To introduce fractals
is for me to introduce the mathematical basis for the I Ching and many other topics that have excited me for forty years. My thinking, primarily visual, looks at the whole basis of physics and biochemistry as connected to life and mind structures by the simple dimensionality of the I Ching as I have reconstructed it.
But first, what are Fractals?
Fractal is a coined word meaning fractional dimension which exist between the dimensions described by whole numbers by our rationality. They are better descriptors of how nature constructs itself, whereas whole number dimensions have more use as directions in maps used by traditional science to divide reality into ratios that can easily be manipulated with mathematics of lines and "rational" numbers. In fact the bias of a world view based on objects that have a definite location and move from place to place would obviously only need such a math. I will discuss this philosophy of knowledge further in other chapters.
Using the checkerboard as a basis for constructing a fractal drawing would mean that we would draw a square 1/4 the size of the original at the midpoint of each side and continue to repeat this process by considering each smaller checkerboard as another example of the original and repeat the process by dividing the newly drawn square into 1/4 and inserting this smaller figure onto the midpoint of the new square ad infinitium. If the figure can be continued forever the outline of the square would be infinitely long. Simple? Not very, because the new figure would be somewhere in between a simple square outline and filling up the entire 2 dimensional paper. In other words it would slowly be approaching a 2 dimensional object even though it started out as a one dimensional line. Yet it could never fill out 2 dimensions, but would land at a point somewhere between 1 and 2 as a fraction 1.xxx.
Another process that creates a fractal is to subtract
or remove from a whole. Thus if the middle of each line of the
square was removed and the same process repeated on the remaining
lines, what is remaining is a Cantor "dust" [named after
the brilliant mathematician] which approaches a collection of
zero dimensional points. The removal can be a change of color
or association or intensity as can be the type of constructed
fractal. Thus the constructed fractal is a set of rules which
determine which points are connected or disconnected at each scale.
And this introduces two important ideas of fractals: scale and
connection. Scale is related to level of resolution and self reference
Here is a quote from a book to describe fractals:
The Mathematical Tourist by Ivars Peterson
Chapter 5 "Ants in Labyrinths"
The flickering flames of a campfire highlight the jagged forms of nearby rocks. The smell of frying fish weaves through the still air. Clusters of gnarled pines crouch on the ragged, stony shore. Patches of golden wildflowers, scattered across a meadow, glow in the last light of a weary sun. At the far end of the lake, a saw-toothed range of towering mountain peaks, crowned with agglomerated ice, tear into the sky. From distant, billowy clouds, a flash of lightning zigzags through the air.
This mountain landscape, like many natural scenes,
has a roughness that's hard to capture in the classical geometry
of lines and planes, circles and spheres, triangles and cones.
Euclidean geometry, created more than 2,000 years ago, best describes
a humanmade world of buildings and other structures based on straight
lines and simple curves. Although smooth curves and regular shapes
represent a powerful abstraction of reality, they can't fully
describe the form of a cloud, a mountain, or a coastline. In the
words of mathematician Benoit B. Mandelbrot, "Clouds are
not spheres, mountains are not cones, coastlines are not circles,
and bark is not smooth, nor does lightning travel in a straight
THE LANGUAGE OF NATURE
A close look shows that many natural forms, despite their irregular or tangled appearance, share a remarkable feature on which a new geometry can be hung. Clouds, mountains, and trees wear their irregularity in an unexpectedly orderly fashion. Nature is full of shapes that repeat themselves on different scales within the same object.
A fragment of rock looks like the mountain from which it was fractured. Clouds keep their distinctive appearance whether viewed from the ground or from an airplane window. A tree's twigs often have the same branching pattern seen at the tree's trunk. Elms, for instance, have two branches coming out of most forks. In a large tree, this repeated pattern, on ever smaller scales, may go through seven levels, from the trunk to the smallest twigs. Similar branching structures can be seen in the human body's system of veins and arteries, and in maps of river systems.
In all these examples, zooming in for a closer view doesn't smooth out the irregularities. Instead, the objects tend to show the same degree of roughness at different levels of magnification. Mandelbrot, the first person to recognize how extraordinarily widespread this type of structure is in nature, introduced the term self-similar to describe such objects and features. No matter how grainy, tangled, or wrinkled they are, the irregularities are still subject to strict rules.
In 1975, Mandelbrot coined the word fractal as a convenient label for irregular and fragmented self-similar shapes. Fractal objects contain structures nested within one another. Each smaller structure is a miniature, though not necessarily identical, version of the larger form. The mathematics of fractals mirrors this relation between patterns seen in the whole and patterns seen in parts of that whole.
Fractals turn out to have some surprising properties, especially in contrast to geometric shapes such as spheres, triangles, and lines. In the world of classical geometry, objects have a dimension expressed as a whole number. Spheres, cubes, and other solids are three-dimensional; squares, triangles, and other plane figures are two-dimensional; lines and curves are one-dimensional; and points are zero-dimensional. Measures of size--volume, area, and length-also reflect this fundamental classification.
Fractal curves can wriggle so much that they fall in the gap between two dimensions. They can have dimensions anywhere between one and two, depending on how much they meander. If the curve more closely resembles a line, then it is rather smooth and has a fractal dimension close to 1. A curve that zigzags wildly and comes close to filling the plane has a fractal dimension nearer to 2.
Similarly, a hilly fractal scene can lie somewhere between the second and third dimensions of classical geometry. A landscape with a fractal dimension close to 2 may show a huge hill with tiny projecting bumps, whereas one with a fractal dimension close to 3 would feature a rough surface with many medium-size hills and a few large ones. A higher fractional dimension means a greater degree of complexity and roughness. But a fractal dimension is never larger than the Euclidean dimension of the space in which the fractal shape is embedded: a hilly scene would never have a dimension greater than 3. In general, fractal geometry fills in the spaces between whole-number dimensions.
Maps of a rugged coastline illustrate another curious property of fractals. Finer and finer scales reveal more and more detail and lead to longer and longer coastline lengths. On a world globe, the eastern coast of the United States looks like a fairly smooth line that stretches somewhere between 2,000 and 3,000 miles. The same coast drawn on an atlas page showing only the United States looks much more ragged. Adding in the lengths of capes and bays, its extent now seems more like 4,000 or 5,000 miles. Piecing together detailed navigational charts to create a giant coastal map reveals an incredibly complex curve that may be 10,000 or 12,000 miles long. A person walking along the coastline, staying within a step of the water's edge, would have to scramble more than 15,000 miles to complete the trip. A determined ant taking the same shoreline expedition but staying only an ant step away from the water may go 30,000 miles. Tinier coastline explorers even closer to the shoreline would have to travel even farther.
This curious result suggests that one consequence of self-similarity is that the simple notion of length no longer provides an adequate measure of size. Although it's reasonable to consider the width of a bookcase as a straight line and to assign to it a single value, a fractal coastline can't be considered in this way. Unlike the curves of Euclidean geometry, which become straight lines when magnified, the fractal crinkles of coastlines, mountains, and clouds do not go away when observed closely. If a coastline's length is measured in smaller and smaller steps or with shorter and shorter measuring sticks, its length grows without bound. Because it wiggles so much, the true length of a fractal coast is infinite. Length, normally applied to one-dimensional objects such as curves, doesn't work for objects with a fractional dimension greater than 1.
Fractal geometry doesn't prove that Euclidean geometry
is wrong. It merely shows that classical geometry is limited in
its ability to represent certain aspects of reality. Classical
geometry is still a handy way to describe salt crystals, which
are cubic, or planets, which are roughly spherical and travel
around the sun in elliptic orbits. Fractal geometry, on the other
hand, introduces a set of abstract forms that can be used to represent
a wide range of irregular objects. It provides mathematicians
and scientists with a new kind of meter stick for measuring and
Quote 1 end
Before continuing, there is a important bias that permeates mechanical science and Western religion: namely that reality is created out of nothing. Chemistry follows this follows when they name the elements based on adding electrons instead of seeing the electromagnetic and other fields as unified wholes out of which emerges electrons and the other particles as a process of breaking the symmetry or wholeness. Of course I am referring to Newtonian physics and chemistry, because in our century science talks directly about that very process, but apparently not consistently. Thus I expect to describe other holo-fields whose symmetry is broken to produce "Matter", and that those fields operate in "higher" dimensions where mind and spirit have synergized explanations with the hologram and physics. When I use the complete set of dimensions in a fractal scaled way, where the symmetry breaking process is "Yin" and the whole is "Yang" the resulting fractal structure fits physical chemistry. This is also a complementary process where the complement or polar opposite is present at all points. Thus chemical "shells" must be complete to be stable, not just matching one for one in the same subshell. But further on this later in my model.
So where is the symmetry or connection broken? Where is the start of a process is like asking where does a circle begin! In a subtractive fractal it depends upon the ration of the "piece" removed to the whole and if it is centered between the endpoints of the connection. If there is only a single endpoint, then the "figure" is closed as a circle or square. Thus what are end points? They re "singularities" meaning places where a change takes place which is not smooth or "linear". This discontinuity can be an angle as the corner of a square or triangle, or it can be the place where a rule is applied.
This brings me to consider time as past / future within a holographic process. It one executes a process in sequence as is expected in everyday life, then time has meaning. Ones place is the present, and where one has been and will be as determined by the process is the past and future. A holographic process takes place simultaneously at all points at once. Thus one does not traverse a path from start to finish, but does many things at once. This can be seen in the everyday action of life forms where a coordination of movement and systems result in a single action. Thus within a context of moving from one state to the next state the future is now. Meaning that a future and past only appear when a complete set of actions is broken into a sequence or a lower dimension. For instance one can paint a picture one color and line at a time or one can splash paint on a canvas, or one can spray the colors thru a template for each color. In terms of fractals, the first is the lowest dimension, the splash is a random or chaotic action close to the next dimension and the spray is somewhere between. More on this later in musical fractals.
Now getting back to where the segment is removed
depends on the start or end point, which is determined by a discontinuity.
The next state of removal can be said to take place at a phase
angle to the base or start/end point. In a holoprocess all the
removals or connections can be said to occur at a phase angle.
In fact a hologram only records these phase angles in reference
[interference pattern] to a coherent source. This in effect records
the differences in arrival time between the reference light and
the "subject" light. The symmetry of the reference light
is broken simultaneously on [at] many scales. Thus the connectivity
is in reference to an original source. [The reader is advised
that connectivity refers ahead to neural networks.]
Scale and level
of resolution and systems, logic as fuzziness, chaos - complexity
and self organization.:
Scale in fractals refers to the application of the holoprocess to the result of the previous state to produce a next state. Thus if a line is divided into 3 parts and a new line is introduced at the middle part, the result of this operation is 4 or more shorter lines between the endpoints of the first line. When the operation is repeated on the shorter lines, it is said to have scaled down to a new level of detail. Each new scale repeats the operation on the resulting state produced by the previous operation. This is called an iterated or repeated function. I propose that the way we experience "time" is an iterated function on a hologram in which the smaller scaled operations are considered "past" and moving to operate on a "larger" scale to produce the next state is our future. This is the inverse of the usual fractal processing where each operation is on a successively smaller "slice" of reality. But when combined with automata theory, each "moment" of time or state is a finished computation whose result is a fractal rule representing or coding a "piece" of the hologram which can be "stored" at succeeding different scales. Thus "consciousness" does not change scale, but stores its results by a holoprocess at a different scale, and operates on the future which is at some scale beyond or "larger than" present time. The operation can be chosen from any previous "stored" operation at a different scale, or continued from present operation. The meaning of larger or more is in a phase space. In ordinary English, this means that we operate on goals and tasks which last longer than a single moment of the present and we continue to orient ourselves within that task as if we were traversing a path laid down by a fractal function, and as we pass each segment of that path we encode the results in terms of the entire function at a different scale or mapping.
