Fractales bartolo luque - curso de introduccion sistemas complejos

Fractales
Bartolo Luque
Gabor
Csordas y
Gabor Papp

3
El efecto Droste
Tal vez la forma más elemental y pri-
mitiva de recursividad sea el efecto
Droste: una imagen que contiene una
réplica en miniatura de sí misma.
El nombre proviene de una popular
marca de chocolates de los Países
Bajos que, a principios del siglo xx,
empleó este efecto en una de sus
imágenes publicitarias. En ella
aparecía una enfermera que portaba,
justamente, una caja de cacao
Droste decorada con una réplica en
miniatura de la imagen original. Así
pues, en la caja aparecía otra vez la
enfermera, la cual llevaba otra caja, y
así sucesivamente.

Visage of War, Salvador Dali (1940)

Geometrical Self-Similarity

The magnified piece of an object is
an exact copy of the whole object.
SierpinskiTriangle.exe

zoom in
and rescale

Cosas raras: el perímetro
Koch snowflake
n
nN 43)( ⋅=
n
nL )3/1()( =
n
nLnNnP )3/4(3)()()( ⋅==
3)0(
1)0(
==
==
nN
nL
∞→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 appropriatefor 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 palabras
de 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 in
Sci-Fi

King Kong (1933) Them (1954) Godzilla (1954)
Record: 120 m
Tarantula (1955)
The deadly
mantis
(1957)

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
~)(
~)(
rrV
rrS
3
2
8~)2(
4~)2(
rrV
rrS
⋅⋅
⋅⋅

ρ
ρ
µ S
L
SL
==
211 −
∝∝ L
SL
ν
µ
ν
T
L2
1
=
Allometry is the study of
the relationship between
size and shape.

Dimension
Topological
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 by
a factor b
length s
Fractal vs. integer dimension
b ·s
b
2·A
area A

intuitive: length, area, volume
rescale by
a factor b
length s
b
2·A
area A
b
1·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 = 2
N = 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
( )
( )r
N
D
/1log
log
=
( )
( )
( )
( )
1
3log
3log
2log
2log
===D
( )
( )
( )
2
3log
9log
2log
)4log(
===D
( )
( )
( )
3
3log
27log
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-k
N = 3k
N = (1/r)D
( )
( )r
N
D
/1log
log
=
( )
( )
( )
( )2log
3log
2log
3log
== 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 straight
line.”
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 by
factor 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
Self-similarity in nature

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 Km
Q(r) = N(r) r = Total Length in Km
r r

Richardson 1961 The problem of contiguity: An Appendix to Statistics
of Deadly Quarrels General Systems Yearbook 6:139-187
Log10(Total Length in Km)
CIRCLE
SOUTH AFRICAN COAST
4.0
3.5
3.0
1.0 1.5 2.0 2.5 3.0 3.5
Log10 (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.

Richardson 1961 The problem of contiguity: An Appendix to Statistics
of Deadly Quarrels General Systems Yearbook 6:139-187
Log10(Total Length in Km)
CIRCLE
SOUTH AFRICAN COAST
4.0
3.5
3.0
1.0 1.5 2.0 2.5 3.0 3.5
Log10 (Length of Line Segments in Km)
25.0
)( −
∝ rrL

Statistical Self-Similarity
In 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
rLarAaraAraL
rArL
⋅=⋅⋅=⋅⋅=⋅
⋅=
−−−−
−

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 Law
measurement
r Log r
Logarithmof
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 law
relationships 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)3
Double the size Octuple Mass
Dimension = 3
Solid Spheres
"Euclidean Object"
3
3
23
4
3
4
~
2
⎟
⎠
⎞
⎜
⎝
⎛
==
=
π
ππρ
π
P
RVM
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)Dimension
log(Mass) Dimension log(Perimeter)

L.H.F. Silva and M.T. Yamashita, “The dimension of the pore space in sponges,”
European Journalof 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 Set
Property 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

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 law
M. E. J. Newman

WWW Nodes: WWW pages
Links: URL links
P(k) ~ k- 2.1
Scale-Free Networks

The Average Depends on the
Amount of Data Analyzed
each piece

The Average Depends on the
Amount of Data Analyzed
or
average
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-Fractal
Log avg
density within
radius r
Log radius r

Fractal
Log avg
density
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
Nature
Electrical
Discharge from
Tesla Coil

Fractals
in
Nature
Lichtenberg Figure
Created by exposing plastic rod to electron beam & injecting charge
into 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 R1
D M2 R2
D

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 boxes
as a function of ∊
(in the limit ∊→0)

Fractals and Chaos.
Larry S. Liebovitch.
Fractals, Chaos,
Power Laws.
Manfred Schroeder

Fractales bartolo luque - curso de introduccion sistemas complejos

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