Fractional Dimension!

Presented by

Sonali SahaSarojini Naidu College for

Women30 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

r

md

ln

ln

Scale factor =3 and no. of copies=2 hence

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.

?

3ln

2lnd

63.3ln

2lnd

Middle third cantor set

Koch curve

Steps to produce Koch Curve

26.13ln

4lnd

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

N

0lim)/1ln(

)(ln

)/1ln(

ln)(ln

asNAN

d

For ϵ=1/3 ; N=8

Hence

89.13ln

8lnd

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 processis the origin of this interesting behaviour

Effect of repeated stretching and folding process

►Dough Flattened and stretch

fold

Re-inject

Repeated stretching and folding process is the origin of this interesting behaviour

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.

we generate a set of very many points {xi; i=1,2,....n} on the attractor considering the system evolve for a long time.

Correlation Dimension

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