Fractals don't have to look exactly the same when you zoom in. They can look almost the same. The famous of all fractals is the Mandelbrot Set.
Hey, here is one of those "copys"
In order to understand the definition of dimention, think of a square with 1 Inch side. This square have an area of 1 square inch .Take this square and double his side (so it will be 2 inchs). Now it's area is 4 square inch. Now take a cube with 1 inch side. This cube has a volume of 1 cubic inch. Double its size, and its volume will be 3 cubic inches.
Now look at the data:
| Dimensions | Side enlargment factor | "Inside" enlargment factor
| 2 - square
| 2
| 4 = 2^2
| 3 - cube
| 2
| 8 = 2^3
| |
Lets find the gasket's dimension. If we take it's side and double it, we coul cover with 3 small gaskets the big one. Therefore, the dimension of the gasket will be the solution of the equation: 2^Dimension=3. Which gives us: 1.58496.
If you think about it, it make sense: removing all those little triangles from the original, leaves us with some kind of line, but this line cover some area, so it's something between line and triangle - between 1 dimensional object to 2 dimensional. So it's 1.58
There are many ways to create fractals:
IFS - Iterated Function System: a set of transformation, which when itrated (apply them once, then aplly them on the resault, then apply them on the new resault and so on till infinity) they produce a fractal. The first time you apply the transformation, you can apply them n any shape (even the Mona Lisa).These transformation are often given in the form of a matrix. For example, the set of transformation that created the sierpenski gasket are:
shrink to 1/4 of the original size and copy it to the bottom, the top right and the top left.
A guy named Mitchel F. Bransley discovered of good and efficiant way to derive the transformation needed to create a fractal from it's picture. This method is called The Collage Theorem.
IFS with probabilities: this time, each transformation has a probability of happening. You start with a single dot and randomly choose one of the transformations. You apply this transformation on the point and again you choose a transformation randomly, applying it on the new point. you go on like this till infinity. To draw the gasket this way, each one of the places to copy the shrinked image to has a probability of 1/3. In order to find the probability of each of the transformation you must use a branch of math called The Measure Theory. The two Ifs Methods can create realy beutifull shapes, like this famous fern:
Iterating a formula: In this method, you use the complex numbers (a number of the form a+b*i, when: a,b are real numbers and i is the square root of -1). For each complex number you iterate a formula. If the resault remain finit then the number is in the fractal, and if the resault is growing to infinity then it's out of the fractal. O course, you dont wait till the resault to reach infinity to discover that it's out of the fractal. Instead, in every iteration, you check if the resault is bigger than a certain limit, if so, you know that this number will reach infinity and you know it's out of the fractal. In the complex plain, put a black dot for every number that is in the fractal and a white dot for every number out of the fractal. The black area is a fractal. Usually, instead of White, each dot is colored in other color acording to the number of iteration it took in order to exceed th limit. The mandelbrot set is drawn using this method when the formula used to create it is:
Z(0)=0
Z(n+1)=Z(n)^2+c (c is the original number)
The limit for the mandelbrot set is 2.
L-system: a set of command, that tells you what to do with an initial shape. Usually the shape is described in turtle graphics command, and the replacement command tells you wich turtle command to replace with which. The replacement procedure is then iterated and the final shape is a feactal. For example, take a rule like this: convert every straight line to two lines which are orthogonal. The first few steps will look like this:
