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8/10/2019 Phy 206 Class Fractal
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Benoit Mandelbrot
- a mathematician
Fractals:
Geometrical objects with
More & more detail
on smaller scales
Self-similarity
(symmetry under scaling
or magnification)
Fractional dimensions
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Power laws with long tails
y =1
x!
! > 0( )
Hyperbola: y =
1
x
2 4 6 8 10x
0.2
0.4
0.6
0.8
1y
Bell curve: y = e!x2
2 4 6 8 10x
0.2
0.4
0.6
0.8
1y
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Separation of scales:
a basis for reductionism
A hierarchy of scales
Microscopic scales ~10!10
m
Macroscopic scales ~ 1 m
-> Usually, no need to worryabout the microscopic level
when we try to understand
macroscopic phenomena
Fractals: no separation of scales
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Symmetry under scaling
or resizingor magnification
Hyperbola
2 4 6 8 10x
0.2
0.4
0.6
0.8
1y
0.2 0.4 0.6 0.8 1x
2
4
6
8
10y
A Gaussian
2 4 6 8 10x
0.2
0.4
0.6
0.8
1y
0.2 0.4 0.6 0.8 1x
0.2
0.4
0.6
0.8
1y
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Zips law
on the English language
A power law (a hyperbola):
Frequency( ) =Constant( )
Ranking( )
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The Cantor setStage Length of
segmentNumber ofsegments
Total length =(# of seg.) ! (length of seg.)
0 1 1 1
1 1/3 2 2 ! (1/ 3)= 2 / 3 = 0.666
2 (1 / 3) ! (1 / 3)
=(1 / 3)2= 1 / 9
2 ! 2 = 22
= 4 4 ! (1 / 9) = (2/ 3)2
= 0.444
3 (1 / 3)3= 1/ 27 2 ! 2
2= 2
3= 8 8 ! (1 / 27) = (2/ 3)
3=0.296
4 (1 / 3)4= 1 / 81 2 ! 2
3= 2
4= 16
2
4! (1 / 3)
4= (2 / 3)
4=0.198
5 (1 / 3)5= 1 / 243 2 ! 2
4= 2
5= 32
25! (1 / 3)
5= (2 / 3)
5=0.132
n (1 / 3)n
2n
2n
! (1 / 3)n
= (2 / 3)n
! 0 ! 0
5 10 15 20 n
0.2
0.4
0.6
0.8
1
Length
At an infinite# of stages:
an infinite# of segments
of an infinitesimallength
but the total length = 0
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Hidden power law
Count the # of gaps for each siz
1 gap of size 1 3 = 0.333
2 gaps of size 1 3( )2
= 0.111
4 gaps of size 1 3( )3
= 0.037
# of gaps with size s( ) =1
2
1
s0.6309
Magnifyxby 3 &yby 2
-> scaling
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The Koch curve
Stage Length of
segment
Number of
segments
Total length or Perimeter =
(# of seg.) ! (length of seg.)
0 1 3 3
1 1/3 3! 4 = 12 12 ! (1 / 3)=4
2 (1 / 3) ! (1 / 3)
= (1/3)2= 1/9
3! 4 ! 4
= 3! 42= 48
48 ! (1/ 9) = 3 ! (4 / 3)2
=16 / 3 = 5.33
3 (1/3)3= 1/27 3! 43= 192 192 ! (1 / 27) = 3 ! (4 / 3)
3
= 64 / 9 = 7.11
4 (1 / 3)4 3! 4
4= 768 3 ! (4 / 3)
4
= 256 / 27 = 9.48
5 (1 / 3)5 3! 4
5= 3072 3 ! (4 / 3)
5
= 1024 / 81 =12.64
n (1/3)n= 1/3n 3! 4n
3 ! (4 / 3)n
=3 ! (1.33)n
! 0 ! !
At an infinite# of stages:
an infinite# of segments
of an infinitesimallength
but the infinitetotal length
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The Sierpinski carpet
Stage Area ofsubsquares
Number ofsubsquares
Total Area =(# of sq.) ! (area of sq.)
0 1 1 1
1 1/9 9 - 1= 8 8/9 = 0.89
2 (1 / 9) ! (1 / 9)
=(1 / 9)2=1 / 81
8! 8= 82= 64 64 / 81= (8 / 9)
2=0.79
3 (1 / 9)3= 1 / 729 8
3= 512 512 / 729 =(8 / 9)
3=0.70
4 (1 / 9)4= 1/ 6561
84= 4096 4096 / 6561=(8/ 9)
4=0.62
5 (1 / 9 )5 8
5 (8 / 9)
5= 0.55
n (1 / 9 )n
8n
(8 / 9)n
= (0.89)n
! 0 ! 0
At an infinite# of stages:
an infinite# of subsquares
of an infinitesimalareabut the total area = 0
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The box-counting dimension
A line segment
Scale factor
x
1/x Number of
boxes,N
1 1 1
1/2 2 2
1/4 4 4
1/8 8 8
1/n n n
# of boxes( ) =
1
Scale factor x( )
"# %&
1
' D = 1
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A square
Scale factor
x
1/x Number of
boxes,N
1 1 1
1/2 2 4
1/4 4 16
1/8 8 64
1/n n n2
# of boxes( ) =1
Scale factor x( )
"#
%&
2
'D
=2
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On a double logarithmic graph
Plot # of boxes vs. 1 x
Fit the data points w/ a line
D ="Rise"
"Run"
Run
RiseN
1/x 1
10
100
1000
1 10 100
N(line)N(square)
N
1/x
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The Cantor set
Scale factorx
1/x Number ofboxes,N
1 1 1
1/3 3 2
1 / 3( )2 32= 9 2
2= 4
1 / 3( )3 33= 27 2
3= 8
1 / 3( )4 3
4= 81 2
4= 16
(1 / 3)n
3n
2n
1
10
100
1000
1 10 100
N (Cantor)N (line)N (square)
N
1/x
D =
ln2
ln3=
0.6931...
1.0986...=0.6309...
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The Sierpinski carpetScale factor
x
1/x Number ofboxes,N
1 1 1
1/3 3 8
1 / 9 32= 9 8
2= 64
1/27 33= 27 8
3= 512
1/81 34= 81 8
4= 4096
1 / 3n
3n
8n
1
10
100
1000
1 10 100
N (Sierpinski carpet)N (line)N (square)
N
1/x
D = ln8ln3
= 2.0794...1.0986...
=1.8928...