Fractals and iteration function systems I Complex systems

Fractals and iteration function systems IComplex systems simulation

Facultad de Informatica (UPM)Sonia Sastre

November 17, 2011

Sonia Sastre () Fractals and iteration function systems November 17, 2011 1 / 37

Classical Fractals. Algorithms. Introduction

Introduction

Many of the fractals and their descriptions and mathematical propertiesgo back to classical mathematics and mathematicians of the pastlike Georg Cantor (1845-1918), Giuseppe Peano (1858-1932), DavidHilbert (1862-1943), Felix Hausdorff (1868-1942), Helge von Koch (1870-1924), Constantin Caratheodory (1873-1950), Henri Leon Lebesgue (1875-1941),Pierre Fatou (1978-1029), Waclaw Sierpinski (1882-1969), AbramSamoilovitch Besicovitch (1891-1970), Gaston Julia (1893-1978) to namejust a few.


Classical Fractals. Algorithms. Introduction

IntroductionWhy Benoit Mandelbrot (1924-2010)is often characterized as the father of fractal geometry?

The Cantor set, the Koch curve, and other fractals were regarded asexceptional objects, as counter examples, as ’mathematical monsters’, butplayed a key role in Mandelbrot’s concept of a new geometry.

Mandelbrot showed that the fractal sets have many features in commonwith shapes found in nature (The Fractal Geometry of Nature, 1982).

The term fractal, derived from the Latin fractus (meaning “broken”), wascoined by Mandelbrot in 1975.A fractal set has some of the following properties

1 It has a fine structure at arbitrarily small scales.2 It is too irregular to be easily described by Euclidean geometry.3 It is self-similar (at least approximately or stochastically).4 It has a simple and recursive definition.5 It has a Hausdorff dimension (dimH) which is greater than its

topological dimension (dim).Sonia Sastre () Fractals and iteration function systems November 17, 2011 3 / 37

Classical Fractals. Algorithms. The Cantor set

The Cantor setThe Cantor set was first published in 1883and emerged as an example of certain exceptional sets.

The Cantor ternary set construction goes as follows1 Start with the unit interval I0 = [0, 1].2 Now take away the (open) interval (1/3, 2/3), this leaves two

intervals I 11 = [0, 1/3] and I 21 = [2/3, 1].3 Now we repeat, we look at the remaining intervals and remove their

middle thirds, which yields four intervals of length 1/9.4 Continue on in this way. After the n-th step we have 2n intervals

I 1n , · · · , I 2n

n of length1

3n.

The Cantor set is the set of points which remain if we carry out theremoval steps infinitely often

C =∞⋂n=1

2n⋃i=1

I in.



The Cantor set

The Cantor set has the following properties1 It is a closed subset and it is also totally bounded and therefore it is

compact.

2 The Cantor set has Lebesgue measure 0.

3 the Cantor set is uncountable.

4 Every point in the Cantor set is an accumulation point of the Cantorset. A closed set in which every point is an accumulation point is alsocalled a perfect set in topology.

5 Any point in the Cantor set is not an interior point. A closed subsetof the interval with no interior points is nowhere dense in the interval.

6 It is self-similar.

7 It has a dimH = log 2log 3 ≈ 0.6309297534 and dim = 0.



Variants of Cantor set

Instead of repeatedly removing the middle third of every piece as in theCantor set, we could also keep removing any other fixed percentage (otherthan 0% and 100% why?) from the middle.

Figure: The Cantor set C 25

The resulting sets also have Lebesgue measure 0.

By removing progressively smaller percentages of the remaining pieces inevery step, one can also construct sets homeomorphic to the Cantor setthat have positive Lebesgue measure, while still being nowhere dense. Thisset is colled Smith-Volterra-Cantor set.



Smith-Volterra-Cantor setThe process begins by removing the middle 1

4 from the interval [0, 1], so theremaining set is

[0, 38]∪[58 , 1].

Next removing subintervals of width 122n

from the middle of each of the 2n−1

remaining intervals, so the leaving set is[0, 5

32

]∪[732 ,

38

]∪[58 ,

2532

]∪[2732 , 1

].

Continuing indefinitely with this removal, the Smith-Volterra-Cantor set isthe set of points that are never removed.

The Smith-Volterra-Cantor set has empty interior, is closed and has positivemeasure of 1

2 . The total length of the intervals removed from [0, 1] is

∞∑n=1

2n−11

22n=

1

4+

1

8+

1

16+ · · · =

1

2.


