1. SPSU, Fall 08, CS6353 Alice In Wonderland! Richard Gharaat

2. Introduction  A fractal is generally “a rough or fragmented geometric shape that can be split into parts, each of which is (at least approximately) a reduced-size copy of the whole”. property called self- similarity.  A fractal is generally “a rough or fragmented geometric shape that can be split into parts, each of which is (at least approximately) a reduced-size copy of the whole”. 1 A property called self- similarity. [1] Mandelbrot, B.B. (1982). The Fractal Geometry of Nature. W.H. Freeman and Company. ISBN 0- 7167-1186-9.

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

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

5.  1872: Karl Weierstrass gave an example of a function with the non-intuitive property of being everywhere continuous but nowhere differentiable.

6.  1904: Helge von Koch, dissatisfied with Weierstrass's very abstract and analytic definition, gave a more geometric definition of a similar function, which is now called the Koch snowflake.

7.  1915: Waclaw Sierpinski constructed his triangle.  1916: Sierpinski constructed his carpet.  Originally, these geometric fractals were described as curves rather than the 2D shapes that they are known as in their modern constructions.  1918: Bertrand Russell had recognized a "supreme beauty" within the mathematics of fractals that was then emerging. 1 [1] Briggs, John. Fractals: The Patterns of Chaos. Thames and Hudson, 148. ISBN 0- 5002-7693-5.

8.  1938: Paul Pierre Lévy described a new fractal curve, the Lévy C curve, in his paper Plane or Space Curves and Surfaces Consisting of Parts Similar to the Whole.

9.  1960s: Benoît Mandelbrot started investigating self-similarity in papers such as: “How Long Is the Coast of Britain? Statistical Self- Similarity and Fractional Dimension”, which built on earlier work by Lewis Fry Richardson.  1975: Mandelbrot coined the word "fractal" to denote an object whose Hausdorff-Besicovitch dimension is greater than its topological dimension. He illustrated this mathematical definition with striking computer-constructed visualizations. These images captured the popular imagination; many of them were based on recursion, leading to the popular meaning of the term "fractal".

10. Properties  Recursive  The traditional geometrical shapes have a straightforward definition.  Mathematically speaking, they have the general form of r(x, y, z) = 0.  They can be categorized as an R 2  R entity which means a surface in space and may be redefined as z = r(x, y), or R  R 2 entity which means a curve in space and may be redefined as y = r 1 (x)  z = r 2 (x).

11.  On the other hand, fractal shapes have a recursive definition. Usually there is a base entity and a substitution pattern that applies over the base entity so that the result is a new-segmented shape, then the pattern applies over the segments and produces sub segments, and it goes on.  Mathematically speaking they have the general form of r(x 0, y 0, z 0 ) = 0  x n+1 = r 1 (x n, y n, z n )  y n+1 = r 2 (x n, y n, z n )  z n+1 = r 3 (x n, y n, z n ) which means they could be an R 3  R 3 entity. The three relations r 1, r 2, and r 3 are the fractal generators (Compare them with the plane generator discussed earlier).

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

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

14.  Exactlessness!  Traditional geometrical entities have a straightforward definition. Usually there is an algebraic formula for each of these entities so they can be constructed at once and it is possible to precisely determine if a given point is at the edge of that entity or not.  On the other hand, because of the recursiveness, fractals cannot be constructed; rather they are developed through infinite recursions. That means YOU CAN NEVER DRAW THE BORDER OF A FARCAL! because they are dynamic entities.

16.  Fractal Dimension (Decimal Dimension)  Euclidian Objects:

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

18.  Von Koch Snowflake:  d = log 3 4 = 1.2618595071429148741990542286855.

19.  Sierpinski Gasket:  d = log 2 3 = 1.5849625007211561814537389439478.

20. Different Types  Deterministic  They have an exact recursive algorithm to be developed, i.e. whenever you render them you will see the same shape.  Stochastic  They are involved with randomization in recursion, i.e. each time you render them you will see a different shape although they follow same structure and they have the same initial entity.

21. Different Structures  Geometric Fractals  These fractals have a fixed geometric replacement rule.  Cantor set, Sierpinski carpet, Sierpinski gasket, Peano curve, Koch snowflake, Harter-Heighway dragon curve, T-Square, Menger sponge, are some examples of such fractals.  They are consideres as deterministic fractals.

22.  The Koch Snowflake

25.  The T-Square Fractal

26.  Escape-Time Fractals (Orbit Fractals)  These are defined by a formula or recurrence relation at each point in a space (such as the complex plane).  Examples of this type are the Mandelbrot set, Julia set, the Burning Ship fractal, the Nova fractal and the Lyapunov fractal.  They are considered as deterministic fractals as well.

27.  The Julia set: z n+1 = z n 2 + c

28.  The Mandelbrot Set: z n+1 = (z n + c) 2

30. The Burning Ship Fractal

33.  Iterated Function Systems (IFS)  They have the general form of f(x, y) = ((ax+by+c) / (gx+hy+i), (dx+ey+f) / (gx+hy+i)) in real space.  They have the general form of f(z) = ((a r +a i i)z + (b r +b i i)) / ((c r +c i i)z + (d r +d i )) in complex space.  In order to make a fractal, you define a set of functions f1, f2, f3, … onto a desired space and choose a random point in that space. Then you let them randomly compete with each other to attract the point.  They are considered as stochastic fractals.

40.  Strange Attractors  “Orbits of dynamical systems, that is, of IFS semigroups generated by a single transformation, may lie on or be attracted to geometrically complicated structures called strange attractors, often by dint of a certain level of complication in the single underlying transformation.” 1 [1] Barnsley, SuperFractals, Cambridge University Press 2006, ISBN: 978-0-521-84493-2.

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

47. Super Fractals  “Superfractals are families of sometimes beautiful fractal objects which can be explored by means of the chaos game and which span the gap between fully ‘random’ fractal objects and deterministic fractal objects.” 1  “A superfractal is associated with a single underlying hyperbolic IFS. It has its own underlying logical structure, called the ‘V- variability’ of the superfractal, for some V  {1, 2, …}, which enable us to sample the superfractal by means of the chaos game and produce generalized fractal objects such as fractal sets, pictures, measures and so on, one after another. The property of V-variability enables us to ‘dance on the superfractal’, sometimes producing wondrous objects in splendid succession.” 1 [1] Barnsley, SuperFractals, Cambridge University Press 2006, ISBN-13: 978-0-521-84493-2

54. Applications  Graphics (Landscape rendering)  Movies (Star Trek, the last season – Star Wars Episode 3)  Image Compression  Fractal antennas  Different application of the chaos theory, especially in economy and statistics.

55. References  en.wikipedia.org  Mandelbrot, The Fractal Geometry of Nature, W.H. Freeman and Company, 1982, ISBN: 0-7167-1186-9.  Briggs, John. Fractals: The Patterns of Chaos. Thames and Hudson. ISBN: 0-5002-7693-5.  Barnsley Michael F., FRACTALS EVERYWHERE, Second Edition, 1993, ISBN: 0-12-079061-0  Barnsley, SuperFractals, Cambridge University Press 2006, ISBN: 978-0-521-84493-2