"Fractal geometry will make you see everything differently. There is a danger in reading further. You risk the loss of your childhood vision of clouds, forests, flowers, galaxies, leaves, feathers, rocks, mountains, torrents of water, carpets, bricks, and much else besides. Never again will your interpretation of these things be quite the same."

                                      -- Michael Barnsley

| Fractal Geometry | Linear Fractal | Non-Linear Fractal | Fractal Dimension | Fractal Used
| Chaos Math |

What is Fractal Geometry? (top)

Figure 2.1 Oak tree, Palo Alto, California © 1989 Michael McGuire
Fractal geometry is the study of shapes made up of smaller repeating patterns. These patterns called fractals repeat themselves through the use of  self-similarity. For instance as seen in Chapter 1, a tree exhibits a similar structure at different levels of magnification. A close examination of the veins of a leaf reveals a branching pattern similar to the whole tree.  Thus, we call the leaf a "self-similar" component of the tree. Just as a tree’s overall pattern is carried by its genetic code, so too is a fractal's shape carried by its mathematical code, the equation. In both cases we refer this generating code as its “seed”.  With trees in theory we can predict its general appearance--the trunk, limbs and leaves--from its DNA.  However, in practice nature's exact positioning of each item becomes unpredictable as the structure is further removed from the original seed in both spatial distance and time. In mathematics we can determine a fractal's every position, allowing randomness only if we so choose. Often mathematicians and computer artist will introduce some randomness to simulate fractal images found in nature.

Birth of Fractal Mathematics.

The field of Fractal Geometry represents a revolution in mathematics rooted deep in the changes begun in the late 19th century. This was a time when art developed forms of Impressionism, Cubism and Modernism, where music found melodies in ragtime and jazz, and architecture developed styles that stressed functionality.

Figure 2.2 The Eiffel Tower exhibiting the changes of the late 19th Century.
A prime example of this adaptation is illustrated in the Eiffel Tower built in 1889. Gustave Eiffel and his engineers purposely incorporated an interwoven design of steel girders supported by a continuing lattice of smaller trusses and beams into his structure, a structure that incorporates self-similarity to make effective use of material and minimize weight [2]

Figure 2.3 The Eiffel Tower Seen at Three Levels of Magnification.

Initially this new branch of mathematics was thought to comprise shapes independent of the physical world, breaking free from the constraints of nature. Armed with irregular shapes, as if to shout out "See this is not all mathematics has to offer", early pioneers explored structures that Euclidean geometry could only describe as "pathological" or "gallery of monsters".  Karl Weierstrass and Helge von Koch discussed curves without tangents, Georg Cantor demonstrated segments without length,  and Giuseppe Peano described a line that made area. Poincaré, Fatou, Julia and others also published works describing odd shapes that were not well defined in mathematics.  Sadly, the enthusiasm generated by the publications of these mathematicians soon faded. It would not be until after their deaths, when a new generation began questioning Euclid's principles, that their pioneering works were grouped into an area of study and given a name: Fractal geometry.

Figure 2.4  Early monsters of self repeating patterns with no smooth edges.

In 1977, the mathematician Benoit Mandelbrot published his manifesto, Fractals: Form, Chance and Dimension.  This time instead of declaring independence form nature, this seminal work would embrace it by showing the similarity between natural objects and irregular shapes, work that earlier pioneers had only explained mathematically. In 1982, the book was revised and given the title used today The Fractal Geometry of Nature.
At the same time, affordable personal computers became relatively accessible. Now all types of explorers could use desktop computers to generate their own fractal images. Publications helped too by popularizing these modern mathematical concepts with beautiful colored images. Two of the most notable publications were the Beauty of Fractals [3] and  Chaos: Making a New Science. [4]
Types of Fractals
Fractal geometry has a large variety of self-similar shapes. Some of these shapes are found in nature: trees, ferns and mountains. Other shapes are created purely from numerical formulas. We use mathematical equations to generate both kinds of fractals. These fractals are generally classified as "linear" or "nonlinear".

What is a Linear Fractal? (top)

Figure 2.5 Collection of Linear fractals.

