SPSU, Fall 08, CS6353SPSU, Fall 08, CS6353

Alice In Alice In Wonderland!Wonderland!Richard GharaatRichard Gharaat

IntroductionIntroduction

A A fractalfractal is generally “a rough or is generally “a rough or fragmented geometric shape that can be fragmented geometric shape that can be split into parts, each of which is (at least split into parts, each of which is (at least approximately) a reduced-size copy of approximately) a reduced-size copy of the whole”.the whole”.1 A property called self- property called self-similarity.similarity.

[1] [1] Mandelbrot, B.B. (1982). The Fractal Geometry of Mandelbrot, B.B. (1982). The Fractal Geometry of Nature. W.H. Freeman and Company. ISBN 0-Nature. W.H. Freeman and Company. ISBN 0-7167-1186-9.7167-1186-9.

HistoryHistory

17th century:17th century: Mathematician and Mathematician and philosopher Leibniz considered recursive philosopher Leibniz considered recursive self-similarity of a straight line.self-similarity of a straight line.

Late 19th and early 20th centuries:Late 19th and early 20th centuries: Henri Poincaré, Felix Klein, Pierre Fatou, Henri Poincaré, Felix Klein, Pierre Fatou, and Gaston Julia investigated iterated and Gaston Julia investigated iterated functions in the complex plane. They functions in the complex plane. They lacked the means to visualize the beauty lacked the means to visualize the beauty of many of the objects that they had of many of the objects that they had discovered.discovered.

1872:1872: Karl Weierstrass gave an example Karl Weierstrass gave an example of a function with the non-intuitive of a function with the non-intuitive property of being everywhere continuous property of being everywhere continuous but nowhere differentiable.but nowhere differentiable.

1904:1904: Helge von Koch, dissatisfied with Helge von Koch, dissatisfied with Weierstrass's very abstract and analytic Weierstrass's very abstract and analytic definition, gave a more geometric definition, gave a more geometric definition of a similar function, which is definition of a similar function, which is now called the Koch snowflake.now called the Koch snowflake.

1915:1915: Waclaw Sierpinski constructed his triangle. Waclaw Sierpinski constructed his triangle.

1916:1916: Sierpinski constructed his carpet. Sierpinski constructed his carpet.

Originally, these geometric fractals were described as curves rather than Originally, these geometric fractals were described as curves rather than the 2D shapes that they are known as in their modern constructions.the 2D shapes that they are known as in their modern constructions.

1918:1918: Bertrand Russell had recognized a "supreme beauty" within the Bertrand Russell had recognized a "supreme beauty" within the mathematics of fractals that was then emerging.mathematics of fractals that was then emerging.11

[1] [1] Briggs, John. Fractals: The Patterns of Chaos. Thames and Hudson, 148. ISBN 0-Briggs, John. Fractals: The Patterns of Chaos. Thames and Hudson, 148. ISBN 0-5002-7693-5.5002-7693-5.

1938:1938: Paul Pierre Lévy described a new Paul Pierre Lévy described a new fractal curve, the Lévy C curve, in his fractal curve, the Lévy C curve, in his paper paper Plane or Space Curves and Plane or Space Curves and Surfaces Consisting of Parts Similar to Surfaces Consisting of Parts Similar to the Wholethe Whole..

1960s:1960s: Benoît Mandelbrot started investigating self-similarity in Benoît Mandelbrot started investigating self-similarity in papers such as: papers such as: “How Long Is the Coast of Britain? Statistical “How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension”Self-Similarity and Fractional Dimension”, which built on earlier , which built on earlier work by Lewis Fry Richardson.work by Lewis Fry Richardson.

1975:1975: Mandelbrot coined the word "fractal" to denote an object Mandelbrot coined the word "fractal" to denote an object whose Hausdorff-Besicovitch dimension is greater than its whose Hausdorff-Besicovitch dimension is greater than its topological dimension. He illustrated this mathematical definition topological dimension. He illustrated this mathematical definition with striking computer-constructed visualizations. These images with striking computer-constructed visualizations. These images captured the popular imagination; many of them were based on captured the popular imagination; many of them were based on recursion, leading to the popular meaning of the term "fractal".recursion, leading to the popular meaning of the term "fractal".

