Fractal diagrams:
A fractal is in some senses like any closed geometrical shape, eg a circle or triangle, in that can be thought of as being made up from a line or lines.
However there is one very important difference:
If you magnify a circle indefinitely any part of it looks more and more like a straight line.
If you magnify a triangle indefinitely, the angles become infinitely far apart.
If you magnify a fractal indefinitely any part of it exhibits the same geometrical structure at all scales.
To illustrate this try the following
:
Start with a square [RED in Fig (i)], construct small squares ,with sides 1/3 of the length of the original square, on the middle of each side [BLACK in Fig (i)]. Repeat this for all the sections of the sides [BLUE in Fig (i)].The outside boundary of this is shown in Fig (ii).
This is a process that can (in theory!) be continued indefinitely and the resulting boundary would be a fractal. Note the following points
:
- However much you enlarged a section of the line you would see the same arrangements of sections of perpendicular lines
- The area enclosed inside the fractal would be finite (In fact it is equal to the area of the large square in Figure (iii) which is twice the area of the original square.
- The length of the fractal will be infinite ! [Each iteration multiplies the length of the boundary by 5/4]
Another example of a fractal is the boundary of the Sierpinski Sieve
Return to Mandelbrot-Plus Return to Sierpinski Sieve
