The term fractal, from the Latin root fractus meaning fragmented, broken and discontinuous, was coined in 1975 by Benoit Mandelbrot(1). However, this does not give us a clear idea of what a fractal actually is.
Fractal definition @ dictionary.com:
A geometric pattern that is repeated at ever smaller scales to produce irregular shapes and surfaces that cannot be represented by classical geometry. Fractals are used especially in computer modeling of irregular patterns and structures in nature.
There are problems with this definition. Many fractals are represented by a finite scaling, but have the additional characteristic that their Hausdorff dimension* is not equal to 1. In fact Benoit Mandelbrot coined the term fractal partially to reference this secondary fact. An example of a finitely iterated fractal is given by Michael Barnsey , one of the leading authorities on the subject, in his treatment of hyperbolic iterated function systems or IFS for short.
Since the subject matter is relatively new and similar matters could prove to be a fertile ground for research, it is probably best to leave the subject loosely defined for now. (This topic has been discussed on ILP in relation to Popper). Barnsley openly and repeatedly refuses to define the subject matter(2).
Examples and some commentary.
a) One of the earliest examples of a fractal was the Cantor Set. This set is constructed by taking a segment of the Real line and removing the middle third and then repeatedly performing this process on the remaining sets. This particular set has the peculiar properties that it exists, the number of elements are uncountable (more of these than all of the Rational numbers (fractions)), and yet it is so “hollow” that it has no Euclidean measure**.
b) There are other constructed sets like the Koch Curve. This curve is constructed by adding an upside down V with the center between the legs of the V removed, to the center of a given line segment. Then indefinitely repeating this operation on each new line segment. These sets resemble snowflakes and can be added together to resemble the coastlines of various countries. (Britain being the most famous). However, it should be noted that the fractal dimension* of the Koch curve and the outline of Britain are not the same(1).
c) The Sierpinski gasket is defined by starting with an equilateral triangle and removing the middle triangle defined by the three points consisting of the mid points of each side of the original triangle. After this step you will have three remaining triangles. Remove the middle sections of these triangles and continue repeating this process indefinitely. (Care should be taken to only remove the interior parts of the triangles (and not their borders)). This fractal was introduced by Vaclav Sierpinski in 1916 but its construction was known to artists for centuries and proto types were sketched by M.C. Escher on the pulpit of the 12 century Ravello cathedral(1).
d) mathworld.wolfram.com/JuliaSet.html
e) math.bu.edu/DYSYS/applets/JuliaIteration.html
f) More commonly recognized sets are examples of Julia Sets, defined below, which resemble ferns, sea shells, bolts of lightning, and many other objects of nature.
g) Some good pictures of fractals can be found in the classic “CHAOS MAKING A NEW SCIENCE” by James Glieick. If you do not have this book you might consider buying it.
How are fractals made?
One method is to start with a familiar shape and then devise a rule to repeatedly remove or add some well defined secondary shape to it. The first three examples were obtained in this way.
Julia sets, which were first studied by Gaston Julia and Pierre Fatou are defined as follows:
For a given Rational Function P/Q where P and Q are polynomials, of a complex variable, with no common divisors, Julia sets are defined as the boundary of the set Z = {z[size=75]0[/size] a [size=100]complex[/size] number such that if z[size=75]n+1[/size] [size=100]=[/size] P(z[size=75]n[/size])[size=100]/[/size]Q(z[size=75]n[/size]) then [size=100]the[/size] limit as n goes to infinity of z[size=75]n[/size] is [size=100]bounded[/size]}. As you might expect, more recently work has also been done on the Trigonometric functions.
By far the greatest amount of work done is on the polynomial P(z)/Q(z) = z^2 + c. In fact the Mandelbrote Set is the set of c such that the Julia sets are not simply dust (“hollow†with no Euclidean measure.)
One final way to generate fractals is to start with a set of points (generally in R X R) and through a system of particular transforms (for example afine transforms in the form of x[size=75]N+1[/size] [size=100]=[/size] ax[size=75]N[/size] + [size=100]b[/size], where a < 1) randomly generate points which can take on the appearance of the common fractals, e.g. ferns, sea shells, et cetera.
Benoit Mandelbrot was the first person to coin the word fractal (which he intuited to be an object with a fractional dimension between 1 and 2) and to popularize the subject. Born in Warsaw in 1924 to a Lithuanian Jewish family, he was a very bright and creative person having taught Economics at Harvard, Engineering at Yale and Physiology at the Einstein School of Medicine. It should be noted that elements of these subjects can be considered as examples of fractals. The family moved to Paris in 1936 where he met and was strongly influenced by his Mathematician uncle Szolem (1) a founding member of the very influence mathematical group known as Nicolas Bourbaki. Later Benoit became a tool maker in Tulle France. After Paris was liberated, he applied to the elite schools Ecole Normale and Ecole Polytechique. Despite the fact that he had little or no formal education, he claims to never have been taught the alphabet or times tables past 5, he passed both exams. He left France and the school before graduating to work at IBM’s Thomas J Watson Research Center(3).