Linear fractals or "classical" fractals are exactly self-similar. If you  look at a very small part of a fractal's overall shape, it looks exactly like the original fractal, only smaller. We call this size difference "the scalability factor" or "scale". These fractals begin with a "seed", a set of lines that form a basic structure. Next you make duplicate copies of the original seed and you use them to replace the lines found in the original seed.  You continue this process at greater levels by replacing line segments with seeds, whose lines in turn get replaced by seeds, so on and so on, forever. Since many of these fractals can be easily drawn, they were the primary types of fractals generated before computers. See Color Plate 11 of computer generated linear fractals.

A poem that illustrates self-similarity is given below.
" Great fleas have little fleas on their backs that bite 'em
And little fleas have lesser fleas, and so ad infinitum,
And the great fleas themselves, in turn, have greater fleas to go on,
While these again have greater still, and so on and so on."
                                                    A. de Morgan [5] (1872)
Plants can be a fine example of self-similarity found in nature. If you look closely at different parts you often will see patterns with the same  structure. Here is a seed grown to different levels, see how this fractal closely resemble shapes found in nature. Linear fractals will be discussed in grater detail in Chapter 3.
Figure 2.6 A Wood Etching of a Cypress Tree [6] .

Figure 2.7 A fractal exhibiting nature's own linear patterns of a Cypress tree.

What is a Non-Linear Fractal? (top)

Figure 2.8 Collection of Non-linear fractals.

Non-linear fractals are fractals that exhibit a self-similar structure, but are not exactly self-similar. The overall appearance of a non-linear fractal closely resembles some of its smaller parts but always with some variation. In some cases smaller parts might look quite similar to the overall fractal and help define the fractal's overall shape, in other cases varying regions appear as twisted or skewed scale copies of the original, while still other regions have shapes that bear no resemblance to the original. See Color Plate 12 of computer generated non-linear fractals.

Figure 2.9 Different parts from the Mandelbrot set.
Non-linear fractals are derived from equations that do not produce straight line segments, which is why we call them non-linear. One of the best known fractals of this type is the Mandelbrot set, named after Benoit Mandelbrot, who first produced its graphical image. Non-linear fractals will be discussed in greater detail in Chapter 6.
What is Fractal Dimension? (top)
Benoit Mandelbrot had originally turned to dimension because length, depth, thickness failed to capture many of the structures found in nature. In his work, he try not to constrain these irregular shapes to the rules of classical geometry, but rather understand them, and where ever possible construct inroads into their development. Gone were the days of trying to fit fractal pegs into a round holes. Now these structures were free to be what they really were, objects with fractal dimension.

How many dimensions are there? There are an infinite number of dimensions, not just the traditional integer dimensions: , , , and , that describe points, lines, planes, and volume.  If an object has a dimension somewhere in-between two integer dimensions, it is said to have a fractional, or fractal dimension. For instance the Koch curve--with an infinite length and no area--contains more information than a line of dimension  but not enough for making a plane dimension ,  the dimension for the Koch curve is   seen in Figure 2.3. Dimensions tell us how much structural information an object contains see Figure 2.9 for segments of increasing dimension.

Figure 2.10 Segments with increasing dimension.

How are these dimensions being used? One area where fractal dimensions are being used is to measure the length of objects at different scales. For instance what is the answer to the question proposed by Mandelbrot, "How long is the coast of Britain?" At first glance this question might appear to have a straight forward solution, get a "ruler" and measure its perimeter. Well, how long a ruler should we use? Say we use a ruler that is 1000 km long, what we get is a measurement that gives a rough outline of Britain's most distinctive features such as Scotland and Wales. If we reduce the scale to lets say 1 km, we can distinguish many of the inlets of its coastal shore. If we reduce our ruler once more , this time to 1 meter we get measurements that include many of the jagged edges of the shore.  Each time our measuring stick is reduced our measured length increases. This rate at which the coast's length increases is related to its fractal dimension. In the case of Great Britain, it has a calculated dimension to be roughly  similar to that of the Koch curve. We will discuss dimension in greater detail in Chapter 4.

Figure 2.11 Measuring the Coast of Great Britain with different Rulers.

Where are Fractals Being Used Today? (top)
Fractals are constantly being applied to new applications. Here is just a short list of the places where fractals are being used.

