Fractals - the ultimate art of mathematics. Adam Kozak

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1 Fratals - the ultimate art of mathematis Adam Koak

2 Outlie What is fratal? Self-similarit dimesio Fratal tpes Iteratio Futio Sstems (IFS) L-sstems Itrodutio to omple umbers Madelbrot sets Julia ad Fatou sets Madelbulbs

3 What is fratal? Wh should I pa attetio to it? Geometri objet with propert of self-similarit i a sale fator i eat maer, approimate or stohasti Similarit dimesio ma be ot equal to topologi dimesio (o-iteger value) Relativel simple reursive defiitios Appliatios: Fratal ompressio Fratal art Ideas i egieerig, eletrois, hemistr, mediie, urba plaig whih have self-similarit patters Fratal atea i mobile phoes apable of apturig muh wider sope of frequeies i muh smaller areas tha lassi atea 3

4 Soure: Wikipedia Fratals i ature Romaeso brooli Fer High voltage breakdow withi a 4 blok of arli Coast with rivers 4

5 Kolmogorov ompleit Evertig what a be desribed, a be desribed as a strig of haraters over a alphabet of sie >. E.g. ifiite strig Ala ma kota, Ala ma kota, Ala ma kota, To eode suh a strig literall we would eed ifiite memor, however we kow that we a rereate its a fiite substrig simpl usig a omputer THIS STRING IS COMPUTABLE Kolmogorov ompleit of a fiite strig is a legth of the shortest omputer program whih rereates the strig (this is a uomputable futio there is o algorithm to evaluate it!) Kolmogorov( Ala ma kota, Ala ma kota, Ala ma kota, Ala ma kota, Ala ma kota, ) = Legth( while (true) prit( Ala ma kota, ); ) It is also alled iformatioal ompleit 5

6 Hausdorff similarit dimesio Similarit dimesio ma be ot equal to topologi dimesio (o-iteger value) N For ormal geometri objet if we sale it b fator (<<), we eed d opies of this objet to fill the area of origial objet where d is dimesio d, lim N d log 3 log d 3 lim log 3 log log log N,

7 Fratal tpes Fratals ma be obtaied from differet oepts: Atrators of Iterated Futio Sstems (IFS) Julia & Fatou sets Madelbrot sets L-sstem (Lidemaer sstem) 7

8 Cotratig mappig Let (X, d) be a metri spae, the f: X X is a otratig mappig if:, : a, a X : d f ( a ), f ( a ) d a a, Baah fied poit theorem: There eists eatl oe poit px suh, that f(p)=p (fied poit of otratig mappig) Reursive eeutio of otratig mappig: f(,)=(/3,/3) lim f os osos osos f os 8

9 Iterated Futio Sstems (IFS) Reursive trasformatios of geometri objet whih sum produt of a set of affie otratig mappigs (ompositios of rotatio, refletio, traslatio ad otratig salig): {F i : X X } (i ) S S Sk i F i Sk S lim Sk k S is a o empt set of poits i a give spae X S is a fratal a attrator of IFS, it s idepedet of iitial S (S is a fied poit of set of otratig mappigs {F i } i metri spae (H, h) where H is set of all ompat subsets of X ad h is Hausdorff distae) 9

10 Iterated Futio Sstems (IFS) A affie otratig mappig F i i spae has the followig formula: os si si os ' ', ' ' i t t F f e d b a 3 4 t t

11 A eample of IFS Sierpiński triagle IFS: {F i : } (i=..3): Sierpiński triagle is a fied poit (attrator) of Iterated Futio Sstem {F, F, F 3 } 4 3, 4, 4, 3 F F F F, F, F, 3

12 IFS workshop Task: loate, out ad defie otratig mappigs Barsle fer with some lues ;) [sr: Wikipedia] Sierpiński arpet [sr: Wikipedia] Sierpiński triagle i 3D spae (pramid) [sr: Wikipedia]

13 L-sstem (Lidemaer sstem) L-sstems are based o reursive grammar with defied variables, ostats, rules, aiom ad geeratig parameters; we a assig some operatios to eah smbol eg.: variables : X F ostats : + [ ] aiom: X rules : (X F-[[X]+X]+F[+FX]-X), (F FF) parameter - agle: 5 Assiged meaig of smbols for above L-sstem: ( F ) draw forward ( - ) tur left 5 ( + ) tur right 5 ( X ) does othig, just otrols evolutio of the urve ( [ ) saves oordiates ad agle o stak (push) ( ] ) reovers oordiates ad agle from stak (pop) Eemplar geerator: 3

