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