From this is emerging a functional / process description of DNA as all the processes needed for "life" stored at the smallest holographic scale.
In general systems terminology, which came before "Fractals" were first described, these "scales" are called "levels of resolution". An illustration of this is clear in modern computers which have different resolution available depending on how many pixels and colors are used to represent the visual image. When the idea of levels of resolution is applied to consciousness, we see that we use many levels from 2 as polarized black/white either/or to other levels which I believe were synchronized with cosmic and planetary cycles.
A human example of levels of resolution with fractal
scales is the street map. It shows superhighways, branching to
expressways, branching to "main" streets, branching
to "side" streets, branching to driveways and homes,
branching to floors as dimensional moves of the fractal, branching
to hallways and rooms, branching to shelves and spaces to put
people [chairs, beds], branching to tables and desks and TV where
maps and pictures and books continue the process, branching to
thoughts and plans! Other examples include trees, rivers, human
and animal internal structures such as the vascular system as
introduced and described in the first quote.
It is important to understand more about dimension and how it is calculated. So I quote from the
The New Golden Age", chapter 4 "Beauty
from Chaos" by Kevin Devlin.
How Long is the Coastline of Britain?
That was the question asked in an epoch-making article of the same title published in the magazine Science in 1967. The author was Benoit Mandelbrot, a brilliant French mathematician working at the IBM Thomas J. Watson Research Center at Yorktown Heights, New York. On the face of it the question seems innocuous enough, and you might expect that a pretty good answer could be obtained either with the aid of a map or by aerial reconnaissance. The only trouble is that no matter how carefully you do it, you will not get the right answer. And for a very good reason: there is no right answer! Mandelbrot arrived at this startling conclusion by reasoning as follows.
Suppose you make your measurement by flying around the coastline in a jet airliner at an altitude of 10 000 meters, taking photographs of the coast all the time, and then, using the appropriate scaling factor, calculating the total length as indicated by the vast collection of photographs you will have accumulated. How accurate is this answer? Not very. From 10 000 meters you will be unable to distinguish a great many small bays and promontories. (Let's assume that your camera is a good but otherwise quite ordinary model.) If you were to repeat the measurement from a small airplane flying at a height of 500 meters, so much extra detail will be visible that the answer you get will be significantly greater than your previous one. What on the first photograph appeared as a smooth stretch of coastline will now be found to consist of numerous little inlets, bays, and promontories.
Now suppose you set off on foot to measure the coastline using a pair of dividers set at a separation of, say, 1 meter. Then features of the coastline not visible from the air will result in an answer which is greater still. If the measurement is repeated with the dividers set at 10 centimeters the result will be even greater. And so on. Each time you make your scale of measurement finer, more detail of the coastline shows up and your answer gets larger. Soon you are measuring around pebbles, then around grains of sand, then molecules, and so on. And all the while your answer keeps on growing.
Of course, in the physical world this process of taking finer and finer measurements must come to an end eventually. Human limitations would probably bring you to a stop with the 1 meter dividers, whilst the physicist might argue that the procedure has a theoretical limit at the atomic level. But from the idealized viewpoint of the mathematician the process of making finer and finer measurements may be continued indefinitely. Since this means that the corresponding sequence of measurements increases indefinitely, it follows that there is no mathematically precise notion of the length of the coastline, only arbitrary choices - choices which are not even approximations to some 'real' answer.
An idealized, mathematical analogue of Mandelbrot's elusive coastline is provided by a geometrical figure first considered by H. yon Koch in 1904, which we shall call Koch's island. Figure 9(i) shows Koch's island as seen from a rocket in outer space. From this distance it looks just like an equilateral triangle. As the rocket approaches Earth, it becomes clear that each of the three straight edges actually contains a central, triangular promontory, forming an equilateral triangle occupying the middle third of the line (Figure 9(ii)). If the perimeter length in Figure 9(i) is 3 units, then that in Figure 9(ii) will be 3 x (4/3) units. Coming in closer still, you see that each of the twelve straight edges you saw before likewise contains a promontory in the shape of an equilateral triangle occupying the middle third (Figure 9(iii)). The perimeter length now is 3 x (4/3) x (4/3) units. Figure 10 shows the island as seen from much closer, after several more levels of detail have unfolded, and gives some indication of the actual (?) shape of Koch's island.
To the mathematician, the nice feature of Koch's
example is the regularity with which successive levels of detail
appear. At each stage, the middle third of every straight-line
segment of the coastline is replaced by two straight-line segments,
each equal in length to that third, as shown in Figure 11.
As you might surmise from an examination of Figures 9 and 10, Koch's island does have a (mathematically) well-defined shape, which Figure 10 approximates quite well as far as the human eye can distinguish. The
mathematically precise coastline of Koch's island
is the 'curve' which is the limit of the infinite sequence of
approximations to it, of which Figure 9 gives the first three.
At this point the mathematics takes over from the human cartographer.
Mathematically, this limit curve is precisely determined, and
like any other curve will consist of an infinitude of points strung
together to form a 'line'. The process of arriving at the limit
curve is analogous to arriving at the number 1/3 as the limit
of the infinite sequence of decimals
0.3, 0.33, 0.333, 0.3333, 0.33333 .....
Since Koch's island is a mathematically defined region of the plane, it will have a definite area. The actual numerical value of its area will depend on the units of measurement being used, of course, but it will certainly be finite. (It may be calculated as a limit of a sequence of numbers, much like the 1/3 example above: it is, in fact, exactly 1.6 times the area of the triangle in Figure 9(i).) What of the length of the coastline surrounding this finite area? Well, each successive stage of the Koch process increases the length of the 'coastline' by a factor of 4/3. By the time the Koch curve (as the limiting coastline is called) is reached, this 4/3 increase will have occurred infinitely often, and so the length of the Koch curve will be infinite.
