Alice In
Wonderland!
Richard Gharaat
SPSU, Fall 08, CS6353
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”.1 A property called selfsimilarity.
[1] Mandelbrot, B.B. (1982). The Fractal Geometry of
Nature. W.H. Freeman and Company. ISBN 07167-1186-9.
History
 17th century: Mathematician and
philosopher Leibniz considered recursive
self-similarity of a straight line.
 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.
 1872: Karl Weierstrass gave an example
of a function with the non-intuitive
property of being everywhere continuous
but nowhere differentiable.
 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.

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 05002-7693-5.
 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.
 1960s: Benoît Mandelbrot started investigating self-similarity in
papers such as: “How Long Is the Coast of Britain? Statistical SelfSimilarity 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".
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 R2R entity which
means a surface in space and may be redefined as
z = r(x, y), or RR2 entity which means a curve in
space and may be redefined as y = r1(x)  z = r2(x).
 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(x0, y0, z0) = 0  xn+1 = r1(xn, yn, zn)  yn+1 = r2(xn,
yn, zn)  zn+1 = r3(xn, yn, zn) which means they could
be an R3R3 entity. The three relations r1, r2, and r3
are the fractal generators (Compare them with the
plane generator discussed earlier).
 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!
 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.
 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.
 Fractal Dimension (Decimal Dimension)
 Euclidian Objects:
 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 = Ld,
we will have: d = logLs
 Von Koch Snowflake:
 d = log34 = 1.2618595071429148741990542286855.
 Sierpinski Gasket:
 d = log23 = 1.5849625007211561814537389439478.
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.
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.
 The Koch Snowflake
 The T-Square Fractal
 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.
 The Julia set: zn+1 = zn2 + c
 The Mandelbrot Set: zn+1 = (zn + c)2
The Burning Ship Fractal
 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) = ((ar+aii)z +
(br+bii)) / ((cr+cii)z + (dr+di)) 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.
 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.
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.
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 ‘Vvariability’ 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
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.
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