How are Fractals formed. is there a particular way???

Answer 1

The definition of a fractal is fairly clear: a structure that is self-similar across scale. This self-similarity gives the fractal it's appealing look.

You have two main ways to create a fractal: additive and reductive.

Additive Example (Koch Snowflake): Take a line. Crumple up the middle third so that where the middle third was, you now have 2 lines, each 1/3 the length of the original line. It looks like a line with a pointy hill in the middle. Now, take each of the 4 line segments and do the same thing. You now have 16 smaller line segments. Repeat for them. And again. And again.

Reductive example: Take a filled triangle. Cut it up into 4 equal triangles: 1 at the top, 2 at the bottom, and 1 i nthe middle pointing down. Remove the middle triangle. You now have a figure with 3 triangles. Do the same thing with each of those triangles. You now have 9 smaller triangles. Repeat. Repeat. Again and again.

The name fractal comes from a "fraction of a dimension". One definition of how to calculate the dimension of a fractal goes something like this:

a) To get a line twice the size of a line, you need 2 copies of the original line, pasted together.

b) To get a square twice the size (measured along each side), you need 4 copies of the square.

c) To get a cube twice the size (measured along each side), you need 8 copies of the cube.

A pattern starts to emerge. In n dimensions, you need 2^n copies to double the size. Another way to put it is: if you need x copies, then you have (log x) / (log 2) dimensions. Take the triangle example from above: to double the size, you need 3 copies of the original. Plug this into our equation:

(log 3) / (log 2) = 1.58496.

This makes sense. It is definitely not 1 dimensional, but it is not quite 2 dimensional. If we had a fractal that "filled" 2 dimensions more, then it would have a higher fractal dimension.

This is just one definition of the fractal dimension. There are several others.

Incidentally, you can build fractals using a random approach too. Try this: Draw a triangle. Put a random dot in the triangle. Now, go halfway to any corner and make another dot. Go halfway from that corner and make another dot. You will probably need a computer to show you what comes out, but I still find it amazing.

One last way to discover a fractal. If you know pascal's triangle, draw out a huge version of the triangle. A computer is probably best again. Mark all the even number is black and all the odd numbers as white. If you have a large enough triangle, you should notice a familiar shape :)

Answer 2

They can be done graphically ormore typically ising a set of complex equations. Here's a write up -

http://en.wikipedia.org/wiki/Fractal

Answer 3

.