How can a finite area have an infinite boundary?
Figures 9 and 10 themselves provide the answer. The boundary curve
twists from side to side along its entire length. For each of
the finite approximations to the final curve, this twisting can
be drawn in full provided you use a suitable scale (magnification),
but for the actual Koch curve the twisting is infinite, and then
something very strange occurs: a new dimension is entered.
The curves that we usually meet in geometry are all one-dimensional: a creature constrained to live on, say, a straight line or a circle can travel in only one direction (if traveling backwards is regarded simply as negative forward movement). The usual geometrical surfaces such as planes or spheres are two-dimensional: there are two independent directions of travel, often referred to in terms of forwards/backwards and left/right. Solid objects are three-dimensional, allowing for three directions of motion. Railways provide an example of motion restricted to one dimension, ships can travel in two dimensions over the surface of the sea, and aircraft can move in three dimensions.
As far as human experience is concerned, there are only three dimensions to the universe we live in (though relativity theory regards time as a 'fourth dimension', and some current physical theories ascribe to the universe eleven dimensions, the three we are physically aware of plus a further eight which manifest themselves as the basic forces of nature, gravity, magnetism, and so on). But for the mathematician there is nothing special about three' dimensions'. 'Spaces' of four or more dimensions may, and routinely are, considered. Though they cannot be realized by traditional geometry, such higher-dimensional spaces can be of real, practical use. (A case in point is the subject known as linear programming, considered in Chapter 11.) But notice that such 'higher dimensions' are still whole numbers.
Where does the Koch coastline fit into all this? Being a curve (in the mathematical sense, though with infinite twisting you could not hope to draw it), you might imagine it is one-dimensional, but this is not so. Although each of the approximations to the Koch curve that are obtained by means of the process described earlier is one-dimensional, the limiting curve is not. With the direction changing infinitely often, we are no longer in a familiar world - indeed, that use of the word 'direction' cannot really be justified. So we cannot hope to decide on the dimensionality of the Koch curve by speaking of 'direction of travel'. What we have to do is find another way of getting at the concept of dimension, one that does not depend on direction.
It makes sense to adopt an approach which is suited to the nature of the Koch curve. The key feature is self-similarity: the parts are similar to the whole (only on a reduced scale).
Suppose we take a D-dimensional figure and divide it into N entirely similar parts. Then the similarity ratio, r, between the entire figure and a single part (i.e. the factor by which the whole exceeds the part in size) will be given by
r = D sq.-root of N
(Since the figure is D-dimensional and r has to be evaluated 'along a dimension', it is necessary to take the Dth root of N.)
For example, suppose we take a straight line and split it into N equal pieces (see Figure 12). Then each piece is exactly 1/N of the length of the whole, so the similarity ratio will be N. This is precisely the value obtained from the above formula when you take D = 1.
Or how about taking a rectangle (so D = 2) and splitting it into N pieces by dividing it horizontally and vertically into k segments (see Figure 13)? Then the entire rectangle is split into exactly N = k2 identical smaller replications of the whole, and the (linear) ratio r of the whole to any one of the parts is given by
r = D sq.-root of N = 2 sq.-root of N = 2 sq.-root of k^2 = k
Again, this is exactly what you would expect.
In both these cases we seem to have been going round in circles, but this was because we were dealing with cases that are very familiar and not at all problematical. When we apply the same analysis to the Koch curve we arrive at an altogether more surprising conclusion. For this curve we do not know D, but the values of N and r are easily determined. All we need to do is look at the replication procedure that produces the curve. We first look at one section of coastline (see Figure 11 (i)) - any one will do since they are all the same. In replication (see Figure 11(ii)) the single line is replaced by four lines (so N = 4), each one-third the length of the original line (so r = 3). Since this is true for any one piece of the coastline, it will be true for the whole Koch curve. So, according to the formula determined above,
3 = N sq.-root of 4
So what is D? Certainly not a whole number. The only way to determine its value is by using logarithms. If you take logarithms of both sides of the above equation you get
log 3 = D log 4.
By reference to a book of logarithm tables (or by use of a calculator which can evaluate logarithms), D may be calculated; to four decimal places it is
D = 1.2618.
So the Koch curve is a mathematical entity whose dimension is fractional.
It is not just 'curves' that can have fractional dimensions. Equally bizarre 'surfaces' and 'solids' may also be constructed using self-replication processes. For instance, by starting with a cube and successively removing 'middles' you arrive eventually (i.e. after infinitely many repetitions) at an object known as the Sierpinski sponge (D = 2.7268), whose construction is shown in Figure 14. This incredible object has zero volume enclosed by an infinite 'surface'. Each external face of the sponge is known as a Sierpinski carpet, and has zero area surrounded by an infinite boundary. The dimension of the Sierpinski carpet is D = 1.2618, the same as for the Koch curve. You should be able to confirm both values of D associated with the sponge by looking at Figure 14 and using the formula
r = D sq.-root of N
or, taking logarithms,
D = log N/log r.
Figures having a fractional dimension were given the name fractals by Mandelbrot in 1977. Fractal geometry is the study of such objects.
The Koch curve and the Sierpinski sponge fractals are highly regular. The self-replication process is the same at every level, and 'zooming in' on a particular part of the figure to see more detail produces no surprises - just more of the same, ad infinitum. Since 1980 computers have been used to examine fractals where the self-replication is continuously changing (though, as will become clear, it can often still be called 'self-replication'). With such figures, zooming in can produce totally unexpected results."
[end of quoted material]
Fractals where different
levels contain reference to different "generators"
and are thus not strictly self similar are like the various
frequencies and phases in a hologram, and a "pure" self
similar fractal is like a hologram of a single conceptual "object".
The hologram is in a convolutional space and the fractal as a
hologram is in a conceptually mapped convolutional space where
turns and angles may have other meanings: can be mapped into proteins
and other objects?
