Fractal geometry extends classical geometry and is more fun too!

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$(i) (ii) etesio of classical geometr such as Euclidea geometr, projective geometr I classical geometries, the geometrical objects are smooth I fractal geometr, the objects are rough For istace, a$

1 Fractal Geometr Reasos for studig this geometr : Fractal geometr eteds classical geometr ad is more fu too! (i) (ii) etesio of classical geometr such as Euclidea geometr, projective geometr I classical geometries, the geometrical objects are smooth I fractal geometr, the objects are rough For istace, a fractal lie is owhere differetiable; I damical sstems, most chaotic regios are fractals What is a fractal? Fractals have self-similar properties If a portio of a fractal object is elarged, the magified portio alwas resembles the origial figure Eg coastal lie of orther Europe, small cloud, fer etc, Cator middle-thirds set How ca we uderstad the geometric structure of fractals? Defiitio of Dimesio

2 Measuremet of coastal lie of HK islad Scale :, Scales of measuremet How to measure a lie segmet of m Scale No of measure Legth m m m m r N rn = m or r N

3 How to measure square of area m? Scale No of measure Legth m m m 6 6 m r N r N m or r N Defiitio of Fractal Dimesio Suppose that r is the scale of measure, N is the umber of measure, L is the total legth The, the fractal dimesio D is defied as as r Therefore, r D N = l D r log N log r lim

4 Simple fractal objects Eample (a) (Koch sowflake) Fig 8 The first four stages i the costructio of Koch sowflake Fig 9 Magificatio of the Koch curve

5 Dimesio Scale No of measure 9 log D log log log log log log s, D log log 6 Eample(b) ( Sierpiski Triagle ) 5

6 Dimesio Scale No of measure log D log log log =58 Iterated Fuctio Sstems (IFS) - method to create ma fractals - Create real life images such as fer - pplied to image compressio Defiitio Let < < Let p,,p be poits i the plae Let i (p) = (p-p i )+p i for each i =, The collectio of fuctios {,, } is called a iterated fuctio sstem If we choose a particular elemet, sa, the repeated iteratio of to a poit p i the plae coverges to p P (p) (p) (p) P 6

7 7 To produce a fractal, we choose a arbitrar iitial poit i the plae ad compute its orbit uder radom iteratio of the i This orbit coverges with probabilit to a specific subset of the plae Defiitio Suppose {,, } is a iterated fuctio sstem The set of poits to which a arbitrar orbit i the plae coverges is the attractor for the sstem Eample Let & i a IFS be defied as Show that a orbit of the IFS teds to the Cator middle-thirds sets Solutio: We ote that the cotractio factor here is, ad the fied poits are located at & alog the -ais Let P be the th poit o the orbit with P, P (p) (p)

8 Sice, the orbit of a poit teds toward the -ais Computatio of -coordiates of poits is as follows: sequece of iteratios ca be described b meas of a sequece of s & s give b (S S S ) where each S j is either or S j = k ( k =, ) k is chose at the j th iteratio S S S The X where X is the iitial -coordiate s, the first term vaishes, implig that X is idepedet of X The remaiig terms ted to a ifiite series which is of the form t i i i where t i is either or This series correspods to poits i the Cator set Eample What is the IFS for the Sierpiski triagle? Solutio: (,) (,) (,) (,) (,) (,) Costructio of Sierpiski triagle 8

$of the bo fractal?$

9 9 The IFS cosists of affie mappigs: st mappig: d mappig: ( The origi is the fied poit) rd mappig: Eample What is the IFS of the bo fractal? Solutio: Costructio of the bo fractal is show below: fied poit fied poit (,) (,) (,) (,) ( /,) ( /,) (,) 5

10 The uit square at the left is mapped to the five smaller squares Therefore, the IFS cosists of five affie mappigs:,,,, 5 We ma iclude rotatio to IFS as follows: cos si si cos P is the fied poit other poit is first cotracted b a factor of toward P ad the rotated b agle about P Eample Fid the attractor of the followig IFS P, P, P = / ad = ½

11 Solutio: The attractor is as follows: IFS code for the geeratio of Fer It cosists of affie mappigs a b e If, c d f The, the code is a b c d e f frequec

MATH 1910 Workshop Solution

MATH 1910 Workshop Solution MATH 90 Workshop Solutio Fractals Itroductio: Fractals are atural pheomea or mathematical sets which exhibit (amog other properties) self similarity: o matter how much we zoom i, the structure remais the