven sizes of triangles that have been cut out in the image. That makes 2187 small triangles already. How much of the original triangle is left at the end? The answer is in here somewhere.
There is another way to draw the triangle. Start from any corner of the triangle. Pick one of the corners at random. Go half way there in a straight line and mark the spot you got to. It's a point in the triangle. Then pick a corner again and do the same thing. Just keep doing it for ever. In fact my computer drew the triangle exactly that way. After 50,000 iterations or so it's already starting to appear. It might be a lot sooner than that, but a PC can do something like this about 1,000,000 times a second.
There is an amazing connection between the Sierpiński triangle and another famous mathematical triangle - Pascal's triangle. Click here to find out more.
Random iteration is often a very good way of constructing fractals. To get really high quality images you need to iterate about 5 million times,for a screen image, and maybe 30,000,000 for a printed image. For more general fractals rather than the "half way to a corner" rule, one of a number of previously chosen contracting affine transformations is selected at random. You may be surprised to learn that there is a good chance you know how to calculate these transformations. All you have to do is remember the matrix multiplication you probably learned at school. A contracting transformation is one where the matrix always brings points closer together. All the computer has to do is keep calculating formulas like this:
x → ax+by+e
y → cx+dy+f
The point this calculates is then plotted (approximately) on the screen or printer. The computer remembers how often it has plotted a point in the pixel being plotted and colours it accordingly.
As you can see from this example, fractals don't always have a simple geometrical form.This, like most Spirofractal fractals, uses many similar transformations, but rotated with respect to one another and reducing the scale of the image by different amounts.
The result of changing the transformations used in a fractal is not that easy to predict.The rule Spirofractal uses for creating this kind of image usually starts out producing something that looks a bit like a birds wing, or a line of trees. But if you add more and more similar transformations after a while you get something like this.
However, sometimes the way transformations combine is predictable: If you use a set of transformations that all map any given point to points that lie on a circle, or that are mirror images of each other, then the fractal will have rotational or mirror symmetry.This can be seen in the next fractal.
It may look more like a Spirograph image than a proper fractal, but it is a fractal. Each ray is a flattened copy of the whole image. Patterns like this arise if the transformations used in the fractal do not contract much in one direction, but contract almost to nothing at right angles. This image is built up from a number of such transformations, all rotations of each other.
This is the end of the Spirofractal tour. However the Spirofractal gallery has lots more information about and picture of fractals, and the other kinds of pattern Spirofractal creates. The Stop button will take you back to the main part of the gallery.
Notice how each branch of the snowflake is a slightly flattened version of the whole thing. In fact, the small white dots are also complete snow flakes. The snow flake is a kind inside-out version of the Sierpiński triangle: if you use almost the same rules that create the Sierpiński triangle but with negative numbers, instead of positive you get a snow flake.
When you are ready please click here to go on with the tour. You'll find out how to draw the triangle, or almost any other fractal, and get a chance to see more fractals.
On this tour all the images are 100% bona fide fractals, created with Spirofractal, which you can download from this website. No mathematical ability is needed to use it, as it will create fractals, and other type of image completely automatically (there is a screen saver too), and most of them are quite nice. Maybe one day I'll make a version that works on a web page but for now it is only available as a Windows program.
What do you think the area of the Sierpiński triangle is? Surprisingly the answer is 0. Each set of triangles that is removed reduces the remaining area by a quarter, and so eventually nothing is left. All the same there are still infinitely many points left in it.
This fern really is a Spirofractal image. There are three different kinds of transformation in the fractal:
The realistic colouring arises because fractals often look like objects in the natural world, and one of the colour schemes in Spirofractal mainly uses natural looking colours.
Many Spirofractal images look like some strange animal or plant. Some resemble something you might see down a microscope. This impression can be enhanced by using one of the nice features of Spirofractal. You can double click on any part of the image, and the fractal will be drawn again with that part in close up.You can do this as often as you like. Unfortunately this inevitably causes it to take much longer to draw the image because the many points that are no longer displayed still must be calculated.
I am not sure you will be able to see why I chose the name. However when you create the fractal with Spirofractal it is fairly obvious as the colours move out as the fractal forms.