Classical Fractals. Algorithms. Cantor dust

Cantor dust

Cantor dust is a multi-dimensional version of the Cantor set. It can beformed by taking a finite Cartesian product of the Cantor set with itself.Like the Cantor set, Cantor dust has zero measure.The Cantor ternary set construction

1 Start with a square Q0 of side equal to 1.

2 Now take away all the open squares of side r (with 0 < r <1

2)but the

four in the corners.3 Now we repeat, in every remained square.4 Continue on in this way. After the n-th step we have 4n squares

Q1n , · · · ,Q4n

n of side rn.The Cantor dust is the set of points which remain if we carry out theremoval steps infinitely often

C 2r =

∞⋂n=1

4n⋃i=1

Q in.


Classical Fractals. Algorithms. Cantor dust

Cantor dust

Figure: Cantor dust C 21/3

Cantor dust C 2r has a dimH = log 4

log 1/r . Note that dimH(C 21/4) = 1


Classical Fractals. Algorithms. The Koch Curve

The Koch Curve

Helge von Koch introduced what is now calledthe Koch curve.

The Koch curve is an example of a curve which there is no way to fit atangent to any of its point.

The geometric construction of the Koch curve:1 Begin with a straight line.

2 Partition it into three equal parts.

3 Then replace the middle third by an equilateral triangle and take awayits base. This completes the basic construction step.

4 Thus, we now repeat, taking each of the resulting line segments,partitioning them into three equal parts, and so on.

5 After the n-th step we have 4n intervals of lenght 13n .

The Koch curve is the limit set if we carry out this process infinitely often.Sonia Sastre () Fractals and iteration function systems November 17, 2011 10 / 37


The Koch Curve

Figure: The Koch CurveSonia Sastre () Fractals and iteration function systems November 17, 2011 11 / 37


The Koch Curve

The Koch Curve has the following properties

1 The Koch curve has an infinite length because it is greater than4n

3nfor all n.

2 The Koch curve is continuous everywhere but differentiable nowhere.3 It is self-similar.4 It has a dimH = log 3

log 4 ≈ 1.261859507 and dim = 1.

Fitting together three suitably rotated copies of the Koch curve producesthe following figure, which is called the snowflake curve or the Koch island

The area of a Koch snowflake is

A =

√3

4(1 + 3

∞∑n=1

4n−1

32n) =

√3

4(1 +

3

4

∞∑n=1

4n

9n) =

2√

3

5.


Classical Fractals. Algorithms. Sierpinski gasket

Sierpinski gasket

The Sierpinski gasket was described byWaclaw Sierpinski in 1915.

The basic geometric construction of the Sierpinski gasket goes as follows:1 Start with a triangle in the plane.

2 The midpoints of its three sides together with the vertices of theoriginal triangle define four triangles.

3 Drop the center one.

4 Now we follow the same procedure with the three remaining triangles.

5 After the n-th step we have 3n intervals of side 12n .

The Sierpinski gasket is the set of points in the plane which remain if onecarries out this process infinitely often.

The Sierpinski Gasket has a dimH = log 3log 2 ≈ 1.584962501 and dim = 1.


Classical Fractals. Algorithms. Sierpinski gasket

Sierpinski gasket

Figure: Sierpinski gasket


Classical Fractals. Algorithms. Sierpinski Carpet

Sierpinski CarpetCould you explain the construction suggested by the following picture?

Figure: Sierpinski CarpetSonia Sastre () Fractals and iteration function systems November 17, 2011 15 / 37

Classical Fractals. Algorithms. Menger sponge

The Menger sponge

The Menger sponge was described byKarl Menger in 1926.

The construction of the Menger sponge goes as follows:1 Begin with a cube.2 Divide the cube into 27 equal smaller cubes.3 Remove the cube at the middle of every face, and remove the cube in

the center.4 Now we follow the same procedure with the 20 remaining squares.

The Menger sponge is the set of points in the plane which remain if onecarries out this process infinitely often. It has the following properties:

1 Menger Sponge is a curve (dimt = 1) and it has adimH = log 20

log 3 ≈ 2.726833.2 It has an infinite surface area and encloses zero volume.3 Every face of the Menger sponge is a Sierpinski Carpet.4 Any intersection of the Menger sponge with a diagonal of the initial

cube is a Cantor set.Sonia Sastre () Fractals and iteration function systems November 17, 2011 16 / 37


The Menger sponge

Figure: The Menger spongeSonia Sastre () Fractals and iteration function systems November 17, 2011 17 / 37


Universality of the Carpet and the Sponge

What Waclaw Sierpinski was trying to accomplish when he developed thecarpet?

He was looking for a kind of super-object which contains all possible one-dimensional objects in a topological sense.

The Sierpinski carpet is universal for all compact one-dimensional objects inthe plane. This includes trees and graphs planes with an arbitrary countablenumber of edges, vertices and closed loops, connected in arbitrary ways.

The branching order is a local concept. It measures the number of brancheswhich meet in a point. It is used to comparer curves in a topological way.All the following spiders with five arms are topologically equivalent.



