ITERATION ALGORITHM TO CREATE FRACTALS UNIVERSITY OF MARYLAND - PowerPoint PPT Presentation

USING THE RANDOM ITERATION ALGORITHM TO CREATE FRACTALS UNIVERSITY OF MARYLAND DIRECTED READING PROGRAM FALL 2015 BY ADAM ANDERSON

THE SIERPINSKI GASKET 2 Stage 1: Stage 0: Stage 2: Stage n: 2 𝑜 2 2 1 3 1 2 A 0 = 2 3 3 2 A n = A 1 = 1 − A 2 = 4 − 2 4 2 2 2 4 1 3 3 9 𝑜 = 1 − 3 = 4 − 16 = = 0 lim A 0 =1 4 16 4 𝑜→∞ 3 2 A 1 = 3 A 2 = 4 4 Sierpinski Gasket has zero area?

CAPACITY DIMENSION AND FRACTALS Let S ⊆ ℝ 𝑜 where n = 1, 2, or 3 z n-dimensional box • n = 1: Closed interval y • n = 2: Square x • n = 3: Cube Let N( ε ) = smallest number of n-dimensional boxes of side length ε necessary to cover S

CAPACITY DIMENSION AND FRACTALS ε n = 1 Boxes of length ε to cover line of length L: Length = L If L = 10cm and ε = 1cm, it takes 10 boxes to cover L If ε = 0.5cm, it takes 20 boxes to cover L … 1 N( ε ) ∝ ε

CAPACITY DIMENSION AND FRACTALS ε n = 2 Boxes of length ε to cover square S of side length L If L = 20cm, area of S = 400 cm 2 ε = 2cm: each box has area 4 cm 2 • It will take 100 boxes to cover S ε = 1cm: each box has area 1cm 2 • It will take 400 boxes to cover S 1 N( ε ) ∝ ε 2

CAPACITY DIMENSION AND FRACTALS 𝐸 1 N( ε ) = 𝐷 𝜁 1 ln ( N(ε)) = D ln 𝜁 + ln(𝐷) ln 𝑂 𝜁 −ln 𝐷 D = C just depends on scaling of S ln 1 𝜁 ln 𝑂 𝜁 Capacity Dimension: dim c S = lim ln 1 𝜁→0 + 𝜁

CAPACITY DIMENSION AND FRACTALS 1 1 ε = 2 Stage n: Stage 1: Stage 2: 1 1 1 If ε = 2 𝑜 , N 𝜁 = 3 𝑜 If ε = 2 , N 𝜁 = 3 If ε = 4 , N 𝜁 = 9 1 𝜁 = 2 𝑜 ln 3 𝑜 ln 𝑂 𝜁 𝑜 ln 3 ln 3 lim = lim ln 2 𝑜 = 𝑜 ln 2 = ln 2 ≈ 1.5849625 ln 1 𝜁→0 + 𝜁→0 + 𝜁

The Sierpinski Gasket is ≈ 1.585 dimensional A set with non-integer capacity dimension is called a fractal.

ITERATED FUNCTION SYSTEMS An Iterated Function System (IFS) F is the union of the contractions T 1 , T 2 … T n THEOREM: Let F be an iterated function system of contractions in ℝ 2 . Then there exists a unique compact subset 𝐵 F in ℝ 2 such that for any compact set B, the sequence of iterates 𝐺 𝑜 𝐶 ∞ converges in 𝑜=1 the Hausdorff metric to 𝐵 F 𝐵 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)

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

RANDOM ITERATION ALGORITHM Drawing an attractor of IFS F : 1. Choose an arbitrary initial point 𝑤 ∈ ℝ 2 2. Randomly select one of the contractions T n in F 3. Plot the point T n ( 𝑤) 4. Let T n ( 𝑤) be the new 𝑤 5. Repeat steps 2-4 to obtain a representation of 𝐵 F 20,000 Iterations 1000 Iterations 5000 Iterations

LET’S DRAW SOME FRACTALS!

ITERATIONS AND FIXED POINTS 𝑔 𝑦 = 𝑦 2 + 1 Iterating = repeating the same procedure Let 𝑔(𝑦) be a function. = 𝑔 2 𝑦 is the second iterate of x under 𝑔. 𝑔 𝑔 𝑦 𝑔 𝑔(𝑔( 𝑦 ) = 𝑔 3 𝑦 is the third iterate of x under 𝑔 Example: 𝑔 𝑦 = 𝑦 2 + 1 𝑔 2 𝑦 = (𝑦 2 +1) 2 + 1 𝑔 5 = 26 = 𝑔 2 5 = 𝑔 26 = 677 𝑔 𝑔 5

ITERATIONS AND FIXED POINTS A point p is a fixed point for a function f if its iterate is itself 𝑔 𝑞 = 𝑞 𝑔 𝑦 = 𝑦 2 Example: 𝑔 𝑦 = 𝑦 2 𝑔 0 = 0 = 𝑔 2 0 = 𝑔 0 = 0 𝑔 𝑔 0 Therefore 0 is a fixed point of f

METRIC SPACES Let S be a set. A metric is a distance function d 𝑦, 𝑧 that satisfies 4 axioms ∀ 𝑦, 𝑧 ∈ S 1. d 𝑦, 𝑧 ≥ 0 2. d 𝑦, 𝑧 = 0 if and only if 𝑦 = 𝑧 3. d 𝑦, 𝑧 = d 𝑧, 𝑦 4. d 𝑦, 𝑨 ≤ d 𝑦, 𝑧 + d 𝑧, 𝑨 x y Example: Absolute Value ℝ d 𝑦, 𝑧 = 𝑦 − 𝑧 ℝ, d is a metric space d 𝑦, 𝑧 = 𝑦 − 𝑧

