Fractales bartolo luque - curso de introduccion sistemas complejos

¿Qué tienen en común los brócolis, las nubes y los cráteres meteoríticos? Todos exhiben fractalidad. Una nueva ciencia como la de los Sistemas Complejos, requería una nueva manera de caracterizar las formas: la geometría fractal. En esta charla aprenderemos qué es un fractal, dónde aparecen, dónde se usan y qué nos desvelan. Veremos que, en el fondo, la invarianza de escala, que va más allá de la geometría, es el concepto crucial.

1. Fractales)
Bartolo)Luque
Gabor
Csordasy*
GaborPapp

3. 3
El)efecto)Droste
Talvezlaformamáselementalypri2
mitivaderecursividadseaelefecto
Droste:unaimagenquecontieneuna
réplicaenminiaturadesímisma.
Elnombreprovienedeunapopular
marcadechocolatesdelosPaíses
Bajosque,aprincipiosdelsigloxx,
empleóesteefectoenunadesus
imágenespublicitarias.Enella
aparecíaunaenfermeraqueportaba,
justamente,unacajadecacao
Drostedecoradaconunaréplicaen
miniaturadelaimagenoriginal.Así
pues,enlacajaaparecíaotravezla
enfermera,lacualllevabaotracaja,y
asísucesivamente.

4. Diseño)publicitario

7. Visage of War,Salvador Dali (1940)

9. Geometrical)Self9Similarity

10. Geometrical)Self9Similarity
The)magnified)piece)of)an)object)is)
an)exact)copy)of)the)whole)object.
SierpinskiTriangle.exe

11. zoomIin
andIrescaleI
Geometrical)Self9Similarity

12. zoomIin
andIrescaleI
Geometrical)Self9Similarity

13. 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. 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 palabralatinafractussignificaquebrado. En palabras
de Benoit Mandelbrot:
Benoit Mandelbrot (1924-2010)

15. TheICantorISetIisItheIdustIofI
pointsIobtainedIasItheIlimitI
ofIthisIsuccessionIofI
segments
ThisIisIalreadyItheIlimitI
ofI
successionIofIiterations

17. Más cosas raras: Curva de Peano
¿TieneIentoncesIlaIcurvaIdimensiónI1IoIdimensiónI2?
¿TieneIsentidoIestaIpregunta?

18. Objects(in(
mirror(are(
closer(than(
they(appear.
Monsters)in
Sci9Fi

19. King(Kong((1933)Them((1954)Godzilla((1954)
Record:I120Im
Tarantula((1955)
Thedeadly
mantis
(1957)

20. 20
?

21. 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'(1564B1642)
¿Qué)se)podemos)deducir)de)la)ley?

22. 3
2
~)(
~)(
rrV
rrS
3
2
8~)2(
4~)2(
rrV
rrS
⋅⋅
⋅⋅

23. 19571958

24. 250)Hz
150)Hz

25. ρ
ρ
µ S
L
SL
==
211
−
∝∝ L
SL
ν
µ
ν
T
L2
1
=
AllometryIisItheIstudyIofI
theIrelationshipIbetweenI
sizeIandIshape.

28. 2−
∝Lν

29. 2−
∝Lν

30. 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.

31. intuitive:)length,)area,)volume
rescaleIby
aIfactorIb
lengthIs
Fractal)vs.)integer)dimension
b·s
b
2
·A
areaIA

32. intuitive:)length,)area,)volume
rescaleIby
aIfactorIb
lengthIs
b
2
·A
areaIA
Fractal)vs.)integer)dimension
b
1
·s
D

33. Dimensions.of.objects
•Consider.objects.in.1,.2.and.3.dimensions:
DI=I1 DI=I2 DI=I3
•Reduce.length.of.ruler.by.factor,.r
rI=I1/2
NI=I2
NI=I4
NI=I8

34. •Quantity.increases.by.N.=.(1/r)
D
rI=I1/2
rI=1/3
NI=I2
NI=I3
NI=I4
NI=I9
NI=I8
NI=I27
()
()r
N
D
/1log
log
=
()
()
()
()
1
3log
3log
2log
2log
===D
()
()
()
2
3log
9log
2log
)4log(
===D
()
()
()
3
3log
27log
2log
)8log(
===D

