USING THE RANDOM ITERATION ALGORITHM TO CREATE FRACTALS

Transcription

1 USING THE RANDOM ITERATION ALGORITHM TO CREATE FRACTALS UNIVERSITY OF MARYLAND DIRECTED READING PROGRAM FALL 205 BY ADAM ANDERSON

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3 THE SIERPINSKI GASKET 2 Stage 0: A 0 = 2 22 A 0 = Stage : A = 2 = 4 A = Stage 2: 2 2 A 2 = = 3 3 = A 2 = Stage n: A n = 3 4 lim n 3 4 n n = 0 Sierpinski Gasket has zero area?

4 CAPACITY DIMENSION AND FRACTALS Let S R n where n =, 2, or 3 n-dimensional bo n = : Closed interval n = 2: Square n = 3: Cube Let N(ε) = smallest number of n-dimensional boes of side length ε necessary to cover S z y

5 CAPACITY DIMENSION AND FRACTALS n = ε Boes of length ε to cover line of length L: Length = L If L = 0cm and ε = cm, it takes 0 boes to cover L If ε = 0.5cm, it takes 20 boes to cover L N(ε) ε

6 CAPACITY DIMENSION AND FRACTALS n = 2 ε Boes of length ε to cover square S of side length L If L = 20cm, area of S = 400cm 2 ε = 2cm: each bo has area 4cm 2 It will take 00 boes to cover S ε = cm: each bo has area cm 2 It will take 400 boes to cover S N(ε) ε 2

7 CAPACITY DIMENSION AND FRACTALS N(ε) = C ε D ln (N(ε)) = D ln ε + ln(c) D = ln N ε ln ε ln C C just depends on scaling of S Capacity Dimension: dim c S = lim ε 0 + ln N ε ln ε

8 CAPACITY DIMENSION AND FRACTALS ε = 2 Stage : If ε = 2, N ε = 3 Stage 2: Stage n: If ε =, N ε = 9 If ε = n, N ε = 3n 4 2 ε = 2n ln N ε lim ε 0 + ln ε = lim ε 0 + ln 3 n n ln 3 ln 2n = = ln 3 n ln 2 ln

9 The Sierpinski Gasket is.585 dimensional A set with non-integer capacity dimension is called a fractal.

10 ITERATED FUNCTION SYSTEMS An Iterated Function System (IFS) F is the union of the contractions T, T 2 T n THEOREM: Let F be an iterated function system of contractions in R 2. Then there eists a unique compact subset A F in R 2 such that for any compact set B, the sequence of iterates F n B n= converges in the Hausdorff metric to A F A F is called the attractor of F. This means that if we iterate any compact set in R 2 under F, we will obtain a unique attractor (attractor depends on the contractions in F)

11 ITERATED FUNCTION SYSTEMS A function T : R 2 R 2 is affine if it is in the form f y = a b c d y + e f = a + by + e c + dy + f (linear function followed by translation) We will deal with iterated function systems of affine contractions.

12 RANDOM ITERATION ALGORITHM Drawing an attractor of IFS F:. Choose an arbitrary initial point v R 2 2. Randomly select one of the contractions T n in F 3. Plot the point T n ( v) 4. Let T n ( v) be the new v 5. Repeat steps 2-4 to obtain a representation of A F 000 Iterations 5000 Iterations 20,000 Iterations

13 LET S DRAW SOME FRACTALS!

14 ITERATIONS AND FIXED POINTS Iterating = repeating the same procedure f = 2 + Let f() be a function. f f = f 2 is the second iterate of under f. f f(f( ) = f 3 is the third iterate of under f Eample: f = 2 + f 5 = 26 f f 5 = f 2 5 = f 26 = 677 f 2 = ( 2 +) 2 +

15 ITERATIONS AND FIXED POINTS A point p is a fied point for a function f if its iterate is itself f p = p Eample: f = 2 f 0 = 0 f f 0 = f 2 0 = f 0 =0 Therefore 0 is a fied point of f f = 2

16 METRIC SPACES Let S be a set. A metric is a distance function d, y that satisfies 4 aioms, y S. d, y 0 2. d, y = 0 if and only if = y 3. d, y = d y, 4. d, z d, y + d y, z y Eample: Absolute Value R d, y = y R, d is a metric space d, y = y

17 METRIC SPACES Let S, d be a metric space. A sequence n n= in S converges to S if lim d n, = 0 n This means that terms of the sequence approach a value s A sequence is Cauchy if for all ε > 0 there eists a positive integer N such that whenever n, m N, d n, m < ε This means that terms of the sequence get closer together S, d is a complete metric space if every Cauchy sequence in S converges to a member of S

18 CONTRACTION MAPPING THEOREM Let S, d be a metric space. A function T: S S is a contraction if q 0, such that d T(), T(y) q d(, y) T d, y d T(), T(y) = q d(, y)

19 CONTRACTION MAPPING THEOREM Contraction Mapping Theorem: If S, d is a complete metric space, and T is a contraction, then as n, T n unique fied point S T T

20 HAUSDORFF METRIC A set S is closed if whenever is the limit of a sequence of members of T, actually is in T. A set S is bounded if it there eists S and r > 0 such that s S, d, s < r Means S is contained by a ball of finite radius A set S R n is compact if it is closed and bounded Let K denote all compact subsets of R 2

21 HAUSDORFF METRIC If B is a nonempty member of K, and v is any point in R 2, the distance from v to B is d v, B = minimum value of v b b B (distance from point to a compact set) B v d v, B

22 HAUSDORFF METRIC If A and B are members of K, then the distance from A to B is d A, B = maimum value of d a, B for a A (distance between compact sets) Means we take the point in A that is most distant from any point in B and find the minimum distance between it an any point in B A B a d A, B

23 HAUSDORFF METRIC The Hausdorff metric on K is defined as: D A, B =maimum of d A, B and d B, A A a d B, A d A, B B b d A, B d B, A D A, B = d B, A

24 Iterated Function System: f y = 2 2 y 2 2 f 2 y = y + 0 Render Details: Point size: p # of iterations: 00,000

25 Iterated Function System: f y = y f 2 y = y f 3 y = 0 /2 /2 0 y + /2 Render Details: Point size: p # of iterations: 00,000

26 Iterated Function System: f f 2 f 3 f 4 f 5 y = y + 2/3 2/3 3 y = y + 2/3 2/3 3 y = y + 2/3 2/3 3 y = y + 2/3 2/ y = y Render Details: Point size: p # of iterations: 00,000

27 Iterated Function System: f y = y f 2 y = y f 3 y = y f 4 y = y Probabilities: f : % f 2 : 85% f 3 : 7% f 4 : 7%

28 Iterated Function System: f y = y f 2 f 3 f 4 f 5 f 6 f 7 y = y = y = y = y = y = y + / 3 /3 y + / 3 /3 y + / 3 /3 y + / 3 /3 y + 0 2/3 y + 0 2/3

29 THANKS Mentor Kasun Fernando Author and Book Provider Denny Gulick Some IFS Formulas from Agnes Scott College Graphs Created with Desmos Graphing Calculator