14 Quik itrodutio to omple umbers There is o a real umber suh, that = Ok, so let s reate a umber whih is two-dimesioal, ad put suh a umber o imagiar ais, let s all it i Comple plae i Imagiar umbers +i - Real umbers -i Let s preserve additio ad multipliatio like for real umbers keepig i mid, that i = : a + bi + + di = a + + b + d i a + bi + di = a + ad + b i + bdi = a bd + ad + b i 4

15 Quik itrodutio to omple umbers But there is aother represetatio! Comple plae Imagiar umbers i r + i = r os + isi = os45 + isi45 = + i - Real umbers -i Now applig the rules for trgoometri futios we see that multipliatio is atuall related to rotatio o a plae! Comple plae is a field. a + bi + di = r os + isi r os + isi = r r os( + ) + isi( + ) 5

16 Riema sphere Let s map whole omple plae oto a spehere, where ifilit orrespods to a oth pole 6

17 Madelbrot sets. Madelbrot sets are defied for ratioal futios over losed set of omple umbers (* orrespods to ifiit). Ratioal futio is a divisio of two polomials: 3. Let W deote a ratioal futio depedet o parameter 4. Let 5. Madelbrot set M(W ) of a ratioal futio W is a set of suh poits that is ot overget to *: 7 ) ( ) ( ) ( b b b b a a a a l w W m m m m k k k k ) ( ) ( W W W () W *} () lim : { W C M W i where bi a C {*} C C C C C W : C

18 Madelbrot sets { C : lim W () *} M W This ma be satisfied i two was: Reursio is overget to some poit lim W () where C Reursio fiall falls ito a le (umber of stable les is related to degree of W) Orbit of poit Orbit of poit 8

19 Orbit of poit Madelbrot set - eample The first ad best kow Madelbrot set was defied for polomial futio ( ) W Thus we eed to hek for eah poit i C if sequee, +, ( +) + +, goes to ifiit or ot Workshop: Chek, if poit =+i belogs to Madelbrot set for this futio W () i i W () i i i 3 W () i i i i i 4 W () i i i i... 9

20 Madelbrot set joure

21 Does Madelbrot set eist? Take a look visual ompleit ver low Kolmogorov ompleit of its image for (it = ; < HEIGHT; ++) { for (it = ; < WIDTH; ++) { double = = ; double X = ( - WIDTH/) / ZOOM; double Y = ( - HEIGHT/) / ZOOM; for (it it = MAX_ITER; * + * < 4 && it > ; it--) { tmp = * - * + X; =. * * + Y; = tmp; } image[][] = olor(it); } }

22 Newto method for fidig futio root f ( f '( ) )

23 Julia ad Fatou sets Are based o the same ratioal futios as Madelbrot sets ad are stritl related to them (Julia set is oeted for parameters belogig to Madelbrot set). Fatou sets are areas i C whih are attrated b some poits (here olors red, blue ad gree) for ratioal futio W() W( ) 3 Julia set is a,,border betwee Fatou set areas whih is attrated b ifiit poit (*). f ( ), 3 f ( ) W ( ) f '( ) 3 i 3, i k,, 3 e 3 k i, Here is Julia/Fatou set for futio W() obtaied from Newto s method for futio f() = 3 -. Thus attratig poits for W () orrespod to roots of f(). Gree olor is attratig basi of, red of, ad blue of. 3

24 Madelbulbs Madelbrot sets i 3D Defied b Daiel White ad Paul Nlader usig double rotatio trasformatio for spherial oordiates, sie there is o 3D equivalee to D omple umbers havig all properties of field 4

25 Thak ou for attetio Referees: T. Mart. Fraktale i obiektowe algortm ih wiualiaji. Nakom, Poań, 996. J. Kudrewi. Fraktale i haos. WNT, Warsawa, 7. P. Prusikiewi ad A. Lidemaer. The Algorithmi Beaut of Plats. The Virtual Laborator Series, Spriger 996. B. Madelbrot. The fratal geometr of ature. W.H. Freeme ad Co. New York, delbulb.html Bakgroud soure: 5

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