35. 1IIIIIIIIIIIIII1I
r N
Sierpinsky)revisited

36. 1IIIIIIIIIIIIII1I
r N
1/2 3
Sierpinsky)revisited

37. 1IIIIIIIIIIIIIIIIII1I
r N
1/2 3
1/4 9
Sierpinsky)revisited

38. 1IIIIIIIIIIIII1I
r N
1/2 3
1/4 9
1/8 27
k
0
1
2
3
rI=I2
2k
NII=I3
k
Sierpinsky)revisited
NI=(1/r)
DI
()
()r
N
D
/1log
log
=
()
()
()
()2log
3log
2log
3log
==
k
k
D

39. Fractal)vs.)integer)dimension
585.1
)2log(
)3log(
D ≈=
“moreIthanIaIlineI–lessIthanIanIarea”
What’sIspecialIaboutIfractalsIisIthatItheI
“dimension”IisInotInecessarilyIaIwholeInumber

40. “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.”
BenoitIB.IMandelbrot
Geometric scale invariance and fractal geometry
«UnIfractalIesIunIobjetoI
matemáticoIcuyaI
dimensiónIdeIHausdorffesI
siempreImayorIaIsuI
dimensiónItopológica».

41. KochIisland:I
scaleIby
factorIb=3
lengthIs
lengthI4Is
2619.1
)3log(
)4log(
D ≈=
Fractal)vs.)integer)dimension

42. N(ε)=I2
k
whereIkIisItheIiterationI
AndIε=(1/3)
k
D=ln(2)/ln(3)I=I0.6309…
N(ε)=I8
k
whereIkIisItheIiterationI
AndIε=(1/3)
k
D=ln(8)/ln(3)I=I1.8927…
TheICantorISetIisItheIdustIofIpointsI
obtainedIasItheIlimitIofIthisIsuccessionI
ofIsegments
ThisIisIalreadyItheIlimitIofI
successionIofIiterations
N.=.(1/r)
D

43. Self9similarity)in)nature

44. Romanesco.–
a.cross.between.broccoli.
and.cauliflower
Self9similarity)in)nature

45. Self9similarity)in)nature

46. FractalIconceptsIcharacterizeI
thoseIobjectsIinIwhichI
properly)scaledportionsIareI
identicalItoItheIoriginalI
object.ICanIbeIidentical)in)
deterministic)or)statistical)
sense.
Self9Similarity:)
Geometrical)and)Statistical

48. LaIgranIolaIdeIKanagawa

49. Scale&Laws...&Power&Laws
α−
⋅=rBrQ)(
.........
2−
∝Lν
Q&(r) Log&Q&(r)
r Log&r
BrrQ loglog)(log +−=α

50. 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.IB.IMandelbrot,IScience’1967
Scale-dependent length.

51. Compass o ruler method:
How)Long)is)the)Coastline)of)Britain?
rI=ILengthIofILineISegmentsIinIKm
Q(r)I=IN(r)IrI=TotalILengthIinIKm
r r

52. How)Long)is)the)Coastline)of)Britain?
RichardsonI1961ITheIproblemIofIcontiguity:IAnIAppendixItoIStatistics*
of*Deadly*Quarrels*General*Systems*Yearbook**6:1392187
Log
10
(Total)Length)in)Km)
CIRCLE
SOUTHIAFRICANIICOAST
4.0
3.5
3.0
1.0 1.5 2.0 2.5 3.0 3.5
Log
10(Length)of)Line)Segments)in)Km)

53. Scaling
The*value&measuredfor*a*property,*
such*as*length,*surface,*or*volume,**
depends&on&the&resolutionat*which*it*
is*measured.*
How*depends*is*called*the*
scaling&relationship.