Magnification is another way of expressing levels of resolution and shows what Hausdorf dimension is present! Thus in the world of reality we look for even dimensions, not rational numbers as ratios. In calculations based on systems we look for numbers as they are constructed in unusual dimensions. This is what happens in computations based on the magnification using the cosmos!
So what are the dimensions of the planets?
N = number of units after magnification: 2, 12, 30, 84, .6 .25
P = object as planet = 1? or?
p = # of earth orbital radii: 1.5, 5, 10, 19,
Mars: logN = .30103 / log.66 = -1.709 or
Jupiter: 1.079 / log.2 [-.698] = -1.54 or 1.54
Sat: 1.462 / log.1 -1 = -1.46 or 1.46
Uranus: 1.92 / log.05 [1.27] = -.782 or 1.5
Neptune: 2.21 / 1.477 = 1.49 inverse as .66
Venus: -.21 /- .142 = 1.50?
mercury -.619 / -.4089 = 1.5156
Pluto: 2.38 / 1.6 = 1.48
No wonder!!!! the fractal dimension of nature!!!!!
This ~1.5 dimension is found in all contexts of life processes. In music, the dimensional measure of what is considered "good" music is halfway between the lower dimension of totally predictable and predetermined such as scales and the higher dimension of "free" choice where there is no self similarity and self reference. This is randomly determined as in much of John Cage and computer music of the 1960's and 70's.
> A side comment on the "problem of evil":
how can evil things happen to "good" people. This is
because we look for 1.5 dimension for free will, yet ignore that
the higher dimension is constructed from "everything that
can happen will happen"!
The importance of understanding how dimension is calculated can give insight into a relationship of the cosmos and nature implied by dimension. Other important Fractals are pointed to by the Star of David and the I Ching. So I quote from another book:
The Mathematical Tourist by Ivars Peterson
Chapter 5 "Ants in Labyrinths"
In 1904, Swedish mathematician Helge von Koch created a mathematically intriguing but disturbing curve. It zigzags so much that a traveler set down anywhere along the curve's path would have no idea in which direction to turn. Like many figures now known to be fractals, von Koch's curve can be generated by a step-by-step procedure that takes a simple initial figure and turns it into an increasingly crinkly form.
The Koch, or "snowflake," curve starts
innocently enough as the outside edge of a large equilateral triangle.
[F I G U R E 5.1 The first four stages in constructing a Koch snowflake. This figure is of
1. a black triangle
2. A full star of David by superimposing 2 opposing triangles. In the next chapter I shall show that this is one of the most significant patterns of the cosmos constructed in the 60 year cycle by the conjunctions and oppositions of Jupiter and Saturn.
3. Shows all the straight lines replaced by the point of a triangle at the middle third!
4. Repeats #3 with each new triangle becoming so
small as to be almost unprintable. As this process repeats any
number of times, it seemingly runs to the infinitely small. But
in the time dimension translated into DNA and cycles, there is
no problem of infinity because the fractal is stretched out in
space. This does not change the self similarity in the time dimension
The addition of an equilateral triangle one-third the size of the original to the middle of each side of the large triangle turns the figure into a six-pointed star. The star's boundary has 12 segments, and the length of its outer edge is four-thirds that of the original triangle's perimeter. In the next stage, 12 smaller triangles are added to the middle of each side of the star. Continuing the process by endlessly adding smaller and smaller triangles to every new side on ever finer scales produces the Koch snowflake. Any portion of the Koch snowflake magnified in scale by a factor of three will look exactly like the original shape.
The boundary of the Koch snowflake, so convoluted that it's impossible to admire in all its fine detail, is continuous but certainly not smooth. It has an infinite number of zigzags between any two points on the curve. A tangent--the unique straight line that touches a curve at only one point--can never be drawn anywhere along its perimeter. Moreover, the length between any two points is infinite, yet the curve bounds a finite area not much bigger than the area of the original equilateral triangle. In fact, despite the figure's wrinkled border, it's possible to compute that the area bounded by the curve is exactly eight-fifths that of the initial triangle.
Such strange mathematical behavior led mathematicians at the turn of the century to label this and several similar curves as mathematical monstrosities. But these monster curves are remarkably easy to generate. Pointing triangles inward rather than outward-subtracting instead of adding them at each step--produces the anti-snowflake curve. This lacy form, too, has an infinitely long outer edge that intersects itself infinitely often, but its area is only two-fifths that of the starting triangle.
The same idea of adding or subtracting pieces that
are successively smaller in size works for a square or any other
polygon. It also works in three dimensions. Dividing each face
of a regular tetrahedron into four equilateral triangles and erecting
a smaller tetrahedron on each face's middle triangle, then continuing
this step-by-step procedure indefinitely creates a prickly, three-dimensional
analog of the Koch snowflake. Its surface area is infinite, but
the figure bounds a finite volume. The supply of monsters seems
No wonder mathematicians were disturbed when they first encountered such bizarre behavior. These pathological curves and surfaces, they believed, were aberrations--skeletons in the closet of otherwise orderly mathematics. To them, such figments of the imagination represented a mathematical pathology having nothing to do with any possible real-world phenomenon and were unlike anything found in nature.
Nevertheless, a few mathematicians took these monster shapes seriously enough to explore their properties in some detail. In 1919, the German mathematician Felix Hausdorff suggested a way to generalize the notion of dimension, which put these disturbing forms into a class of their own. He came up with the idea of fractional dimension, a concept that is now one of several methods used to characterize a fractal.
The idea of fractal dimension extends the concept of dimension normally used for describing ordinary, regular objects such as squares and cubes. The idea is to figure out how many small objects or units of size p are needed to cover a large object of size P. In the case of a line segment, say, 8 meters long, it takes eight 1-meter lengths to cover the whole line. If the measuring unit were 10 centimeters long, then it would take 80 such units to cover the 8-meter length. The ratio between the two results, 80 and 8, is 10:l or 10^1, which is the ratio of the measuring-stick lengths: 1 meter and l0 centimeters. The exponent 1 matches the dimension of a line. Similarly, in the case of area, a square 1 meter by 1 meter fits into an 8-square-meter area eight times, whereas a measuring square 10 centimeters by 10 centimeters fits into the same area 800 times. The ratio between the two results, 800 and 8, is 100, or 10^2. The ratio of the measuring-square unit areas (1^2 m:10^2 cm) is also 10^2, and the exponent 2 matches the dimension normally associated with area. Volume can be handled in the same way, using measuring cubes of different sizes. In every case, the dimension of the object appears as the exponent of the ratio of the length scale of the measuring units.
