fractals and self-similarity

Characteristics of a fractal object

1.    Infinite detail at every point

2.    Self similarity between object parts

Types of self similarity

ü Exact self similarity

ü Statistical self similarity

Exact self similarity

     if a region of a curve is enlarged the enlargement looks exactly like the original

Statistical self similarity

     if a region is enlarged the enlargement on an average looks like the original

Successive refinement of curves

     by repeatedly refining a simple curve very complex curves can be fashioned

Ex. Koch curve

Koch curve produces an infinitely long line within a region of finite area

Generations

Successive generations are denoted by K0 , K1, K2 , …

The zeroth generation shape  K0  is a horizontal line of unit length

The curve  K1 is generated by dividing  K0 into three equal parts and replacing the middle section with a triangular bump

The bump should have sides of length 1/3 so the length of the line is now 4/3

The second order curve  K2 is generated by building a bump on each of the 4 line segments of  K1

Void drawKoch(double dir , double len , int n)

           {          doubledirRad = 0.0174533 * dir  ;  // direction in radians

                       if ( n==0)

                       lineRel(len * Cos(dirRad) , len * Sin(dirRad));

                       else     {       n--;  // reduce the order & length

                       len  /=3;

drawKoch(dir , len ,n);

                       dir  += 60;

                       drawKoch(dir , len ,n);

                       dir  -= 120;

                       drawKoch(dir , len ,n);

                       dir  += 60;

                       drawKoch(dir , len ,n);

                                  }

              }      

Fractal Dimension

Estimated by box covering method

                 D  =  log(N) / log(1/r)

   N = no. of equal segments

    r = 1/N

For Koch curve the fractal dimension is in between 1 & 2

For Peanocurve  D is 2