54. How)Long)is)the)Coastline)of)Britain?
RichardsonI1961ITheIproblemIofIcontiguity:IAnIAppendixItoIStatistics*
of*Deadly*Quarrels*General*Systems*Yearbook**6:1392187
Log
10
(Total)Length)in)Km)
CIRCLE
SOUTHIAFRICANIICOAST
4.0
3.5
3.0
1.0 1.5 2.0 2.5 3.0 3.5
Log
10(Length)of)Line)Segments)in)Km)
25.0
)(
−
∝rrL

55. Statistical)Self9Similarity
InIrealIworldIareIusuallyInotIexactIsmallerIcopiesIofItheI
wholeIobject.ITheIvalueIofIstatisticalIpropertyIQ(r)I
measuredIatIresolutionIr,IisIproportionalItoItheIvalueIQ(ar)I
measuredIatIresolutionIar.
Q(ar))=)kQ(r)
pdfI[Q(ar)]I=IpdfI[kQ(r)]
d
)()()(
;)(
25.025.025.025.0
25.0
rLarAaraAraL
rArL
⋅=⋅⋅=⋅⋅=⋅
⋅=
−−−−
−

56. Self9SimilarityImplies)a)Scaling)Relationship
Q)(r))=)B)r
b
Q)(ar))=)k)Q(r) Q)(r))=)B)r
b
Self9Similarity)can)be)satisfied)by)the)power)
law)scaling,)the)simplest)and)most)common)
form)of)the)scaling)relationship:
Proof:IusingItheIscalingIrelationshipItoIevaluateIQ(r)IandIQ(ar)
Q)(r))=)B)r
b)
Q)(ar))=)B)a
b
r
b
if))))k)=)a
b
then))))Q)(ar))=)k)Q)(r)

57. Power)Law
measurement
r Log&r
Logarithm
ofI
theImeasuremnt
ResolutionIusedItoImakeI
theImeasurement
LogarithmIofItheIresolutionI
usedItoImakeItheI
measurement
Suchpowerlawscalingrelationshipsare.characteristicof.fractals.Powerlaw
relationshipsare.foundso.oftenbecauseso.manythingsin.natureare.fractal.
ScaleLawsand&PowerLaws
α−
⋅=rBrQ)( BrrQ loglog)(log +−=α

58. Mass (Perimeter)
3
DoubleItheIsizeI OctupleIMass
DimensionI=I3
SolidISpheres
"EuclideanIObject"
3
3
23
4
3
4
~
2
!
"
#
$
%
&
==
=
π
ππρ
π
P
RVM
RP

59. CrumbledIPaperIBalls
"Non2EuclideanIObjects"
M.A.F.Gomes,I“FractalIgeometryIinIcrumpledIpaperIballs”
Am.J.Phys.I55,I6492650I(1987).
R.H.KoandIC.P.Bean,I“AIsimpleIexperimentIthatI
demonstratesIfractalIbehavior”,IPhys.*Teach.29,I78I(1991).

60. CrumbledIPaperIBalls
"Non2EuclideanIObjects"
Mass (Perimeter)
Dimension
log(Mass)I Dimensionlog(Perimeter)

64. L.H.F.ISilvaIandIM.T.IYamashita,I“TheIdimensionIofItheIporeIspaceIinIsponges,”I
EuropeanIJournalofIPhysicsI30:I1352137,I2009I.
PorIcierto,IlosIgeólogosIsuelenIutilizarI
esteItipoIdeIideaIparaIcaracterizarIlaI
porosidadIdeIrocasIyIsuIpermeabilidadI
(AlexisIMojica,ILeomarAcosta,I“LaI
porosidadIdeIlasIrocasIyIsuInaturalezaI
fractal,”IInvet.Ipens.Icrit.I4:I88293,I2006I).
SeIrecortanImuchosIcubitosIdeIesponjaI
deIladoIprogresivamenteImayor,IporI
ejemplo,desdeI1IcmIdeIlado,I2Icm,I3I
cm,IhastaIdondeIpodamos.IPesamosIlasI
esponjasIconIunaIbalanza,IluegoIlasI
sumergimosIenIaguaIyIlasIvolvemosIaI
pesar.ILaIdiferenciaIdeImasaIentreIlaI
esponjaIsecaIyIlaImojada.IDibujandoI
estaIdiferenciaIenIfunciónIdelIladoIenI
escalaIdoblementeIlogarítmicaIseI
observaráIqueIlaIdimensiónIfractalIdeIlaI
esponjaIesIDI=I2.95,ImenorIqueI3,I
resultadoIdeIlaIexistenciaIdeIlosIporos.