The whole process of finding the dimension of an object can be turned into a mathematical operation of taking logarithms of the appropriate ratio. Tripling the width of a square creates a new square that contains nine of the original squares. Its dimension is calculated by taking logarithms of the size ratio, or magnification: log 9/log 3 -log 3^2/log 3 -- 2 log 3/log 3 -- 2. Thus a square is two-dimensional. Doubling the size of a cube produces a new cube that contains eight of the original cubes. Its dimension is log 8/log 2 = log 2^3/log 2 = 3 log 2/log 2 = 3. It's no surprise that a cube is three-dimensional.
In general, for any fractal object of size P, constructed
of smaller units of size p, the number, N, of units that fits
into the object is the size ratio raised to a power, and that
exponent, d, is called the Hausdorff dimension. In mathematical
terms, this can be written as
N = (P/p)^d or d = log N/log (P/p).
This way of defining dimension shows that familiar objects, such as the line, square, and cube are also fractals, although mathematically they count as "trivial" cases. The line contains within itself little line segments, the square contains little squares, and the cube little cubes.
Applying the concept of the Hausdorff dimension to the Koch curve gives a fractional dimension. Suppose at the first stage of its construction, the snowflake curve is 1 centimeter on a side. With a resolution of 1 centimeter, the curve is seen as a triangle made up of 3 line segments. Finer wrinkles aren't visible. If the resolution is improved to 1/3 centimeter, 12 segments, each 1/3 centimeter in length, become evident. Every time the unit of measurement is cut by a factor of one-third, the number of visible segments increases four times. In this case, N = 4 and P/p = 3. Hence, 3^d = 4, so log 3^d = log 4, d log 3 = log 4, and d = log 4/log 3. The Hausdorff dimension of the fractal Koch curve is 1.2618 .... The strange properties of the snowflake curve stem from the fact that it is not a one-dimensional object. It belongs in the wonderland of fractional dimensions.
Fractals in nature typically lack the regularity evident in a Koch curve, but natural fractals are often self-similar in a statistical sense. With a large enough collection of examples, a magnified portion of one sample will closely match some other sample in the collection. The fractal dimension of these shapes can be determined only by taking the average of the fractal dimensions at many different length scales. Although the Koch curve's peninsulas and bays occur with absolute precision, they bear a striking resemblance to the shape of a rugged coastline. Although the bends and wiggles of a coastline are not in precisely the same locations on all scales, the coastline's general shape looks the same no matter what scale is used for measurement.
Another significant difference between a Koch curve
and an actual coastline is that the curve is an idealized mathematical
form with structure at infinitely many levels. A physical coastline
isn't really an infinitely complex line. However, for many purposes,
a fractal is a better, though not necessarily perfect, model for
a coastline than is a smoother, finitely complex curve. The similarity
between an abstract object like the Koch curve and the typical
characteristics of a coastline is enough to define an approximate
fractal dimension for actual ocean coastlines. Studying maps on
different scales shows that ocean coastlines vary in dimension
but generally range from 1.15 to 1.25 not quite as rough as a
One of the joys of fractal geometry is the opportunity to create new monster curves and other pathological forms. Usually, that means starting with some basic shape or figure (the Initiator), then applying a rule (the generator) that step-by-step makes the figure more irregular, tangled, or wrinkled on ever smaller scales in an endlessly 1ooping process.
One strange form, called the Sierpifiski gasket, starts as a triangle, like the Koch snowflake, but all the drawing and cutting take place within the figure. Marking the midpoints of the three sides and joining those points creates a new triangle embedded within the original triangle. This construction breaks up the large triangle into four smaller triangular pieces: one central and three corner triangles. Cutting out the central triangle leaves the three corner triangles. Then the process of finding midpoints and drawing in a triangle is repeated within each of the remaining triangles. The central triangle that sits within each of these corner pieces is cut out, leaving nine small triangles. The pattern seen in the first step is thus duplicated in each of the corner triangles within the figure. The process continues indefinitely to generate an arrangement of triangles nested within triangles nested within triangles, and so on, resulting in a two-dimensional sieve punctured by an infinite number of holes (see Figure 5.2).
At each step in the process, the length of a triangle's side is cut in half, and three times as many triangles appear. Therefore, the Sierpinski gasket has a Hausdorff dimension of log 3/log 2 = 1.584 .... It also happens to have zero area.
A similar procedure can be applied to a square. Dividing each side of a square into three parts to create a 3 by 3 grid and removing the central square is the first stage in generating an infinitely moth-eaten Sierpinski carpet (see Figure 5.3). Its fractal dimension is log 8/log 3 = 1.8928 .... showing that this figure is actually more like a bumpy curve than a real carpet.
The same operation can take place in three dimensions. Starting with a cube, dividing it into 27 smaller cubes, and removing the central cube as well as the cubes lying at the center of each face of the original cube (7 in all) leads inexorably, step by step, to a novel form called the Menger sponge (see Figure 5.4). Its dimension is log 20/log 3 = 2.727 .... Because its dimension is closer to three than to two, the Menger sponge is more a solid body than a smooth surface.
At the opposite dimensional extreme, a similar removal process can be applied to a one-dimensional line segment. In this case, a section is removed from the middle of a line. Then a corresponding section is removed from the middle of the two remaining pieces, and so on, until the line falls apart in a shower of dimensionless fragments. This extremely simple fractal is an example of a Cantor set, named for mathematician Georg Cantor. All Cantor sets created by breaking up a line have a dimension between zero and one.