Benoit Mandelbrot working with computers at IBM was able to generate a large enough set to get a general idea of what the kernel of a particular (and simple) Julia set looked like as an image of the iterative process.
So What? (A tale of failure, greed, intrigue, life, death, sex, redemption, and a greater power. I bet you never thought that math could be so fun!)
A couple of criticisms of the study of fractals come to mind immediately. Why do we need these relatively complex descriptions that may or may not look like naturally occurring objects? And just because we can describe many naturally occurring objects in this manner what good does it do? It does not appear to be causal like a force or the consequence of genetics.
To answer the first question, we need to look at trying to describe these objects with classical geometric objects. Rectangles, Ellipses, conic sections generally, and Differentiable functions will all fail to describe these objects. Additionally, Michael Barnsley has devised a method to create specific shapes in a relatively simple manner. The Collage Theorem tells us that a class of fractals can be constructed by “pasting together small sets of similar fractal”(4). Eventually Barnsely found a way to take a given shape and devise a simple formula to recreate that shape.
And why is this important? Entities like the US government, Microsoft, Mitsubishi, Multi Comm, and Virgin have paid millions to use it’s results to compress digital images to 1/10 of the size of any previous compaction of digital data(4)! Imagine the time that they can save down loading files. Additionally, small fuzzy pictures can be enlarged to more clear images! I SPY.
To answer the second question we will look at the sets generated by the function P(z)/Q(z) = rz(1-z) = -rz^2 + rz. This function, which is generally restricted to the Real variable x, describes population growth of an ecological system with both growth and decay factors; and is called the logistic function(3). This function does in fact generate fractals for various values of r but the Hausdorff dimension is less than 1(5). A number of books fail to specify the domain that this function works on and the nature of the constant r, and we will look more closely at this function.
x[size=75]0[/size] is [size=100]restricted[/size] to a number between 0 and 1 and can be thought of as yin as it ranges between nothingness and unity. Then (1-x[size=75]0[/size]) [size=100]can[/size] be thought of as yang, as it ranges between unity and nothingness. The sum of yin and yang is unity.
When yin and yang are coupled in the form of x[size=75]0/size they produce a new generation of [size=100]yin[/size] and yang. x[size=75]1[/size] [size=100]=[/size] x[size=75]0/size. [size=100]Oh[/size] boy sex!!!
This coupling always reduces the magnitude of x[size=75]n+1[/size] for any n because [size=100]the[/size] magnitude of yang is always less than one.
So to save a population governed by this dynamic from ultimate destruction, there must be a restorative or redemptive factor r with a power (magnitude) greater than 1 to redeem the population.
Finally to get some idea of what restrictions there might be on this restorative or redemptive power of r, oh the hubris of this author, we notice that the largest value that x[size=75]n/size [size=100]must[/size] be .25. Those of you with a Calculus background will notice that the function defined by f(x) = x(1-x) has a derivative df(x)/dx = 1 - 2x and since the maximum/minimum occur when df(x)/dx = 0 we must have 1 - 2x = 0 or 1 = 2x or ½ = x.
Therefore if r is greater than 4 and x[size=75]n[/size] [size=100]=[/size] .5, then rx[size=75]n/size > 1. [size=100]Thus[/size] x[size=75]n+1[/size] is outside [size=100]the[/size] range of possibilities (the population is greater than 100%) and we get an error. This is not to say that r can not be greater than 4, only that in these cases, x[size=75]0[/size] must [size=100]start[/size] small enough to avoid this problem.
While this description of the logistics function is generally presented without the accompanying gratuitous anthropomorphic overtones, I think it adds color to the palette and has powerful psychological implications.
Raw Speculation:
The Universe itself is a fractal that is still being iterated, and it is a result of a dynamical system analogous to the logistics function.
How can this be? Circumstantial data supporting the speculation includes the fact the Universe appears to be experiencing an expansion(6)*** similar to the effect of antigravity. This gives the feel of a yin and yang type of dynamic. Furthermore, there is evidence that there is new galaxy formation relatively close to us and that there are old galaxies at the edges of our Universe where the Big Bang was supposed to happen(7)(8)***. Finally, the background noise, which is supposed to confirm the remnants of the Big Bang could be the residual elements of an unbounded set falling away to infinity as we see when we look at the examples of the various fractals.
The book Introducing Fractal Geometry by Nigel Lesmoir, Will Rood, and Ralph Edney goes into the fractal nature of the Universe to some extent starting with the section entitled the “The Great Wall” on pages 134 through 138.
(1) “Introducing Fractal Geometry” by Nigel Lesmoir, Will Rood, and Ralph Edney
(2) “Fractals Everywhere” Michael Barnsey Second Edition.
(3) “CHAOS Making A New Science” by James GLeick
(4) PBS broadcast of “The colors of Infinity” with Arthur Clark, Benoit Mandelbrot, Michael Barnsley et al.
