Fractional Dimension!

Fractional Dimension! Presented by SonaliSaha Sarojini Naidu College for Women 30 Jessore Road, Kolkata 700028

Fractal • Objects having self similarity Self similarity means on scaling down the object repeats onto itself Mountain, coastal area, blood vessels, brocouli``

On zooming up it repeats onto itself

On magnification it does not produce any regular shape i.e. any finite combination of 0,1, 2 and 3 dimensional objects In Eucledian geometry we considered some axioms point has 0 dimensionline has 1 dimensionand so on ……………………

How to Quantify dimension? • Scale down the line by factor 2 No. of copies m=2, scale factor r=2 We can check for r=3; m will be 3

Here sale factor r =2 and no. of copies m=4 We can also check for r =3 Then m will be 9 Conclusion: m=rd where d=similarity dimension

Middle third cantor set It is not an 1D pattern as length goes to zero after infinite no. of steps Not 0D as we cannot filled up the pattern by finite no. of points. ? Scale factor =3 and no. of copies=2 hence

Koch curve

Steps to produce Koch Curve

Fractals are the objects having fractional dimension. • In general they are self similar or nearly self similar or having similarity in statistical distributiuon Similarity dimension is not applicable for nearly self similar body

Various methods have been proposed where irregularities within a range ϵ have ignored and the effect on the result at zero limit has been considered Box dimension is one of them No. of boxes N(ϵ) = L/ϵ No. of boxes N(ϵ) = A/ϵ2

For ϵ=1/3 ; N=8 Hence

d=(ln 13/ln 3)=2.33 Scale factor r= 3; No. of copies = 13

Attractors • Where all neighbouring trajectories converge. It may be a point or line or so on. • Accordingly it is 0D, 1D and so on…….. • When it is strongly dependent on initial conditions, they are called Strange Attractors. Strange attractors have fractal pattern Trajectories of Strange attractors remains bound in phase space yet their separation increases exponentially Repeated stretching and folding process is the origin of this interesting behaviour

Effect of repeated stretching and folding process Repeated stretching and folding process is the origin of this interesting behaviour • Dough Flattened and stretch Re-inject fold

Effect of the Process

S  is the product of a smooth curve with a cantor set. • The process of repeated stretching and folding produce fractal patterns. Correlation Dimension we generate a set of very many points {xi; i=1,2,....n} on the attractor considering the system evolve for a long time.

fix a point x on the attractor • Nx() is the no. of points inside a ball of radious  about x Nx() will increase with increase of  Nx()  d d is point wise dimension We take average on many x C()  d d is correlation dimension

There is no unique method to calculate the dimension of fractals Thank You

Fractional Dimension!