Fractal Geometry
Euclidean Geometry and Fractal Geometry:
The conventional geometry we are familiar with is Euclidean Geometry that deals with 0,1, 2 and 3 dimensional figures. But what about a figure having a dimension 2.5 or 1.67. Is this really possible? Yes, and the branch of mathematics that deals with such figures is called Fractal Geometry. It was first developed by Benoit B.Mandelbrot.
Consider a simple case of some amount of cotton. We consider it to be made up of one-dimensional fibers. But millions of such fibers together makes up an approximate 3D object. So its dimension is supposed to lie between 1 and 3. But it is evidently not a 2D structure. In fact its dimension is something between 2 and 3, i.e. it has a fractal dimension.
Other examples include The Fern Leaf (1<D<2), Cantor Dust (D = ln2/ln3), Koch Curve (D = ln4/ln3), etc.
Dimension and fundamental properties of a fractal structure:
Though not much required for FC 2.0 users, the following mathematical properties of fractal structures deserve mention:
Dimension measurement of a fractal structure: Consider a geometric figure of dimension D drawn in a B dimensional space (B = 1,2 or 3). We take a B dimensional sphere (i.e. a line segment, circle or sphere according as B=1,2 or3) of radius r. Let N(r) = the minimum number of such spheres required to completely cover (or include) all the points of the figure. Then the dimension D of the figure is given by solving the Limit as r tends to 0, (r^D) . N(r) = a finite real positive number (not equal to 0).
Transformations used to create fractal structures: Most of the self-similar fractal structures are created with the help of a number of transformations (i.e. 'function of points' or 'vector functions' which are such functions which accepts the coordinates of a point and returns coordinates of another point as the functional value). A fractal structure consists of the set of points which when put into a transformation that creates it returns a point that lies in the fractal set (or fractal structure) itself. Thus the fractal set remains invariant under transformations that create it.
For more mathematical details you can refer to books on Fractal Geometry.