65. Object) Set
Property) Distribution
Mean)size)o)characteristic)size

66. 66
WhatIisItheInormalI
lengthIofIaIpenis?I

67. 67
WhileIresultsIvaryIacrossI
studies,ItheIconsensusIisIthatI
theImeanIhumanIpenisIisI
approximatelyI12.9)–15)cminI
lengthIwithIaI95%IconfidenceI
intervalIofI(10.7Icm,I19.1Icm).

68. Mean
Non)9Fractal
More)Data

69. 69
Fractal?

71. Self-similarity in geology
From:ID.ISornette,ICriticalIPhenomenaIinINaturalISciencesI(2000)

72. Self-similarity in geology
From:ID.ISornette,ICriticalIPhenomenaIinINaturalISciencesI(2000)

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

75. Power)laws,)Pareto)distributions)and)Zipf’s)law
M.IE.IJ.INewman

76. WWW Nodes:WWW pages
Links:URL links
P(k)'~'k
B2.1
Scale9Free)Networks

77. 77
?

78. The)Average)Depends)on)the)
Amount)of)Data)Analyzed
eachIpiece

79. The)Average)Depends)on)the)
Amount)of)Data)Analyzed
or
average)
size
number)of)pieces)
included
average)
size
number)of)pieces)
included
ContributionsItoItheImeanIdominatedI
byItheInumberIofIsmallestIsizes.I
ContributionsItoItheImeanIdominatedI
byItheInumberIofIbiggestIsizes.I
0→µ
∞→µ

80. Non9Fractal
LogIavg
densityIwithinI
radiusIr
LogIradiusIr

81. Fractal
LogIavg
densityI
withinI
radiusIr
LogIradiusIr
.5
91.0
92.0
91.5
.51.01.52.02.53.03.54.04.55.05.56.00
92.5
0
Meakin)1986)In)On&Growth&and&Form:&Fractal&and&Non@Fractal&Patterns&in&
Physics&Ed.)Stanley)&)Ostrowsky,)Martinus)Nijoff)Pub.,)pp.)1119135
When)the)moments,)such)as)the)mean)and)variance,)
don’t)exist,)what)should)I)measure?)The)exponent...

82. Fractals.in.Nature
Electrical
Discharge from
Tesla Coil

83. 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

84. Viscous)fingering
Electrodeposition

85. Diffusion-limited aggregation (DLA)
T.A.IWitten,IL.M.ISanderI1981I

86. Statistical scale invariance of DLA
P.IMeakin,IFractals,IscalingIandIIgrowthIfarIfromIequilibrium

87. Mass9length)relation)
M
1 R
1
D)))))
M
2 R
2
D)

88. Fractal
LogIavg
densityI
withinI
radiusIr
LogIradiusIr
.5
91.0
92.0
91.5
.51.01.52.02.53.03.54.04.55.05.56.00
92.5
0
Meakin)1986)In)On&Growth&and&Form:&Fractal&and&Non@Fractal&Patterns&in&
Physics&Ed.)Stanley)&)Ostrowsky,)Martinus)Nijoff)Pub.,)pp.)1119135
When)the)moments,)such)as)the)mean)and)variance,)
don’t)exist,)what)should)I)measure?)The)exponent...

89. Box9counting
∙CoverItheIobjectIbyI
boxesIofIsizeII∊
<II∊>
∙IcountInon2emptyIboxes
∙IrepeatIforImanyII∊

90. Measuring)fractal)dimension
∙coverItheIobjectIbyI
boxesIofIsizeII∊
<∊>
∙IcountInon2emptyIboxes
∙IrepeatIforImanyII∊
box9counting:Iresolution2dependentImeasurementI

91. Measuring)fractal)dimension
∙coverItheIobjectIbyI
boxesIofIsizeII∊
∙IcountInon2emptyIboxes
∙IrepeatIforImanyII∊
box9counting:Iresolution2dependentImeasurementI
∙IconsiderItheInumberI
nIofInon2emptyIboxes
asIaIfunctionIofI∊
(inItheIlimitI∊→0)

93. Fractalsand)Chaos.
LarryIS.ILiebovitch.
Fractals,)Chaos,)
Power)Laws.
ManfredISchroederI