Box dimension
Different ways to define fractal dimensions usually lead to the same result.
Important: different ways usually lead to different methods to calculate the fractal
dimension, in particular, in random fractals.
We define the box dimension:
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Given a set of points in d-dimensions.
Calculate the number of boxes of linear sizeε needed to cover the set.
If N (ε ) is the number of boxes of sizeε and there exists the relation
A
N (ε ) = d ( for ε → 0)
ε f
log N (ε )
( for ε → 0)
Then d f =
1
log
ε
is the fractal dimension of the set.
Box dimension
For a line section: N (ε ) =
For a square: N (ε ) =
A
ε
⇒ d f =1
ε
ε
A
ε2
⇒ df = 2
A
ε
ε3
That is for integer dimensions d f = d as expected !
⇒ df =3
For a cube: N (ε ) =
Triadic Cantor set
ε = 1/ 3
ε = (1/ 3)2
ε = (1/ 3)3
k
1
For boxes of size  
ε 
The number is: N (ε ) = 2k
A
ln N (ε ) ln 2k ln 2
k
=
=
≅ 0.6309
Solving 2 = d ⇒ d f =
k
f
1
ln 3
ln 3
ε
ln
ε
More fractal dimensions
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-
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The common fractal dimension d f cannot fully characterize the fractal
⇒
Given fractal
fractal dimension
⇐
More fractal dimensions are needed!
How many dimensions are needed – no answer today
Shortest path (chemical distance) dimension - d min
d min
l ( L)
- The fractal dimension of the shortest path defined by l (bL ) = b
d
1  1
1 f
d f : M  L  = M ( L) =   M ( L )
Example: modified Koch curve
4  7
4
log 7
d
df =
≅ 1.404, M ( L ) = AL f
log 4
d
1  1
 1  min
d min : l  L  = l ( L ) =   l ( L)
4  5
4
log 5
d min =
≅ 1.161, l = BLdmin
log 4
Shortest path dimension - d min
For Koch curve: the shortest path is the line itself
d
1  1
1 f
l  L  = l ( L) =   l ( L)
3  4
 3
log 4
df =
log3
1/3 L
Chemical dimension - dl - “how the mass scales with the shortest path”
d
Defined by: M (bl ) = b l M (l ) ,
For the modified Koch curve
M (l ) = Cl dl
d
l
1  1
1
M  l  = M (l ) =   M (l )
5  7
5
log 7
dl =
≅ 1.209, M (l ) = Cl dl
log 5
Is there a relation between dl , d min and d f ?
M ( L) = AL
df
from l = BLdmin
follows L : l1/ dmin
d /d
= A′l f min
= Cl dl ⇒ dl = d f / d min
More characteristics of fractals include: backbone, external perimeter, red bonds, etc.
3. Self-affinity
Self-similarity or scale invariance is an isotropic property, the change of scale is the
same in every direction in space.
uuuuuuur
x → 2x
Example: Sierpinski gasket
y → 2y
Self-affinity – include anisotropic symmetry magnifying x in different scale than y.
Example:
n=0
n=2
-
n=1
n=0
n=1
n=3
Here we see that to get the same picture we need to magnify the x axis by
4 and y axis by 2, x → 4 x, y → 2 y
1  1
1
M  Lx , L y  = M ( Lx , Ly )
2  4
4
f
Generalization of self-similar fractals: M (bL) = b M (L)
d
3.1 Fractal dimension – self-affine structures
Here we need to define two fractal dimensions
d xf
M (aLx , bLy ) = a M ( Lx , Ly )
d fy
= b M ( Lx , Ly )
Example:
d xf
1  1
1
1
M  Lx , Ly  = M ( Lx , L y ) =   M ( Lx , Ly )
2  4
4
 4
d fy
1  1
1
1
M  Lx , Ly  = M ( Lx , L y ) =   M ( Lx , Ly )
2  4
4
 2
1
 
4
d xf
1
 
2
d fy
=
1
⇒ d xf = 1,
4
=
1
⇒ d fy = 2
4
Example: self-affine Sierpinski carpet
d xf
1  1
1
1
M  Lx , Ly  = M ( Lx , Ly ) =   M ( Lx , Ly )
2  3
3
 3
d fy
1
=   M ( Lx , Ly )
2
d xf = 1, d fy =
log 3
log 2
Self affine fractals
Here also
d xf
1  1
1
1
M  Lx , Ly  = M ( Lx , Ly ) =   M ( Lx , Ly )
2  3
3
 3
dy
1 f
=   M ( Lx , Ly ),
2
Generalization:
Start with a square of unit size:
(a)
Divide x axis to b1 and y axis to b2
(b)
We get rectangulars of size (1/ b1 ) × (1/ b2 )
(c)
Number of rectangulars b1 × b2
(d)
Keep n rectangulars and remove b1 × b2 − n of them (above: n=3,b2
d fx = 1, d fy =
=2, b1 =3)
(e) To each rectangular left full, apply the same rule.
The fractal dimension:
{

