Fractales bartolo luque - curso de introduccion sistemas complejos

Fractales

Bartolo Luque

GaborCsordas y Gabor Papp

3

El efecto Droste

Tal vez la forma más elemental y pri-mitiva de recursividad sea el efectoDroste: una imagen que contiene unaréplica en miniatura de sí misma.

El nombre proviene de una popularmarca de chocolates de los PaísesBajos que, a principios del siglo xx,empleó este efecto en una de susimágenes publicitarias. En ellaaparecía una enfermera que portaba,justamente, una caja de cacaoDroste decorada con una réplica enminiatura de la imagen original. Asípues, en la caja aparecía otra vez laenfermera, la cual llevaba otra caja, yasí sucesivamente.

Diseño publicitario

Visage of War, Salvador Dali (1940)

Geometrical Self-Similarity

Geometrical Self-SimilarityThe magnified piece of an object is an exact copy of the whole object.

SierpinskiTriangle.exe

zoom in

and rescale


zoom in

and rescale


Cosas raras: el perímetroKoch snowflake

nnN 43)( ⋅=

nnL )3/1()( =

nnLnNnP )3/4(3)()()( ⋅==

3)0(1)0(==

==

nNnL

∞→n ∞

KochCurve.exe

14

"I coined fractal from the Latin adjective fractus. The corresponding Latin verb frangere means "to break": to create irregular fragments. It is therefore sensible - and how appropriate for our needs! - that, in addition to "fragmented"(as in fraction or refraction), fractus should also mean "irregular", both meanings being preserved in fragment."

(The Fractal Geometry of Nature)

La palabra latina fractus significa quebrado. En palabrasde Benoit Mandelbrot:

Benoit Mandelbrot (1924-2010)

The Cantor Set is the dust of points obtained as the limit of this succession of

segments

This is already the limit of

succession of iterations

Más cosas raras: Curva de Peano

¿Tiene entonces la curva dimensión 1 o dimensión 2?¿Tiene sentido esta pregunta?

Objects in mirror are closer than they appear.

Monsters inSci-Fi

King Kong (1933) Them (1954) Godzilla (1954)Record: 120 m

Tarantula (1955)

The deadlymantis(1957)

20

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Ley cuadrado cúbica

Cuando un objeto crece sin cambiar de forma, su superficie crece como el

cuadrado de alguna longitud característica

(por ejemplo, su altura) mientras que el volumen crece como el cubo de dicha

cantidad.

Galileo (1564-1642)

¿Qué se podemos deducir de la ley?

3

2

~)(~)(rrVrrS

3

2

8~)2(4~)2(rrVrrS⋅⋅

⋅⋅

19571958

250 Hz150 Hz

ρρ

µ SLSL

==

211 −∝∝ LSL

ν

µν

TL21

=

Allometry is the study of the relationship between size and shape.

2−∝ Lν

2−∝ Lν

DimensionTopological Dimension

• Points (or disconnected collections of them) have topological dimension 0.

• Lines and curves have topological dimension 1.• 2-‐D things (think filled in square) have topological dimension 2.• 3-‐D things (a solid cube) have topological dimension 3.

intuitive: length, area, volume

rescale bya factor b

length s

Fractal vs. integer dimension

b ·s

b2·Aarea A

intuitive: length, area, volume

rescale bya factor b

length s

b2·Aarea A


b1·s

D

Dimensions of objects• Consider objects in 1, 2 and 3 dimensions:

D = 1 D = 2 D = 3

• Reduce length of ruler by factor, r

r = 1/2

N = 2N = 4

N = 8

• Quantity increases by N = (1/r)D

r = 1/2

r =1/3

N = 2

N = 3

N = 4

N = 9

N = 8

N = 27

( )( )rND/1log

log=

( )( )

( )( )

13log3log

2log2log

===D ( )( )( )

23log9log

2log)4log(

===D( )

( )( )

33log27log

2log)8log(

===D

1 1

r N

Sierpinsky revisited

1 1

r N

1/2 3


1 1

r N

1/2 3

1/4 9


1 1

r N

1/2 3

1/4 9

1/8 27

k

0

1

2

3

r = 2-kN = 3k

Sierpinsky revisitedN = (1/r)D

( )( )rND/1log

log=

( )( )

( )( )2log3log

2log3log

== k

k

D


585.1)2log()3log( D ≈=

“more than a line – less than an area”

What’s special about fractals is that the “dimension” is not necessarily a whole number

“Clouds are not spheres,mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straightline.”