Universality of the Carpet and the Sponge

Though visually, Sierpinski gaket and carpet seems to be not much of adifference, but Sierpinski gasket has points with branching order of only 2(if P is a corner point), 4 (if P is touchimg point) and 3 (if P is any otherpoint).

In a similar way, the Sierpinski carpet is a universal curve for all curvesthat can be drawn on the two-dimensional plane, the Menger sponge isuniversal for all compact one-dimensional objects.


Classical Fractals. Algorithms. The Multiple Reduction Copy Machine algorithm

The Multiple Reduction Copy Machine algorithm

Consider a copy machine equipped with a reduction feature. If we take animage, put it on the machine we obtain a reduced copy. The same operationis carried out repeatedly, the output of one iteration being the input for thenext one. After some ten or so cycles any initial image would be reduced tojust a point.

Consider a modification of the copy machine consists of the choice of thenumber of lenses, the reduction factors and the placements of the reducedimages. We call this machine a Multiple Reduction Copy Machine (MRCM).

Consider an MRCM with three lens systems, each of which is set to reduceby a factor of 1/2. The resulting copies are assembled in the configurationof an equilateral triangle.

What is the effect of the MRCM beginning with different initial images?Sonia Sastre () Fractals and iteration function systems November 17, 2011 20 / 37



1 The Sierpinski Gasket is the attractor of the iteration MRCM process:No matter which initial image we take and run the MRCM with, weobtain a sequence of images which always tends towards theSierpinski gasket.

2 The Sierpinski Gasket is invariant or fixed:When we start the machine with the Sierpinski gasket, then it is leftinvariant or fixed.






The Multiple Reduction Copy Machine algorithmThis MRCM provides a good metaphor for what is known as deterministiciterated function systems (IFS) in mathematics.

A set of contraction functions is called an IFS. The next similarities

w1(x , y) =

(1/2 0

0 1/2

)(xy

)w2(x , y) =

(1/2 0

0 1/2

)(xy

)+

(1/2

0

)w3(x , y) =

(1/2 0

0 1/2

)(xy

)+

(1/4√3/4

)is an IFS which have the Sierpinski gasket T as invariant, i.e.

T = w1(T ) ∪ w2(T ) ∪ w3(T ).A set which is invariant for an IFS of similarities is called self-similar.



The Multiple Reduction Copy Machine algorithmThe MRCM algorithm for the Sierpinski Gasket is the following

1 Pick a random Z image.2 Plot Z .3 For i = 1 to N:

i) Delete Z .ii) Calculate w1(Z ),w2(Z ),w3(Z )iii) Do Z = w1(Z ) ∪ w2(Z ) ∪ w3(Z )iv) Plot Z

4 End

If we have an IFS Φ = {f1, f2, . . . , fk}, we can modify the MRCM1 Number of lens systems equal to k .2 Setting of reduction factor for each lens system individually (according to

those of fi ).

3 Configuration of lens systems for the assembly of copies (according to of fi ).

Note that the algorithm is the same: to generate each possible sequenceof functions up to a given maximum length, and then to plot the results ofapplying each of these sequences of functions to an initial image.



ExercisesImplement the MRCM algorithm for:

1 The Sierpinski Gasket, the Cantor dust and the Koch curve.

2 the following IFS’s.

a b c d e f

0 -0.5 0.5 0 0.5 00 0.5 -0.5 0 0.5 0.50.5 0 0 0.5 0.25 0.5

a b c d e f

0.387 0.43 0.43 -0.387 0.256 0.5220.441 -0.091 -0.009 -0.322 0.4219 0.5059-0.468 0.02 -0.113 0.015 0.4 0.4

a b c d e f

0.5 0 0 0.75 0.25 00.25 -0.2 0.1 0.3 0.25 0.50.25 0.2 -0.1 0.3 0.5 0.40.2 0 0 0.3 0.4 0.55

a b c d e f

0.849 0.037 -0.037 0.849 0.075 0.1830.197 -0.226 0.226 0.197 0.4 0.049-0.15 0.283 0.26 0.237 0.575 -0.0840 0 0 0.16 0.5 0

a b c d e f

0.382 0 0 0.382 0.3072 0.6190.382 0 0 0.382 0.6033 0.40440.382 0 0 0.382 0.0139 0.40440.382 0 0 0.382 0.1253 0.05950.382 0 0 0.382 0.4920 0.595

a b c d e f

0.195 -0.488 0.344 0.443 0.4431 0.24520.462 0.414 -0.252 0.361 0.2511 0.5692-0.058 -0.07 0.453 -0.111 0.5976 0.0969-0.035 0.07 -0.469 -0.022 0.4884 0.5069-0.637 0 0 0.501 0.8562 0.2513