  • aggregation growths
  • art
  • cardiovascular system analysis
  • computer graphics
  • cellular automata modeling
  • chemistry
  • city planning
  • crystallography
  • data compression
  • dielectric processes
  • electro-chemistry
  • epidemics
  • geology
  • image rendering
  • image processing
  • lung analysis
  • kidney structures analysis
  • material science
  • mammography
  • meteorology
  • metallurgy
  • music
  • network structure
  • optimum path transmission models
  • peace and conflict studies
  • psychology
  • plant structures
  • population biology
  • semiconductors
  • stellar formation
  • structural engineering
  • traffic flow
What is Chaos Mathematics? (top)
Figure 2.12 Susceptibility to initial conditions illustrated by The Butterfly Effect.
The basic premise behind chaos mathematics is that small changes at the beginning of a process can cause great differences further on, sometimes causing deterministic systems to exhibit apparent random behavior. This premise is called sensitivity to initial conditions..   A classic example of this is the butterfly effect [7]
where the slightest change in how a butterfly flaps his wings influences weather patterns all over the world at some future time.
Often we can examine a shapes complexity by looking at its basic fractal structures. For instance, if we look closely at a cross section [8]
of a chaotic model called a strange attractor seen in Figure 2.13 , it closely resemble the dust-like features of fractals such as the Cantor set discussed further in Chapter 3. In constructing a strange attractor an initial value is chosen. This value is then put into a set of equations producing a new value, which in turn is reentered into the original equation producing yet another new value. This continuous process is call iteration, which we will discuss it further in Chapter 5. The plotted points of these values is what we see when looking at a strange attractor.

Figure 2.13 Cross section of the Lorenz attractor and a corresponding illustration of the Cantor set.
Often functions in Chaos Mathematics will exhibit self-similar structure. Here are some mathematical examples generated by the Logistic map and the Hénon map.

Figure 2.14 Self similarity shown in the Logistic map.
Figure 2.15 Self similarity shown in the Hénon map.

Here is a listing of some applications where chaos is being used:
  • brain feed back control
  • chemistry reactions
  • ecology models
  • economic models
  • electronic circuit design
  • epidemic control
  • global conflict studies
  • heart attack prediction
  • law principles [9]
  • mathematics
  • music synthesis and feed back
  • pattern recognition
  • planetary motion
  • population models
  • societal behavior (relationships)
  • stock market analysis
  • turbulence models (fluid dynamics)
  • weather forecasting (meteorology)
For a listing of chaos models and the equations that produced them see Chapter 7.

Fractals Everywhere, Michael Barnsley,  Academic Press 1988.
The Eiffel Tower weighs 7000 metric tons, this translates to a  weight of 7 grams for a 30 cm high scale  model.
A book by H. O. Peitgen and P. H. Richter , Springer Verlag 1986.  The popularity of this book was largely based on a spectacular array of pictures since the mathematics in this book  is only intended for the expert. The cover of the August 1985 issue of Scientific American should also be noted as a periodical that was widely received. Other Scientific American articles on fractals include February 1967 on the Dragon curve, articles December 1976 by Martin Gardner discussing Mandelbrot's book French and English edisions of "Fractals" published in 1975 and 1977 respectively, December 1985 on Chaos , and February 1990 on Chaos.
A book by James Gleick , Viking Press chronicling the adventures of the modern discoveries of chaos and fractals.
According to Mandelbrot this 1872 work is probably derived in part from Jonathan Swift's 1773 work, lines 337-340 ," So Nat'ralists observe, a Flea hath smaller Fleas that on him prey, And these have smaller Fleas to bit 'em, And so proceed ad infinitum."  This poem in turn inspired Lewis Richardson to write his 1922 poem in which describes turbulence as a process of continually smaller wholes and eddies. The poem by Richardson can be found in Chapter 4.
Cypress, Mattioli's Commentaries, Lyons, 1579.
This term may have its origins from the short story 'A Sound of Thunder' in the book The Golden Apples of the Sun, by Ray Bradbury  (1953) Rupert Hart-Davis Ltd. in which he discriobes a world profusily effected by a time traveler, who's uncontious act of stepping on a butterfly has unforeseen repercussions.
Also known as a Poincaré cut named after the French mathematician Jules Henri Poincaré.
Chaos and Law, by Andrew W. Hayes (1990) is a 174 page report, on how the principles of chaos are applied to the field of law.