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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 .

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slide1

Modelling and Simulation 2008

A brief introduction to self-similar fractals

slide2

Outline

Motivation:

- examples of self-similarity

Fractal objects:

- iterative construction of geometrical fractals

- self-similarity and scale invariance

Fractal dimension:

- conventional vs. fractal dimension

- a working definition

- the box-counting method

slide3

Self-similarity in nature

identical/similar

structures repeat

over a wide range

of length scales

slide5

Self-similarity in art

mosaic from the cathedral of Anagni / Italy

slide6

Self-similarity in computer graphics

an artificial, fractal landscape

slide7

Self-similarity in physics

Diffusion limited aggregation

Clusters of Pt atoms

slide8

Self-similar time series

medicine:

further examples:

heart beat intervals

economy (e.g. stock market)

weather/climate

seismic activity

chaotic systems

random walks

Heart beat intervals

time

beat number

slide9

Fractal objects: iterative construction

The Sierpinsky construction

∙initialization: one filled triangle

∙ iteration step:

remove an upside-down

triangle from the center of

every filled triangle

∙repeat the step ...

( 3 )

( 2 )

( 1 )

slide10

Fractal objects: iterative construction

The fractal is defined in the

mathematical limit of

infinitely many iterations

( ∞ )

( 8 )

slide11

Fractal objects: properties

(a) self-similarity

∙ exactly the same structures

repeat all over the fractal

zoom in

and rescale

slide12

Fractal objects: properties

(a) self-similarity

∙exactly the same structures

repeat all over the fractal

zoom in

and rescale

slide13

Fractal objects: properties

(b) scale invariance:

∙there is no typical …

… size of objects

… length scale

Sierpinsky:

contains triangles of

all possible sizes

apart from “practical” limitations:

- size of the entire object

- finite number of iterations (“resolution”)

slide15

D=1

D=2

Fractal vs. integer dimension

Embedding dimensiond

in a d-dimensional space:

d numbers specify a point

y

D=0

Example: d=2

Dimension (of an object) D

x

in a d-dimensional space,

all objects have a dimension D ≤ d

slide16

area A

b·s

b2·A

Fractal vs. integer dimension

intuitive: length, area, volume

rescale by

a factor b

length s

slide17

b2·A

b1·s

Fractal vs. integer dimension

intuitive: length, area, volume

rescale by

a factor b

length s

D

area A

slide18

dimension D of aspect A(Q)

Fractal vs. integer dimension

working definition of dimension D:

  • object Q, embedded in a d-dimensional space
  • measure aspect A(Q), e.g. perimeter, area, volume,…
  • compare results

A(Q) = A1 in the original space

A(Q) = Ab after rescaling all d directions by b

slide19

b=2

Fractal vs. integer dimension

aspect: black area

“more than a line – less than an area”

slide20

Fractal vs. integer dimension

another (famous) example: Koch islands

∙ initialization: 3 lines forming a triangle

∙iteration: replace every straight line

by a, e.g. a spike

first iteration:

slide22

Fractal vs. integer dimension

Koch island:

length s

scale by

factor b=3

length 4 s

slide23

Summary

∙ introduction: self-similar objects

∙ construction of example fractals:

- the Sierpinsky construction

- Koch islands

∙ qualitative properties of fractal objects:

- self-similarity

- scale invariance

∙ quantitative characterization of fractals:

- fractal dimension (vs. integer dimension)

- working definition / measurement

slide24

Problems

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 ?
slide25

Stochastic fractals

repeating structures of equal statistical properties

length scale ?

slide26

Measuring fractal dimension

Box-counting: resolution-dependent measurement of D

∙ cover the object by

boxes of size ∊

∙ count non-empty boxes

∙ repeat for many ∊

< ∊ >

slide27

Measuring fractal dimension

box-counting: resolution-dependent measurement

∙ cover the object by

boxes of size ∊

∙ count non-empty boxes

∙ repeat for many ∊

<∊>

slide28

Measuring fractal dimension

box-counting: resolution-dependent measurement

∙ cover the object by

boxes of size ∊

∙ count non-empty boxes

∙ repeat for many ∊

∙ consider the number

n of non-empty boxes

as a function of ∊

(in the limit ∊→0)

slide29

Measuring fractal dimension

n ~ ( 1/∊ ) D ( as ∊→0 )

obtain D from

D = log(n) /log(1/∊)

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

area A

slide30

Sierpinsky revisited

suitable shape of boxes ?

slide31

n

1 1

Sierpinsky revisited

slide32

n

1 1

Sierpinsky revisited

1/2

3

slide33

n

1 1

Sierpinsky revisited

1/2

3

1/4

9

slide34

n

k

0

1 1

1

2

3

1/∊ =2 k

n =3 k

k log(3)

k log(2)

D=

Sierpinsky revisited

n ~ (1/∊)D

1/2

3

1/4

9

1/8

27

slide35

Remarks / Outlook

in practice: linear regression ln(n) vs. ln(1/∊)

for a range of box sizes

-

  • Box-counting is only one method for estimating D,
  • widely applicable, but costly to realize
  • important alternatives: Sandbox-method
  • correlation functions
  • 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
slide36

Outlook

  • 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