Adaptive Cut Method

Note that the algorithm gives kN images and most of them are much smallerthan necessary.We can reduce the number of images by covering of the attractor by setswhich have a diameter lower than ε.The attractor A of an IFS Φ = {f1, f2, . . . , fk} verifiesA∞ = f1(A) ∪ f2(A) ∪ · · · ∪ fn(A). Then, we can cover A by

1 k2 sets: A∞ = f1f1(A) ∪ f1f2(A) ∪ · · · f2f1(A) ∪ · · · ∪ fk fk(A).2 km sets after m iterations of the form :

fs1 fs2 . . . fsm(A) with si ∈ {1, 2, ..., k}The Adaptive Cut Method is applied when the maps of the IFS havedifferent reduction factor:In the nth level of subdivision we obtain subsets of the form:

fs1fs2 . . . fsn−1f1, fs1fs2 . . . fsn−1f2, fs1fs2 . . . fsn−1fk ,

keep those subsets which are small enough and subdivide the others.All fsi are contractions then, after m iterations the diameter of all sets islower than ε.



Adaptive Cut Method

Example. Consider an IFS Φ = {f1, f2}.In the following graph Ai1i2...in = f11fi2 . . . fin(A).The box minds that the diameter of the set in is lower than ε.

A1111 A1112

�� @@

A111 A112

�� @@

A121 A122

�� @@

A211 A212

�� @@

A11 A12

��

HHH

H

A21 A22

��

HHH

H

A1 A2

((((((((

hhhhhhhhA



Adaptive Cut Method

Next pictures show on the left, 9 iterations with the MRCM starting with apoint (49 = 262.144 points). On the right the adaptive cut method using198.541 points.


Classical Fractals. Algorithms. The Chaos Game

The Chaos Game

This algorithm was originally described byM.Barnsley in 1988.

1 Take 3 points in a plane (vertex of an equilateral triangle).2 Randomly select any point and consider that your current position.3 Plot the current position.4 For i from 1 to N

1 Randomly select any one of the 3 vertex points.2 Move half the distance from your current position to the selected

vertex. Consider that your new current position.3 Plot the current position.

5 End



The Chaos Game

Apparently, this chaos game will give a chaotic cloud of points but thefollowing picture shows the result with different number of iterations.

Figure: The Chaos Game



The Chaos Game





The Chaos Game





The Chaos Game





The Chaos Game

The chaos game has been generalized to a method of generating theattractor of any IFS Φ = {f1, f2, . . . , fk}.Starting with any point x0, successive iterations are formed as xn+1 = fi (xn),where fi is a member of the IFS randomly selected for each iteration. Theiterations converge to the fixed point of the IFS. If x0 belongs to the attractorof the IFS, all iterations xn stay inside the attractor.

Figure: The Chaos Game. Barnsley’s fren (left) and Sierpinski Carpet*



The Chaos Game. Probabilities

In the Chaos Game for Sierpinski Gasket, every map has the same probabilityto be selected. When all the maps of an IFS have the same reduction factors,the result is worse when we take different probabilities. In the next picture,we take 50%, 30%, 20% (100 points on the left and 10.000 on the right).

When the maps of an IFS have different reduction factors, it is usual to takedifferent probabilities. Next picture shows the Barnsley’s fern with 100.000points (equal probabilities on the left and different ones on the right).



ExercisesImplement the Chaos Game algorithm for:

1 The Sierpinski Gasket and the Cantor dust, taking probabilities of 13 for the

first one and 14 for the second one.

2 The following fractal ferns. Last column indicate the probability.

a b c d e f P

0.85 0.04 -0.04 0.85 0 1.6 0.840.2 -0.26 0.23 0.22 0 1.6 0.07-0.15 0.28 0.26 0.24 0 0.44 0.070 0 0 0.16 0 0 0.02

a b c d e f P

0.95 0.002 -0.002 0.93 -0.002 .5 0.840.035 -0.11 0.27 0.01 -0.05 0.005 0.07-0.04 0.11 0.27 0.01 0.047 0.06 0.070 0 0 0.25 0 -0.4 0.02

a b c d e f P

0.85 0.02 -0.02 0.83 0 1 0.840.09 -0.28 0.3 0.11 0 0.6 0.07-0.09 0.28 0.3 0.09 0 0.7 0.070 0 0 0.25 0 -0.14 0.02

a b c d e f P

0.95 0.005 -0.005 0.93 -0.002 0.5 0.840.035 -0.2 0.16 0.04 -0.09 0.02 0.07-0.4 0.2 0.16 0.04 0.083 0.12 0.070 0 0 0.25 0 -0.4 0.02



Home work

Make a search in the web to find curiosities, algorithms, applications oranything related with fractals.Send to

[email protected]

an e-mail with at least the following

1 Your name.

2 Five questions, comments or curiosities that surprised you.

3 Three links, with comments describing why have you selected them.

Deadline: wednesday 23 at 8:00 am


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Fractals and iteration function systems I Complex systems