b2 =3

b1 =5
d xf
1
1
1  1
M  Lx , L y  = M ( Lx , L y ) =   M ( Lx , L y )
b2  n
 b1
 b1 
d fy
1
=   M ( Lx , Ly ),
 b2 
log 3
log 2
d xf =
log b1
log b2
, d fy =
log n
log n
3.2 Local dimension – box dimension
Alternative definition of dimension is self-affine using box dimension
Example:
1 1
1
, 2 ,K n
- Chose a square box of linear size
3 3
3
- How many boxes are needed to cover the fractal?
For size
1
we need
3
1 1
3⋅ ⋅
3 2 = rectangular area × number of rectangulars
2
box area
1
 
3
1 1
⋅ k
k
1
3 2 boxes. More general: if we divide to
In general for k we need N (ε ) =
2
3
 1
 k
3 
b1 × b2 rectangulars and leave n of them full, we obtain a box of size ε = b1− k and the
−k
−k
k b1 ⋅ b2
number of boxes N (ε ) = n
−k 2
b1
3k ⋅
( )
nb1
nb
ln 1
ln N (ε )
b2
b1
−dl
=
=
The local box dimension: N (ε ) = ε f , d lf =
1
k ln b1
ln b1
ln
ε
k ln
Self affine curves – single valued
Example:
Alternative definition of dimension:
Denote L – linear scale in x-direction
Denote W – linear scale in y-direction
W
α
Dimension α defined by W (bL) = b W ( L)
The dimension α is also called roughness exponent
For the above fractal
α
1  1
1
W  L  = W ( L) =   W ( L)
4  2
4
log 2 1
⇒α =
=
log 4 2
For the fractal
L
α
1  1
1
W  L  = W ( L) =   W ( L)
5  3
5
log 3
α=
≅ 0.683
log 5
Random Fractals
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Fractals do not have to be deterministic
One can generate random fractals
Instead of always removing the central square, we remove randomly one of
the 9 squares
Random Sierpinski carpet
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Deterministic Sierpinski carpet
The fractal dimension of the random Sierpinski carpet is the same as the
d
log8
1  1
1 f
≅ 1.893
deterministic: M  L  = M ( L) =   M ( L), d f =
3
8
3
log
3


 
The self-similarity is not exact – valid statistically
Random Fractals – Fractal Dimension
Methods: (a) sand box; (b) box counting; (c) correlations.
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4.1 Sand Box method
Choose a site on the fractal – origin
plot circles of several radiuses r = Rmax
Rmax : radius of the fractal
count the number of sites inside r
repeat the measurements for several origins
average over all results for each r - M (r )
plot M (r ) vs r on log-log plot
the slope is d f of the fractal
M ( r ) = Ar f , log M ( r ) = log A + d f log r
d
This method is analogous to the determination of d f in deterministic fractals.
How the mass M scales with the linear metric r .
4.2 Box counting method
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Draw a lattice of squares of different sizesε
For each ε count the number of boxes N (ε )
needed to cover the fractal
N (ε ) increases with decreasing ε
The fractal dimension is obtained from
N (ε ) = Aε
−d f
log N (ε ) = log A − d f log ε
-
Plotting N (ε ) vs ε on log-log graph –
the slope is −d f
4.3 Correlation method
Measurements of the density-density autocorrelation function
1
C (r ) = ρ (r ′ ) ρ (r ′ + r ) r ′ = ∑ ρ (r ′ ) ρ (r ′ + r )
V r′
1 if at r′ there is a site of the fractal
′
ρ (r ) = 
 0 if at r′ there is no site
The volume V = ∑ ρ (r′) .
r′
C (r ) is the average density at distancer from a site on a fractal.
−α
For isotropic fractals we expect C (r ) = C (r ) = Ar .
The mass within a radius R is:
R
M ( R) = ∫ C (r )d d r = R −α + d ≡ R
0
⇒ α =d −df
Thus, from measuring α one can determine d f .
df
4.4 Experimental method
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Scattering experiments like x-rays, neutron scattering etc. with different
wave vectors is proportional to the structure factor.
The structure factor is the Fourier transform of the density-density
correlation function.
For fractals – the structure factor is
−d
S (q ) = S ( q ) = q f
4π
q=
sin ϑ is the wave vector.
λ
Since physical fractals have lower and upper bounds length scales
( λ− and λ+ )
4π
4π
sin α < q <
sin ϑ , we obtain d f
λ+
λ−
Measurements of S (q ) yields d f
Example: polymers.
It follows that only for
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