Mandelbrot, who initiated much of the research on fractals, has been particularly inventive in generating novel fractals with unique properties. Some of these fractals resemble natural shapes, such as mountains. Others have geometric features that make them useful as idealized models for physical processes, such as the seeping of oil through porous rock or the way in which a material becomes magnetized. Until Mandelbrot recognized fractal forms as a potentially rewarding tool for analyzing a variety of physical phenomena, these forms remained bizarre and useless curiosities.
The first application of fractals was to solve the problem of noise during data transmission. Like the irregular crackling sound sometimes heard during a radio broadcast, electrical disturbances Interrupt and confuse the flow of data over telephone lines and other transmission channels. By looking at the pattern of errors that occur when computer data are transmitted electrically as a group of on/off signals, Mandelbrot noticed that the errors seemed to show up in bursts. Examining these bursts more closely, he found that each burst was itself intermittent. Those shorter bursts and their associated gaps also had a similar structure. The best mathematical model for these characteristic bursts appeared to be the Cantor set.
The Cantor set can be extended to three dimensions to become what Mandelbrot calls a Cantor dust. This fractal dust starts off as a solid block of matter, which is divided into stacks of smaller blocks. Some of the smaller blocks are randomly removed. The remaining blocks are subdivided further, and even more matter is randomly removed. Gradually, this Swiss-cheese structure comes to resemble water droplets scattered in a cloud or even the clusters of stars and galaxies dispersed throughout space. Ultimately, it becomes a dust of totally disconnected points.
Using the Cantor-dust concept, Mandelbrot concocted
distributions for stars in a galaxy and galaxies in the universe.
Although the model is a forgery and uses no real astronomical
data, it closely resembles the hitherto inexplicable distributions
of stellar matter seen by astronomers. Astrophysicists have now
confirmed that as the model predicts, mass within the universe
is distributed throughout space like a three-dimensional Cantor
set, with large regions of space left empty. Cantor-set fractals
describe not only the way matter clusters in space but also the
way it clusters in time. Cantor sets seem to describe cars on
a crowded highway, cotton price fluctuations since the nineteenth
century, and the rising and falling of the River Nile over more
than 2,000 years.
[The Cantor dust is the "ideal" version of the I Ching and it is found to model more than the stars!]
As the search for fractals branches into ever more exotic trails, the concept of fractal is also shifting. Mathematicians are finding that the idea of self-similarity alone doesn't cover all possible fractals. In 1980, Mandelbrot himself described a fractal set that John Hubbard later named the Mandelbrot set. It has the appearance of a snowman with a bad case of warts and comes up when simple mathematical expressions such as x^2 -- 3x are repeatedly evaluated, starting with an initial value for x, then substituting the resulting answer back into the original equation, and so on. On superficial inspection, the Mandelbrot set looks like a self-similar fractal, with infinitely many copies of itself within itself. On detailed investigation, however, the set is extraordinarily complicated. The baby Mandelbrot sets within the parent Mandelbrot set are fuzzier than the original. They have more "hair" and other curious features. Their own baby Mandelbrot sets are fuzzier still. This hairiness increases at finer and finer scales until it seems to carpet whole areas.
Fractals such as the Mandelbrot set are called nonlinear fractals. For self-similar fractals, lines that show up within a figure, whether blown up or reduced in size, remain lines. For nonlinear fractals, such a change in scale doesn't necessarily preserve the straightness of individual lines. Instead, changes in scale reveal a host of fascinating new features, many of which make an appearance in the next chapter.
Mathematicians have come across a broad range of
objects that have the property of infinite irregularity characteristic
of fractals. Such forms ultimately raise a not yet satisfactorily
resolved theoretical question: What should or should not be included
within the compass of the term "fractal"?
PACKING IT IN
Fractal geometry and computer graphics are inextricably linked. Computer graphics provides a convenient way of picturing and exploring fractal objects, and fractal geometry is a useful tool for creating computer images. The simple, repeated operations that go into the construction of a fractal are ideally suited to the way a computer functions. The computer patiently performs the same set of operations over and over again to generate a particular fractal object. Of course, the fractal image that appears on a computer's video display isn't really, in a mathematical sense, a true fractal. A computer generating a Sierpinski gasket, for instance, can display no triangles smaller than the tiny spots of light, or pixels, that make up the screen. The true Sierpinski gasket is filled with even smaller triangles. Computer graphics therefore provides a useful but only approximate picture of fractal forms.
Fractal geometry also provides an elegant, efficient way to draw realistic natural objects on a video screen. It overcomes some of the disadvantages that crop up when more conventional geometric techniques for creating a video image are used. One common conventional method is to piece the image together from simple Euclidean shapes such as squares, triangles, and circles. Such constructions work well for angular, human-made objects such as bridges, robots, and spacecraft, but they can't capture the enormous detail and irregularity evident in clouds or natural terrain.
Specifying all that detail is a monumental computing task requiring either a lengthy computer program or one that would need huge amounts of time to compute a given scene. Fractals suggest a simpler solution. The fractal approach to drawing pictures involves feeding into a computer a small set of numbers that generates a basic shape, such as a triangle. That shape is then recreated many times on smaller and smaller scales within the original figure. Random variations are thrown in to make the image look a little rougher and, as a result, more realistic. Such artificial landscapes can be mathematically magnified to reveal more detail, just as a close-up lens probes deeply into a natural scene.
One way to build a mountain using loosely based fractal geometry concepts is to start with a triangle. The computer first finds the midpoint of each of the triangle's sides. Each midpoint is then displaced along its corresponding edge through a distance determined by a random-number generator. Joining the displaced midpoints generates a new triangle and divides the original figure into four smaller triangles. Unlike the triangles in the Sierpinski gasket, these component triangles aren't necessarily identical or even equilateral. The same procedure is applied in turn to each of the four new triangles, generating sixteen triangles, to each of which the procedure is applied again, and so on. The process continues until the individual triangles are so small that their edges can no longer be distinguished. Although the algorithm for subdividing the triangles is straightforward, the resulting figure is a complex polygonal surface. The addition of color and shadow turns this figure into a reasonable facsimile of a mountain (see Figure 5.5).