Benoit B. Mandelbrot

Geometric scale invariance and fractal geometry

«Un fractal es un objeto matemático cuya dimensión de Hausdorff es siempre mayor a su dimensión topológica».

Koch island:

scale byfactor b=3

length s

length 4 s

2619.1)3log()4log( D ≈=


N(ε) = 2k where k is the iteration And ε =(1/3)k

D=ln(2)/ln(3) = 0.6309…

N(ε) = 8k where k is the iteration And ε =(1/3)k

D=ln(8)/ln(3) = 1.8927…

The Cantor Set is the dust of points obtained as the limit of this succession

of segments

This is already the limit of succession of iterations

N = (1/r)D

Self-similarity in nature

Romanesco –a cross between broccoli

and cauliflower



Fractal concepts characterize those objects in which properly scaled portions are identical to the original object. Can be identical in deterministic or statistical sense.

Self-Similarity: Geometrical and Statistical

La gran ola de Kanagawa

Scale Laws... Power Laws

α−⋅= rBrQ )(

......... 2−∝ Lν

Q (r) Log Q (r)

r Log r

BrrQ loglog)(log +−= α

How long is the coast of Britain?

Suppose the coast of Britain is measured using a 200 km ruler, specifying that both ends of the ruler must touch the coast. Now cut the ruler in half and repeat the measurement, then repeat again:

B. B. Mandelbrot, Science’1967

Scale-dependent length.

Compass o ruler method:

How Long is the Coastline of Britain?

r = Length of Line Segments in KmQ(r) = N(r) r = Total Length in Km

r r

How Long is the Coastline of Britain?Richardson 1961 The problem of contiguity: An Appendix to Statistics

of Deadly Quarrels General Systems Yearbook 6:139-187

Log 10(Total Length in Km)

CIRCLE

SOUTH AFRICAN COAST

4.0

3.5

3.0

1.0 1.5 2.0 2.5 3.0 3.5Log10 (Length of Line Segments in Km)

Scaling

The value measured for a property, such as length, surface, or volume, depends on the resolution at which it is measured.

How depends is called the scaling relationship.

How Long is the Coastline of Britain?Richardson 1961 The problem of contiguity: An Appendix to Statistics

of Deadly Quarrels General Systems Yearbook 6:139-187

Log 10(Total Length in Km)

CIRCLE

SOUTH AFRICAN COAST

4.0

3.5

3.0

1.0 1.5 2.0 2.5 3.0 3.5Log10 (Length of Line Segments in Km)

25.0)( −∝ rrL

Statistical Self-SimilarityIn real world are usually not exact smaller copies of the whole object. The value of statistical property Q(r) measured at resolution r, is proportional to the value Q(ar) measured at resolution ar.

Q(ar) = kQ(r)pdf [Q(ar)] = pdf [kQ(r)]d

)()()(;)(

25.025.025.025.0

25.0

rLarAaraAraLrArL

⋅=⋅⋅=⋅⋅=⋅

⋅=−−−−

−

Self-Similarity Implies a Scaling Relationship

Q (r) = B rb

Q (ar) = k Q(r) Q (r) = B rb

Self-Similarity can be satisfied by the power law scaling, the simplest and most common form of the scaling relationship:

Proof: using the scaling relationship to evaluate Q(r) and Q(ar)

Q (r) = B rb Q (ar) = B ab rb

if k = ab then Q (ar) = k Q (r)

Power Lawmeasurement

r Log rLogarithm

of

the measuremnt

Resolution used to make the measurement

Logarithm of the resolution used to make the measurement

Such power law scaling relationships are characteristic of fractals. Power lawrelationships are found so often because so many things in nature are fractal.

Scale Laws and Power Laws

α−⋅= rBrQ )( BrrQ loglog)(log +−= α

Mass�(Perimeter)3Double the size � Octuple MassDimension = 3

Solid Spheres"Euclidean Object"

33

234

34~

2

⎟⎠

⎞⎜⎝

⎛==

=

πππρ

π

PRVM

RP

Crumbled Paper Balls"Non-Euclidean Objects"

M.A.F. Gomes, “Fractal geometry in crumpled paper balls”Am.J.Phys. 55, 649-650 (1987).