(4) “Introducing Fractal Geometry” by Nigel Lesmoir, Will Rood, and Ralph Edney
(5) “Fractal Geometry Mathematical Foundations and Applications” by Kenneth Falconer
(6) pbs.org/newshour/bb/science/ … _2-27.html
(7) usatoday.com/news/science/20 … usat_x.htm
(8) seds.org/~spider/spider/LG/Add/and8pr.html
- The Hausdorff dimension of a given set S can be defined as the limit as n goes to infinity of log(B[size=75]n+1[/size][size=100]/[/size]B[size=75]n[/size])[size=100]/[/size][size=100]log/size [size=100]where[/size] B[size=75]n[/size] [size=100]and[/size] B[size=75]n+1[/size]are [size=100]the[/size] minimal number of boxes required to cover S and P[size=75]x[/size] is the Projection of the boxes on the x axis.
Examples from “Introducing Fractal Geometry” by Nigel Lesmoir, Will Rood, and Ralph Edney are:
Straight Line:
Fractal dimension: [log(20boxes/10boxes)]/[log(20 units in the x dimension/10units in the x dimension)] = 1
Julia Set:
Fractal dimension: [log(60 Boxes/27 Boxes)]/[log(20 units in the x dimension/10units in the x dimension)] =1.152… To complete the calculation, successively more detailed grids must be considered.
** The subject of Measure Theory is interesting and Lebesque measure and integration provides constructive results.
*** These facts appear to violate the big bang theory.
How does the Fractal Model compare to the Gravitational Model and the Non Euclidean Geometry Model?
The Gravitational Model is a causal model and it is described by using a specific differential equation. Differential Equations assume that “Physical Reality” is continuous and Euclidean. While the Non Euclidean model does not make recourse to causality, it too uses the embedded Euclidean metric to describe the surfaces studied. Due to the fact that the objects studied in fractal geometry can be very irregular, different metrics have been devised.
Metric Spaces (More math! How much fun can we have!; and yes there is something wrong with me. You can skip ahead, if you want, to Skip Ahead).
In general a metric d on a set S has the following properties. d is a function mapping S X S to the Real numbers such that 0 < d(x,y) < infinity, where x is not y, d(x,y) =0 if and only if x = y, d(x,y) = d(y,x), and d(x,z) <= d(x,y) + d(y,z) for all x,y, and z in S. It should be noted that not all sets can necessarily have a metric. (In my Humanities class I did show that St. Augustine’s concepts of good and evil could have a metric and we could speak of the greater good).
Examples a metrics on R X R are:
d(x,y) = |x2 - y2| + |x1 - y1|, d = ad1 where a is a Real number greater than 0, and d1 is any existing metric, d(x,y) = square root of ((y2-x2)^2n + (y1-x1)^2n). Generally the square roots are unnecessary for a function to be a metric.
If we consider R X R - {(0,0)} then d(x,y) = |r1 - r2| + |Theta| where r1 and r2 are the Euclidean distances from {(0,0)} and Theta is the internal angle between the rays from {(0,0)} to x and y respectively.
In studying fractals the spaces generally studied are subsets of (R X R, d), and subsets of (Riemann Sphere, spherical). Here the Riemannian Sphere = C U {infinity}. Where C is the complex plane and the spherical metric measures the length between two points along the geodesics that connects them. Visually, the Riemannian sphere can be pictured as a sphere sitting on top of the origin of R X R and points on the sphere can be identified by projecting a line from the North Pole through a particular point to its corresponding point on the R X R plane.
The space of subsets of these spaces are generally designated (H,h) where H is the collection of the subsets of R X R or the Riemannian Sphere and h is the Hausdorff metric defined below. H also has the technical restrictions of being a complete, closed, and bounded subset.
To define the Hausdorff metric, h, we start by defining d(x,A) where A is in H. Then d(x,A) = shortest distance from x to A. Now we can define d(B,A) where B and A are in H by d(B,A) = the maximum number of d(x,A) such that x is in B. The reader should notice that d(B,A) is NOT equal to d(A,B). At this point the Hausdorff measure is defined to the greater of d(B,A) and d(A,B)(1). It can be fun to prove that the Hausdorff metric is in fact a true metric.
Skip Ahead:
Barnsely wrote: “We would like the reader to think of the space itself as ‘lying above’ the coordinate system…”. Our “Physical Reality” itself could be a collection of dynamical systems represented by {Si,fi} where Si is some N dimensional space, embedded in a 2*N +1 dimensional Euclidean space (we can not shake Nash’s Embedding Theorem) and fi maps into the 2N+1 Euclidean space. Dependent on i, these spaces could be discontinuous and dust at various points that correspond to our embedding in a Euclidean measure; and continuous and connected at other points. When the reader considers Quantum Mechanics, and our inabilities to solve even simple “Real Life” physical problems, and balances that view with the fact that in some well defined domains our universe appears to be predictable and orderly, this view might appear to be enlightening.
In any case the world of fractals does not depend on the assumption that our physical reality matches the continuous smooth world of a 3 or 4 dimensional Euclidean space.