modelling and simulation 2008

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


Self-similarity in nature

identical/similarstructures repeatover a wide rangeof length scales


Self-similarity in nature


mosaic from the cathedral of Anagni / Italy

Self-similarity in art


an artificial, fractal landscape

Self-similarity in computer graphics


Self-similarity in physics

Clusters of Pt atoms Diffusion limited aggregation


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


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 )


The fractal is defined in the

mathematical limit of

infinitely many iterations

Fractal objects: iterative construction

( 8 )( ∞ )


Fractal objects: properties

(a) self-similarity

∙ exactly the same structures

repeat all over the fractal

zoom inand rescale



(a) self-similarity

∙ exactly the same structures

repeat all over the fractal

zoom inand rescale



(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”)


Scale invariance

1m


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


intuitive: length, area, volume

rescale bya factor b

length s


b ·s

b2·Aarea A


intuitive: length, area, volume

rescale bya factor b

length s

b2·Aarea A


b1·s

D


working definition of dimension D:


- 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)



b=2

aspect: black area

D1b 23 A / A

585.1)2log(

)3log( D

“more than a line – less than an area”



∙ initialization: 3 lines forming a triangle

another (famous) example: Koch islands

∙ iteration: replace every straight line

by a, e.g. a spike

first iteration:


Koch island:



Koch island:

scale byfactor b=3

length s

length 4 s

D1b 34 A / A

2619.1)3log(

)4log( D



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


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


Stochastic fractals

repeating structures of equal statistical properties

leng

th s

cale

?


Measuring fractal dimension

Box-counting: resolution-dependent measurement of D

∙ cover the object by boxes of size ∊

< ∊ >

∙ count non-empty boxes

∙ repeat for many ∊




<∊>



box-counting: resolution-dependent measurement






box-counting: resolution-dependent measurement

∙ consider the number n of non-empty boxes as a function of ∊ (in the limit ∊→0)


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/∊)


area A


Sierpinsky revisited

suitable shape of boxes ?


1 1

∊ n



1 1

∊ n

1/2 3



1 1

∊ n

1/2 3

1/4 9



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=


n ~ (1/∊)D


- 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


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

modelling and simulation 2008

Documents