R.H.Ko and C.P.Bean, “A simple experiment that demonstrates fractal behavior”, Phys. Teach. 29, 78 (1991).

Crumbled Paper Balls"Non-Euclidean Objects"

Mass�(Perimeter)Dimensionlog(Mass) �Dimension log(Perimeter)

L.H.F. Silva and M.T. Yamashita, “The dimension of the pore space in sponges,” European Journal of Physics 30: 135-137, 2009 .

Por cierto, los geólogos suelen utilizar este tipo de idea para caracterizar la porosidad de rocas y su permeabilidad (Alexis Mojica, Leomar Acosta, “La porosidad de las rocas y su naturaleza fractal,” Invet. pens. crit. 4: 88-93, 2006 ).

Se recortan muchos cubitos de esponja de lado progresivamente mayor, por ejemplo, desde 1 cm de lado, 2 cm, 3 cm, hasta donde podamos. Pesamos las esponjas con una balanza, luego las sumergimos en agua y las volvemos a pesar. La diferencia de masa entre la esponja seca y la mojada. Dibujando esta diferencia en función del lado en escala doblemente logarítmica se observará que la dimensión fractal de la esponja es D = 2.95, menor que 3, resultado de la existencia de los poros.

Object � SetProperty � Distribution

Mean size o characteristic size

66

What is the normal length of a penis?

67

While results vary across studies, the consensus is that the mean human penis is approximately 12.9 – 15 cm in length with a 95% confidence interval of (10.7 cm, 19.1 cm).

Mean

Non - Fractal

More Data

69

Fractal?

Self-similarity in geology

From: D. Sornette, Critical Phenomena in Natural Sciences (2000)

Self-similarity in geology

From: D. Sornette, Critical Phenomena in Natural Sciences (2000)

Cloud perimeters over 5 decades yield D ≈ 1.35 (Lovejoy, 1982)

Power laws, Pareto distributions and Zipf’s lawM. E. J. Newman

WWW Nodes: WWW pages

Links: URL linksP(k) ~ k- 2.1Scale-Free Networks

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The Average Depends on the Amount of Data Analyzed

each piece

The Average Depends on the Amount of Data Analyzed

oraverage size

number of pieces included

average size

number of pieces included

Contributions to the mean dominated by the number of smallest sizes.

Contributions to the mean dominated by the number of biggest sizes.

0→µ ∞→µ

Non-FractalLog avg

density within radius r

Log radius r

Fractal

Log avgdensity within radius r

Log radius r

.5

-1.0

-2.0

-1.5

.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.00-2.5

0

Meakin 1986 In On Growth and Form: Fractal and Non-Fractal Patterns in Physics Ed. Stanley & Ostrowsky, Martinus Nijoff Pub., pp. 111-135

When the moments, such as the mean and variance, don’t exist, what should I measure? The exponent...

Fractals in NatureElectrical Discharge from Tesla Coil

Fractals in Nature

Lichtenberg Figure

Created by exposing plastic rod to electron beam & injecting chargeinto material. Discharged by touching earth connector to left hand end

Viscous fingeringElectrodeposition

Diffusion-limited aggregation (DLA)

T.A. Witten, L.M. Sander 1981

Statistical scale invariance of DLA

P. Meakin, Fractals, scaling and growth far from equilibrium

Mass-length relation M1�R1D M2�R2D

Fractal

Log avgdensity within radius r

Log radius r

.5

-1.0

-2.0

-1.5

.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.00-2.5

0

Meakin 1986 In On Growth and Form: Fractal and Non-Fractal Patterns in Physics Ed. Stanley & Ostrowsky, Martinus Nijoff Pub., pp. 111-135

When the moments, such as the mean and variance, don’t exist, what should I measure? The exponent...

Box-counting

∙ Cover the object by boxes of size ∊

< ∊ >

∙ count non-empty boxes

∙ repeat for many ∊

Measuring fractal dimension

∙ cover the object by boxes of size ∊

<∊>



box-counting: resolution-dependent measurement

Measuring fractal dimension

∙ cover the object by boxes of size ∊



box-counting: resolution-dependent measurement

∙ consider the number n of non-empty boxesas a function of ∊

(in the limit ∊→0)

Fractals and Chaos.Larry S. Liebovitch.

Fractals, Chaos, Power Laws.Manfred Schroeder

Education

Fractales bartolo luque - curso de introduccion sistemas complejos