PropertiesProperties

RecursiveRecursive The traditional geometrical shapes have a The traditional geometrical shapes have a

straightforward definition.straightforward definition. Mathematically speaking, they have the general Mathematically speaking, they have the general

form of r(x, y, z) = 0.form of r(x, y, z) = 0. They can be categorized as an RThey can be categorized as an R22R entity which R entity which

means a surface in space and may be redefined as means a surface in space and may be redefined as z = r(x, y), or Rz = r(x, y), or RRR22 entity which means a curve in entity which means a curve in space and may be redefined as y = rspace and may be redefined as y = r11(x) (x) z = r z = r22(x).(x).

On the other hand, fractal shapes have a recursive On the other hand, fractal shapes have a recursive definition. Usually there is a base entity and a definition. Usually there is a base entity and a substitution pattern that applies over the base entity substitution pattern that applies over the base entity so that the result is a new-segmented shape, then so that the result is a new-segmented shape, then the pattern applies over the segments and produces the pattern applies over the segments and produces sub segments, and it goes on.sub segments, and it goes on.

Mathematically speaking they have the general form Mathematically speaking they have the general form of r(xof r(x00, y, y00, z, z00) = 0 ) = 0 x xn+1n+1 = r = r11(x(xnn, y, ynn, z, znn) ) y yn+1n+1 = r = r22(x(xnn, y, ynn, , zznn) ) z zn+1n+1 = r = r33(x(xnn, y, ynn, z, znn) which means they could be ) which means they could be an Ran R33RR33 entity. The three relations r entity. The three relations r11, r, r22, and r, and r33 are are the fractal generators (Compare them with the plane the fractal generators (Compare them with the plane generator discussed earlier).generator discussed earlier).

InfiniteInfinite The main difference between regular shapes The main difference between regular shapes

and fractals is the recursiveness of fractals. and fractals is the recursiveness of fractals. Recursiveness brings infinity with it. If there Recursiveness brings infinity with it. If there is no terminator, then any recursive process is no terminator, then any recursive process will continue forever. That is exactly what will continue forever. That is exactly what happens for a fractal. As it is said, there is happens for a fractal. As it is said, there is no stop for iteration. What that means is NO no stop for iteration. What that means is NO ONE CAN SEE THE FINAL FRACTAL or ONE CAN SEE THE FINAL FRACTAL or you have to be immortal!you have to be immortal!

Self SimilarSelf Similar Fractals have a recursive definition. This is a Fractals have a recursive definition. This is a

necessary condition but it is not enough. For necessary condition but it is not enough. For a recursive definition to be a fractal, it should a recursive definition to be a fractal, it should be in such a way that produces intermediate be in such a way that produces intermediate sub entities that have the same properties of sub entities that have the same properties of the initial entity. the initial entity.

Exactlessness!Exactlessness! Traditional geometrical entities have a Traditional geometrical entities have a

straightforward definition. Usually there is an straightforward definition. Usually there is an algebraic formula for each of these entities so algebraic formula for each of these entities so they they can be constructed at oncecan be constructed at once and it is possible to and it is possible to precisely determine if a given point is at the edge of precisely determine if a given point is at the edge of that entity or not.that entity or not.

On the other hand, because of the recursiveness, On the other hand, because of the recursiveness, fractals cannot be constructed;fractals cannot be constructed; rather they are rather they are developed through infinite recursions. developed through infinite recursions. That That means YOU CAN NEVER DRAW THE BORDER means YOU CAN NEVER DRAW THE BORDER OF A FARCAL! because they are OF A FARCAL! because they are dynamic entitiesdynamic entities..

