1 Fract als Coined by Benoit Mandelbr ot To dif f erent iat e f - - PDF document

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1 Fract als Coined by Benoit Mandelbr ot To dif f erent iat e f - - PDF document

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5/11 - 04 Emergent Systems, Jonny Pettersson, UmU 1 Last t ime Concept s Emergence, emer gent syst ems, Lif e Real lif e Ar t if icial lif e Topics 5/11 - 04 Emergent Systems, Jonny Pettersson, UmU 2 Out line f

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5/11 - 04 1 Emergent Systems, Jonny Pettersson, UmU 5/11 - 04 2 Emergent Systems, Jonny Pettersson, UmU

Last t ime

❒ Concept s

❍ Emergence, emer gent syst ems, …

❒ Lif e

❍ Real lif e ❍ Ar t if icial lif e

❒ Topics

5/11 - 04 3 Emergent Systems, Jonny Pettersson, UmU

Out line f or t oday

❒ Fract als ❒ Net Logo ❒ Assignment 1

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Fract als

❒ Coined by Benoit Mandelbr ot ❒ To dif f erent iat e f rom pure geomet ric

f igures

❒ Two int erest ing qualit ies

❍ Self -similar on mult iple scales ❍ Fr act ional dimension

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Examples of Fract als

❒ The Cant or Set ❒ The Koch Cur ve ❒ The P

eano Cur ve

❒ Fr act ional dimension

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Random Fract als

❒ Random pr ocesses in nat ur e ar e of t en self -similar on

var ying t empor al and spat ial scale

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Brownian Mot ion and Whit e Noice

❒ Br ownian Mot ion

❍ Part icles in liquids

❒ Whit e Noice

❍ Decribe t hings belived

t o be f ormed by r andom walk-like processes

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Dif f usion Limit ed Aggregat ion

❒ Par t icles wit h Br ownian mot ion st op moving when

t hey t ouch st at ionar y obj ect s

❒ 2-dimensional ❒ 3-dimensional

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Fract al growt h

❒ Fract als ar e ef f ect ive at compressing inf o ❒ Nat ural f ract als ❒ Must gr ow!

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Lindenmayer Syst ems

❒ Arist id Lindenmayer , 1968 ❒ Mat hemat ical descript ion of plant growt h ❒ Very compact ❒ Axiom: seed cell ❒ Product ion r ules: describe gr owt h ❒ St rings can be int erpret ed

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Turt le Graphics

❒ Seymour Papert ❒ A simple comput er language t hat children

could use t o draw graphical pict ures

❒ Net Logo is an ext ent ion of t his

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L-syst ems

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Linear Algebr a

❒ Translat ion ❒ Scaling ❒ Ref lect ion ❒ Rot at ion ❒ Composing

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The Mult iple Reduct ion Copy Machine Algorit hm

❒ Uses 2 or mor e

linear t r ansf ormat ions

❒ P

r oblem:

❍ n = # t ransf orm ❍ d = dept h ❍ nd

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I t erat ed Funct ional Syst ems

❒ Michael Barnsley

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Nonlinear Fr act als

❒ I t erat ive dynamical syst ems ❒ Complex number s

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The Mandelbrot Set

❒ xt + 1 = xt 2 + c, x0 = 0 + i0 = 0 ❒ Quest ions:

❍ Wit h c = const ant complex number , what

happens t o xt when t goes t o inf init y?

❍ What values of c makes xt diver ges? ❍ (I f a2 + b2 >

4, t hen xt diver ges)

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Mandelbrot - Algorit hm

  • For each number , c, in a subset or t he complex plane
  • Set x0 = 0
  • For t = 1 t o t max
  • Comput e xt = xt

2 + c

  • I f | xt| >

2, t hen break out of loop

  • I f t <

t max, t hen color point c whit e

  • I f t = t max, t hen color point c black

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Mandelbrot - I nf init y

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Mandelbrot – Self -similar

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The Mast er J ulia Set

  • Set c t o some const ant complex value
  • For each number, x0 , in a subset of t he complex plane
  • For t = 1 t o t max
  • Comput e xt = xt

2 + c

  • I f |xt| >

2, t hen break out of loop

  • I f t <

t max, t hen color point c whit e

  • I f t = t max, t hen color point c black

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A Myst ery of t he M-set

❒ David Boll, 1991 ❒ Want ed t o conf ir m

t hat t he “neck” of t he M-set at c = -3/ 4 + 0i is 0 in t hickness

❒ Test ed: c = -3/ 4 + εi ❒ What is π doing t here?

31415928 00000001 3141593 0.000001 314160 0.00001 31417 0.0001 3143 0.001 315 0.01 33 0.1 I t erat ions ε

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Out line

❒ ❒ ❒ Fract als

Fract als Fract als

❒ Net Logo ❒ Assignment 1

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Net Logo

❒ A mult i-agent modeling language ❒ A parallel ext ension of Logo

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Assignment 1

❒ The assignment ❒ The report ❒ Rules f or assignment s

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Rules f or assignment s

❒ ht t p:/ / www.cs.umu.se/ inf ormat ion/ Labregl

erV3.ht m

❒ Grades: G, O, U, (K, F) ❒ I n t he case of O – only one chance t o f ix

❍ The t ime f or cor rect ing an O is set individually

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Assignment s – Mor al and knowledge

❒ What is knowledge?

❍ I nf ormat ion – Knowledge - Skills

❒ How do one get knowledge? ❒ How do one get skills? ❒ How t o share t he work?

❍ Responsibilit ies t o each ot her ❍ Responsibilit ies t o one self

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Summar y

❒ Fract als ❒ Net Logo ❒ Assignment 1

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Next t ime

❒ Chaos ❒ Producer-consumer dynamics