modelling and simulation 2008
DESCRIPTION
Modelling and Simulation 2008. A brief introduction to self-similar fractals. Outline. Motivation: - examples of self-similarity. Fractal objects: - iterative construction of geometrical fractals - self-similarity and scale invariance. - PowerPoint PPT PresentationTRANSCRIPT
Modelling and Simulation 2008
A brief introduction to self-similar fractals
Michael Biehl, Modelling and Simulation 2008/09 2
Fractal objects: - iterative construction of geometrical fractals - self-similarity and scale invariance
Outline
Fractal dimension:
- conventional vs. fractal dimension
- a working definition
- the box-counting method
Motivation: - examples of self-similarity
Michael Biehl, Modelling and Simulation 2008/09 3
Self-similarity in nature
identical/similarstructures repeatover a wide rangeof length scales
Michael Biehl, Modelling and Simulation 2008/09 4
Self-similarity in nature
Michael Biehl, Modelling and Simulation 2008/09 5
mosaic from the cathedral of Anagni / Italy
Self-similarity in art
Michael Biehl, Modelling and Simulation 2008/09 6
an artificial, fractal landscape
Self-similarity in computer graphics
Michael Biehl, Modelling and Simulation 2008/09 7
Self-similarity in physics
Clusters of Pt atoms Diffusion limited aggregation
Michael Biehl, Modelling and Simulation 2008/09 8
Heart
beat
inte
rvals
Self-similar time series
heart beat intervals
time beat number
medicine: further examples:
economy (e.g. stock market)
weather/climate
seismic activity
chaotic systems
random walks
Michael Biehl, Modelling and Simulation 2008/09 9
Fractal objects: iterative construction
∙ initialization: one filled triangle
The Sierpinsky construction
remove an upside-down
triangle from the center of
every filled triangle
∙ iteration step:
( 1 )∙ repeat the step ... ( 2 ) ( 3 )
Michael Biehl, Modelling and Simulation 2008/09 10
The fractal is defined in the
mathematical limit of
infinitely many iterations
Fractal objects: iterative construction
( 8 )( ∞ )
Michael Biehl, Modelling and Simulation 2008/09 11
Fractal objects: properties
(a) self-similarity
∙ exactly the same structures
repeat all over the fractal
zoom inand rescale
Michael Biehl, Modelling and Simulation 2008/09 12
Fractal objects: properties
(a) self-similarity
∙ exactly the same structures
repeat all over the fractal
zoom inand rescale
Michael Biehl, Modelling and Simulation 2008/09 13
Fractal objects: properties
(b) scale invariance:
∙ there is no typical …
… size of objects
… length scale
Sierpinsky:contains triangles ofall possible sizes
apart from “practical” limitations: - size of the entire object- finite number of iterations (“resolution”)
Michael Biehl, Modelling and Simulation 2008/09 14
Scale invariance
1m
Michael Biehl, Modelling and Simulation 2008/09 15
Fractal vs. integer dimension
Embedding dimension d
in a d-dimensional space:d numbers specify a point
x
y
Dimension (of an object) D
in a d-dimensional space,
all objects have a dimension D ≤ d
Example: d=2
D=1
D=2
D=0
Michael Biehl, Modelling and Simulation 2008/09 16
intuitive: length, area, volume
rescale bya factor b
length s
Fractal vs. integer dimension
b ·s
b2·Aarea A
Michael Biehl, Modelling and Simulation 2008/09 17
intuitive: length, area, volume
rescale bya factor b
length s
b2·Aarea A
Fractal vs. integer dimension
b1·s
D
Michael Biehl, Modelling and Simulation 2008/09 18
working definition of dimension D:
Fractal vs. integer dimension
- object Q, embedded in a d-dimensional space- measure aspect A(Q), e.g. perimeter, area, volume,…
A(Q) = A1 in the original space
A(Q) = Ab after rescaling all d directions by b
- compare results
)blog(
)A / Alog( D 1b
D1b b A / A
dimension D of aspect A(Q)
Michael Biehl, Modelling and Simulation 2008/09 19
Fractal vs. integer dimension
b=2
aspect: black area
D1b 23 A / A
585.1)2log(
)3log( D
“more than a line – less than an area”
Michael Biehl, Modelling and Simulation 2008/09 20
Fractal vs. integer dimension
∙ initialization: 3 lines forming a triangle
another (famous) example: Koch islands
∙ iteration: replace every straight line
by a, e.g. a spike
first iteration:
Michael Biehl, Modelling and Simulation 2008/09 21
Koch island:
Fractal vs. integer dimension
Michael Biehl, Modelling and Simulation 2008/09 22
Koch island:
scale byfactor b=3
length s
length 4 s
D1b 34 A / A
2619.1)3log(
)4log( D
Fractal vs. integer dimension
Michael Biehl, Modelling and Simulation 2008/09 23
Summary
∙ qualitative properties of fractal objects: - self-similarity - scale invariance
∙ construction of example fractals: - the Sierpinsky construction - Koch islands
∙ quantitative characterization of fractals: - fractal dimension (vs. integer dimension) - working definition / measurement
∙ introduction: self-similar objects
Michael Biehl, Modelling and Simulation 2008/09 24
Problems with the working definition
- we measure, e.g., the black area in the Sierpinsky
fractal, only to conclude that it has no area !?