Fractal Dimension (Decimal Dimension)Fractal Dimension (Decimal Dimension) Euclidian Objects:Euclidian Objects:

Instead of having dimension (d) and Instead of having dimension (d) and division factor (L) as input and calculating division factor (L) as input and calculating the number of segments (s) as output, let the number of segments (s) as output, let us say we know the number of segments us say we know the number of segments and division factor and want to calculate and division factor and want to calculate the dimension. Using the equation s = Lthe dimension. Using the equation s = Ldd, , we will have: d = logwe will have: d = logLLss

Von Koch Snowflake:Von Koch Snowflake: d = logd = log334 = 1.2618595071429148741990542286855.4 = 1.2618595071429148741990542286855.

Sierpinski Gasket:Sierpinski Gasket: d = logd = log223 = 1.5849625007211561814537389439478.3 = 1.5849625007211561814537389439478.

Different TypesDifferent Types

DeterministicDeterministic They have an exact recursive algorithm to be They have an exact recursive algorithm to be

developed, i.e. whenever you render them you will developed, i.e. whenever you render them you will see the same shape.see the same shape.

StochasticStochastic They are involved with randomization in recursion, They are involved with randomization in recursion,

i.e. each time you render them you will see a i.e. each time you render them you will see a different shape although they follow same structure different shape although they follow same structure and they have the same initial entity.and they have the same initial entity.

Different StructuresDifferent Structures

Geometric FractalsGeometric Fractals These fractals have a fixed geometric replacement These fractals have a fixed geometric replacement

rule.rule. Cantor set, Sierpinski carpet, Sierpinski gasket, Cantor set, Sierpinski carpet, Sierpinski gasket,

Peano curve, Koch snowflake, Harter-Heighway Peano curve, Koch snowflake, Harter-Heighway dragon curve, T-Square, Menger sponge, are some dragon curve, T-Square, Menger sponge, are some examples of such fractals.examples of such fractals.

They are consideres as deterministic fractals.They are consideres as deterministic fractals.

The Koch SnowflakeThe Koch Snowflake

The T-Square FractalThe T-Square Fractal

Escape-Time Fractals (Orbit Fractals)Escape-Time Fractals (Orbit Fractals) These are defined by a formula or These are defined by a formula or

recurrence relation at each point in a space recurrence relation at each point in a space (such as the complex plane).(such as the complex plane).

Examples of this type are the Mandelbrot Examples of this type are the Mandelbrot set, Julia set, the Burning Ship fractal, the set, Julia set, the Burning Ship fractal, the Nova fractal and the Lyapunov fractal.Nova fractal and the Lyapunov fractal.

They are considered as deterministic fractals They are considered as deterministic fractals as well.as well.

The Julia set: zThe Julia set: zn+1n+1 = z = znn22 + c + c

The Mandelbrot Set: zThe Mandelbrot Set: zn+1n+1 = (z = (znn + c) + c)22

The Burning Ship FractalThe Burning Ship Fractal

Iterated Function Systems (IFS)Iterated Function Systems (IFS) They have the general form of f(x, y) = ((ax+by+c) / They have the general form of f(x, y) = ((ax+by+c) /

(gx+hy+i), (dx+ey+f) / (gx+hy+i)) in real space.(gx+hy+i), (dx+ey+f) / (gx+hy+i)) in real space. They have the general form of f(z) = ((aThey have the general form of f(z) = ((arr+a+aiii)z + i)z +

(b(brr+b+biii)) / ((ci)) / ((crr+c+ciii)z + (di)z + (drr+d+dii)) in complex space.)) in complex space. In order to make a fractal, you define a set of In order to make a fractal, you define a set of

functions f1, f2, f3, … onto a desired space and functions f1, f2, f3, … onto a desired space and choose a random point in that space. Then you let choose a random point in that space. Then you let them randomly compete with each other to attract them randomly compete with each other to attract the point.the point.

They are considered as stochastic fractals.They are considered as stochastic fractals.