METRIC SPACES Let S, d be a metric space. A sequence 𝑦 𝑜 𝑜=1 ∞ in S converges to x ∈ S if lim 𝑜→∞ d 𝑦 𝑜 , 𝑦 = 0 This means that terms of the sequence approach a value s A sequence is Cauchy if for all 𝜁 > 0 there exists a positive integer N such that whenever n, m ≥ N , d 𝑦 𝑜 , 𝑦 𝑛 < 𝜁 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

CONTRACTION MAPPING THEOREM Let S, d be a metric space. A function T: S → S is a contraction if ∃ 𝑟 ∈ 0,1 such that d T(𝑦), T(𝑧) ≤ q ∗ d(𝑦, 𝑧) T d T(𝑦), T(𝑧) = q ∗ d(𝑦, 𝑧) d 𝑦, 𝑧

CONTRACTION MAPPING THEOREM Contraction Mapping Theorem: If S, d is a complete metric space, and T is a contraction, then as n → ∞, T 𝑜 𝑦 → unique fixed point 𝑦 ∗ ∀ 𝑦 ∈ S T T

HAUSDORFF METRIC A set S is closed if whenever x is the limit of a sequence of members of T, x actually is in T. A set S is bounded if it there exists x ∈ S and r > 0 such that ∀ 𝑡 ∈ S , d x, 𝑡 < r Means S is contained by a “ball” of finite radius A set S ⊆ ℝ 𝑜 is compact if it is closed and bounded Let K denote all compact subsets of ℝ 2

HAUSDORFF METRIC If B is a nonempty member of K , and 𝑤 is any point in ℝ 2 , the distance from 𝑤 to B is 𝑤, 𝐶 = minimum value of d 𝑤 − 𝑐 ∀ 𝑐 ∈ 𝐶 (distance from point to a compact set) B 𝑤 d 𝑤, 𝐶

HAUSDORFF METRIC If A and B are members of K , then the distance from A to B is d 𝐵, 𝐶 = maximum value of d 𝑏, 𝐶 for 𝑏 ∈ 𝐵 (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 B A 𝑏 d 𝐵, 𝐶

HAUSDORFF METRIC The Hausdorff metric on K is defined as: D 𝐵, 𝐶 =maximum of d 𝐵, 𝐶 and d 𝐶, 𝐵 B d 𝐶, 𝐵 d 𝐵, 𝐶 𝑐 d 𝐶, 𝐵 A D 𝐵, 𝐶 = d 𝐶, 𝐵 𝑏 d 𝐵, 𝐶

Iterated Function System: 1 2 1 2 − 𝑦 𝑦 𝑔 𝑧 = 1 𝑧 1 2 1 2 1 2 1 2 − − 𝑦 𝑦 𝑧 + 1 𝑔 𝑧 = 2 1 2 1 2 0 − Render Details: Point size: 1px # of iterations: 100,000

Iterated Function System: 1 2 0 𝑦 𝑦 𝑔 𝑧 = 1 𝑧 1 2 0 1 2 0 1 2 𝑦 𝑦 𝑔 𝑧 = 𝑧 + 2 1 2 0 0 𝑦 𝑦 0 −1/2 1 𝑔 𝑧 = 𝑧 + 3 1/2 0 1/2 Render Details: Point size: 1px # of iterations: 100,000

Iterated Function System: 1 3 0 𝑦 𝑦 𝑧 + 2/3 𝑔 𝑧 = 1 1 3 2/3 0 1 3 0 𝑦 𝑦 2/3 𝑔 𝑧 = 𝑧 + 2 1 3 −2/3 0 1 3 0 𝑦 𝑦 𝑧 + −2/3 𝑔 𝑧 = 3 1 3 2/3 0 1 3 0 𝑦 𝑦 𝑧 + −2/3 𝑔 𝑧 = 4 1 3 −2/3 0 13 40 13 40 𝑦 𝑦 𝑔 𝑧 = 5 𝑧 −13 40 13 40 Render Details: Point size: 1px # of iterations: 100,000

Iterated Function System: 𝑦 𝑦 𝑧 = 0 0 𝑔 1 𝑧 0 .16 𝑦 𝑦 .85 .04 0 𝑔 𝑧 = 𝑧 + 2 1.6 −.04 .85 𝑦 𝑦 .2 −.26 0 𝑔 𝑧 = 𝑧 + 3 .23 .22 1.6 𝑦 𝑦 𝑧 = −.15 .28 0 𝑔 𝑧 + 4 .26 .24 .44 Probabilities: f 1 : 1% f 2 : 85% f 3 : 7% f 4 : 7%

Iterated Function System: − 3 1 2 𝑦 𝑦 6 𝑔 𝑧 = 1 𝑧 3 1 2 6 1 3 0 𝑦 𝑦 𝑧 + 1/ 3 𝑔 𝑧 = 2 1 3 1/3 0 1 3 0 𝑦 𝑦 𝑧 + 1/ 3 𝑔 𝑧 = 3 1 3 −1/3 0 1 3 0 𝑦 𝑦 𝑧 + −1/ 3 𝑔 𝑧 = 4 1 3 1/3 0 1 3 0 𝑦 𝑦 𝑧 + −1/ 3 𝑔 𝑧 = 5 1 3 −1/3 0 1 3 0 𝑦 𝑦 0 𝑔 𝑧 = 𝑧 + 6 1 3 2/3 0 1 3 0 𝑦 𝑦 0 𝑔 𝑧 = 𝑧 + 7 −2/3 1 3 0

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