- implicitly we make use of the construction scheme,
what about “observed” fractals like the following ?
Problems
Michael Biehl, Modelling and Simulation 2008/09 25
Stochastic fractals
repeating structures of equal statistical properties
leng
th s
cale
?
Michael Biehl, Modelling and Simulation 2008/09 26
Measuring fractal dimension
Box-counting: resolution-dependent measurement of D
∙ cover the object by boxes of size ∊
< ∊ >
∙ count non-empty boxes
∙ repeat for many ∊
Michael Biehl, Modelling and Simulation 2008/09 27
Measuring fractal dimension
∙ cover the object by boxes of size ∊
<∊>
∙ count non-empty boxes
∙ repeat for many ∊
box-counting: resolution-dependent measurement
Michael Biehl, Modelling and Simulation 2008/09 28
Measuring fractal dimension
∙ cover the object by boxes of size ∊
∙ count non-empty boxes
∙ repeat for many ∊
box-counting: resolution-dependent measurement
∙ consider the number n of non-empty boxes as a function of ∊ (in the limit ∊→0)
Michael Biehl, Modelling and Simulation 2008/09 29
n ~ ( 1/∊ ) D ( as ∊→0 ) obtain D from
integer dimensional objects?
as the grid gets finer (∊→0),the shape is more accurately approximated and we obtain
n → A/∊2 i.e. D=2
D = log(n) /log(1/∊)
Measuring fractal dimension
area A
Michael Biehl, Modelling and Simulation 2008/09 30
Sierpinsky revisited
suitable shape of boxes ?
Michael Biehl, Modelling and Simulation 2008/09 31
1 1
∊ n
Sierpinsky revisited
Michael Biehl, Modelling and Simulation 2008/09 32
1 1
∊ n
1/2 3
Sierpinsky revisited
Michael Biehl, Modelling and Simulation 2008/09 33
1 1
∊ n
1/2 3
1/4 9
Sierpinsky revisited
Michael Biehl, Modelling and Simulation 2008/09 34
1 1
∊ n
1/2 3
1/4 9
1/8 27
k
0
1
2
3
1/∊ =2 k
n =3 k
k log(3) k log(2)
D=
Sierpinsky revisited
n ~ (1/∊)D
Michael Biehl, Modelling and Simulation 2008/09 35
- Box-counting is only one method for estimating D,
widely applicable, but costly to realize
- important alternatives: Sandbox-method correlation functions
Remarks / Outlook
- in deterministic self-similar fractals, all these methods yield the same D
- for “real world fractals”, results can differ significantly
- further topics: self-affine fractals, multi-fractals
- in practice: linear regression ln(n) vs. ln(1/∊)
for a range of box sizes
Michael Biehl, Modelling and Simulation 2008/09 36
Diffusion Limited Aggregation- simple, random growth process- model of various real world processes - yields self-similar aggregates with 1 < D <2- quantitative study in terms of fractal dimension
Outlook