Strange AttractorsStrange Attractors ““Orbits of dynamical systems, that is, of IFS Orbits of dynamical systems, that is, of IFS

semigroups generated by a single transformation, semigroups generated by a single transformation, may lie on or be attracted to geometrically may lie on or be attracted to geometrically complicated structures called complicated structures called strange attractorsstrange attractors, , often by dint of a certain level of complication in the often by dint of a certain level of complication in the single underlying transformation.”single underlying transformation.”11

[1] Barnsley, SuperFractals, Cambridge University Press [1] Barnsley, SuperFractals, Cambridge University Press 2006, ISBN: 978-0-521-84493-2.2006, ISBN: 978-0-521-84493-2.

Semi FractalsSemi Fractals

It was implicitly emphasized that fractal It was implicitly emphasized that fractal generator is defined in such a way that when it generator is defined in such a way that when it is applied over the entity, it results in segments is applied over the entity, it results in segments that are that are BOTH similar to each other AND to BOTH similar to each other AND to the original entity.the original entity. Fractals of this kind are Fractals of this kind are considered complete fractals. Now if we just considered complete fractals. Now if we just waive the second part of the condition, we get waive the second part of the condition, we get into a class of fractals considered as semi into a class of fractals considered as semi fractals.fractals.

Super FractalsSuper Fractals

““Superfractals are families of sometimes beautiful fractal objects Superfractals are families of sometimes beautiful fractal objects which can be explored by means of the chaos game and which which can be explored by means of the chaos game and which span the gap between fully ‘random’ fractal objects and span the gap between fully ‘random’ fractal objects and deterministic fractal objects.”deterministic fractal objects.”11

““A superfractal is associated with a single underlying hyperbolic A superfractal is associated with a single underlying hyperbolic IFS. It has its own underlying logical structure, called the ‘V-IFS. It has its own underlying logical structure, called the ‘V-variability’ of the superfractal, for some V variability’ of the superfractal, for some V {1, 2, …}, which {1, 2, …}, which enable us to sample the superfractal by means of the chaos game enable us to sample the superfractal by means of the chaos game and produce generalized fractal objects such as fractal sets, and produce generalized fractal objects such as fractal sets, pictures, measures and so on, one after another. The property of pictures, measures and so on, one after another. The property of V-variability enables us to ‘dance on the superfractal’, sometimes V-variability enables us to ‘dance on the superfractal’, sometimes producing wondrous objects in splendid succession.”producing wondrous objects in splendid succession.”11

[1] Barnsley, SuperFractals, Cambridge University Press 2006, ISBN-13: [1] Barnsley, SuperFractals, Cambridge University Press 2006, ISBN-13: 978-0-521-84493-2978-0-521-84493-2

ApplicationsApplications

Graphics (Landscape rendering)Graphics (Landscape rendering) Movies (Star Trek, the last season – Star Movies (Star Trek, the last season – Star

Wars Episode 3)Wars Episode 3) Image CompressionImage Compression Fractal antennasFractal antennas Different application of the chaos theory, Different application of the chaos theory,

especially in economy and statistics.especially in economy and statistics.

ReferencesReferences

en.wikipedia.orgen.wikipedia.org Mandelbrot, The Fractal Geometry of Nature, W.H. Mandelbrot, The Fractal Geometry of Nature, W.H.

Freeman and Company, 1982, ISBN: 0-7167-1186-9.Freeman and Company, 1982, ISBN: 0-7167-1186-9.

Briggs, John. Fractals: The Patterns of Chaos. Thames Briggs, John. Fractals: The Patterns of Chaos. Thames and Hudson. ISBN: 0-5002-7693-5. and Hudson. ISBN: 0-5002-7693-5.

Barnsley Michael F., FRACTALS EVERYWHERE, Barnsley Michael F., FRACTALS EVERYWHERE, Second Edition, 1993, ISBN: 0-12-079061-0Second Edition, 1993, ISBN: 0-12-079061-0

Barnsley, SuperFractals, Cambridge University Press Barnsley, SuperFractals, Cambridge University Press 2006, ISBN: 978-0-521-84493-22006, ISBN: 978-0-521-84493-2