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

Mar 03, 2024 •198 likes •321 views

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

  1. 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 or t oday ❒ Fract als ❒ Net Logo ❒ Assignment 1 5/11 - 04 Emergent Systems, Jonny Pettersson, UmU 3 1

  2. 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 5/11 - 04 Emergent Systems, Jonny Pettersson, UmU 4 Examples of Fract als ❒ The Cant or Set ❒ The Koch Cur ve ❒ The P eano Cur ve ❒ Fr act ional dimension 5/11 - 04 Emergent Systems, Jonny Pettersson, UmU 5 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 5/11 - 04 Emergent Systems, Jonny Pettersson, UmU 6 2

  3. 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 5/11 - 04 Emergent Systems, Jonny Pettersson, UmU 7 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 5/11 - 04 Emergent Systems, Jonny Pettersson, UmU 8 Fract al growt h ❒ Fract als ar e ef f ect ive at compressing inf o ❒ Nat ural f ract als ❒ Must gr ow! 5/11 - 04 Emergent Systems, Jonny Pettersson, UmU 9 3

  4. 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 5/11 - 04 Emergent Systems, Jonny Pettersson, UmU 10 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 5/11 - 04 Emergent Systems, Jonny Pettersson, UmU 11 L-syst ems 5/11 - 04 Emergent Systems, Jonny Pettersson, UmU 12 4

  5. 5/11 - 04 Emergent Systems, Jonny Pettersson, UmU 13 Linear Algebr a ❒ Translat ion ❒ Scaling ❒ Ref lect ion ❒ Rot at ion ❒ Composing 5/11 - 04 Emergent Systems, Jonny Pettersson, UmU 14 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 ❍ � n d 5/11 - 04 Emergent Systems, Jonny Pettersson, UmU 15 5

  6. I t erat ed Funct ional Syst ems ❒ Michael Barnsley 5/11 - 04 Emergent Systems, Jonny Pettersson, UmU 16 Nonlinear Fr act als ❒ I t erat ive dynamical syst ems ❒ Complex number s 5/11 - 04 Emergent Systems, Jonny Pettersson, UmU 17 The Mandelbrot Set ❒ x t + 1 = x t 2 + c, x 0 = 0 + i0 = 0 ❒ Quest ions: ❍ Wit h c = const ant complex number , what happens t o x t when t goes t o inf init y? ❍ What values of c makes x t diver ges? ❍ (I f a 2 + b 2 > 4, t hen x t diver ges) 5/11 - 04 Emergent Systems, Jonny Pettersson, UmU 18 6

  7. Mandelbrot - Algorit hm • For each number , c , in a subset or t he complex plane Set x 0 = 0 • For t = 1 t o t max • 2 + c • Comput e x t = x t • I f | x t | > 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 • 5/11 - 04 Emergent Systems, Jonny Pettersson, UmU 19 Mandelbrot - I nf init y 5/11 - 04 Emergent Systems, Jonny Pettersson, UmU 20 Mandelbrot – Self -similar 5/11 - 04 Emergent Systems, Jonny Pettersson, UmU 21 7

  8. The Mast er J ulia Set Set c t o some const ant complex value • For each number, x 0 , in a subset of t he complex plane • For t = 1 t o t max • 2 + c • Comput e x t = x t • I f | x t | > 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 • 5/11 - 04 Emergent Systems, Jonny Pettersson, UmU 22 5/11 - 04 Emergent Systems, Jonny Pettersson, UmU 23 A Myst ery of t he M-set ε ❒ David Boll, 1991 I t erat ions ❒ Want ed t o conf ir m 0.1 33 t hat t he “neck” of t he M-set at c = -3/ 4 + 0 i 0.01 315 is 0 in t hickness 0.001 3143 ❒ Test ed: c = -3/ 4 + ε i ❒ What is π doing t here? 0.0001 31417 0.00001 314160 0.000001 3141593 00000001 31415928 5/11 - 04 Emergent Systems, Jonny Pettersson, UmU 24 8

  9. Out line Fract als ❒ ❒ ❒ Fract als Fract als ❒ Net Logo ❒ Assignment 1 5/11 - 04 Emergent Systems, Jonny Pettersson, UmU 25 Net Logo ❒ A mult i-agent modeling language ❒ A parallel ext ension of Logo 5/11 - 04 Emergent Systems, Jonny Pettersson, UmU 26 Assignment 1 ❒ The assignment ❒ The report ❒ Rules f or assignment s 5/11 - 04 Emergent Systems, Jonny Pettersson, UmU 27 9

  10. 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 5/11 - 04 Emergent Systems, Jonny Pettersson, UmU 28 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 5/11 - 04 Emergent Systems, Jonny Pettersson, UmU 29 Summar y ❒ Fract als ❒ Net Logo ❒ Assignment 1 5/11 - 04 Emergent Systems, Jonny Pettersson, UmU 30 10

  11. Next t ime ❒ Chaos ❒ Producer-consumer dynamics 5/11 - 04 Emergent Systems, Jonny Pettersson, UmU 31 11

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