To generate a fractal landscape, the designer can enter into the computer a set of elevations, specifying the locations of mountain peaks and valleys. The computer then connects these points to produce a complicated polygon. That figure is further subdivided into a mesh of simple triangles. Now the computer subdivides each triangle into smaller triangles, using the same technique as for drawing a single mountain. In the end, the computer screen shows an irregular terrain made up of a large number of triangular facets. Given sufficient computing time, the facets can be subdivided to the point where their edges are too small to be distinguished.
This particular recipe for building a mountain landscape is one of the simplest of several possible construction schemes. Mandelbrot's own scheme, which sticks more closely to the true characteristics of a fractal, generates somewhat more realistic scenes--ones that wouldn't be mistaken for a mound of crumpled paper. (The price of this greater realism, however, is longer computation time to generate the image.)
Mandelbrot and his colleagues start with a mathematical construct that closely resembles a random walk-- a sequence of up and down steps, determined by the toss of a coin. A set of these jagged, vertical cross sections is then assembled to form a markedly rough landscape. A final mathematical smoothing operation gives the constructed scene a more natural look. Any points on a random-walk cross section that happen to go below an arbitrarily set "zero" level are automatically reset to zero. Collections of these points appear as depressions filled with water (see Figure 5.6 ant/Color Plate 5).
Clouds can be created by putting together pictures
of selected components of white noise, which corresponds to the
hiss heard between FM radio stations. White noise, which can be
described by fractal geometry, consists of fluctuations spread
evenly throughout the radio-frequency spectrum. Selecting a fractal
dimension of 3.2 puts together a combination of long-wave and
short-wave components that, when plotted, produce the proper puffiness
for a cloud. With the addition of the correct lighting and coloring,
the result is a soft-looking, wispy object.
Fractal coastlines can be generated in much the same way that clouds are created. Both clouds and coastlines have the same basic, computer-generated relief or outline. The key difference lies in the type of lighting applied to the fractal outline. Light reflects directly off the surface of a landscape, but it partially penetrates a cloud and scatters, or softly diffuses. A computer program can be written to mimic these different lighting effects.
Snowflakes can be made by using a fractal branching program to generate a tree structure. The treelike form is then reproduced six times, in the same way that a kaleidoscope creates a picture with sixfold symmetry. By changing certain parameters in the branching routine, a wide variety of different snowflake designs can be created (see Figure 5.7). However, because a snowflake Isn't really a true fractal, the images don't always look quite right.
Attempts to generate realistic fractal models of trees have had only limited success so far. As trees grow, twigs and branches tend to avoid one another and to die off when severely overshadowed. The combination of randomness and a strong degree of self-interaction greatly complicates the generation of acceptable fractal images of trees.
The fractal images just described have been created largely by trial and error and are the result of computer doodlers' stumbling accidentally on mathematical procedures, or algorithms, that happen to lead to drawings that look like natural objects. The trial-and-error period, however, is reaching an end as more systematic approaches to creating fractal images are developed.
Michael Barnsley of the Georgia Institute of Technology has tackled the problem of finding a specific fractal to fit a given natural object. He and his coworkers have been studying how a scene's geometry can be analyzed to generate an appropriate set of rules that can then be used to recreate the scene. Because fractal mathematics is a compact way to store the characteristics of an object, this approach would compress the content of an image into just a few equations.
The idea is to start with a digitized picture. Such a picture may consist of a 1,000 by 1,000 grid of dots or pixels. Each pixel is assigned, say, eight bits of data to represent 256 different shades of gray or an equal number of different colors. Thus, the entire picture can be thought of as a string of 8 million ones and zeros, one digit for each bit. If this string of digits can be encoded in some way to produce a new, shorter string of digits, then the image is said to have been compressed. Of course, the compressed string should be able to reproduce, pixel for pixel, the original picture.
Barnsley and his colleagues believe that the key to image compression is in the redundancy found in natural forms. For instance, one pine needle is more or less like any other pine needle. That means that there's really no need to describe each needle over and over again. It's sufficient to describe just one. Fractals, in which similar structures are repeated at smaller scales within an object, also capture much of the redundancy found in nature.
Several important mathematical concepts lie at the heart of Barnsley's scheme. His procedure relies on mathematical operations called affine transformations. An affine transformation behaves somewhat like a drafting machine that takes in a drawing--that is, takes in the coordinates for all the points making up the lines in the drawing--then shrinks, enlarges, shifts, rotates, or skews the picture and finally spews out a distorted version of the original.
Affine transformations can be applied to any type
of object, including triangles, leaves, mountains, ferns, chimneys,
clouds, or even the space in which an object may sit. In the case
of a leaf, the idea is to find smaller, distorted copies of the
leaf that, when fitted together and piled one on top of another
so that they partially overlap, form a "collage," which
approximately adds up to the original, full leaf. Each distorted,
shrunken copy is defined by a particular affine transformation,
or "contractive map," as it's called, of the whole leaf.
If it takes four miniature copies of the leaf to approximate the
whole leaf, then there will be four such transformations
Now the original image or "target," whether leaf or cloud, can be thrown away, leaving only the corresponding collection of affine transformations. These can be used to recreate the original image, essentially by molding a piece of space. That's done by starting with a point somewhere on a computer screen and applying one of the available transformations to shift the point to a new spot. That spot is marked. Again, randomly applying one of the transformations shifts the point to another location. The new spot is colored in, and the process is repeated again and again.
Amazingly, although the designated point appears to hop about aimlessly, first pulled one way then another, a pattern gradually emerges. The point's colored tracks add up to an image called an attractor--a concept discussed in more detail in Chapter 6. In the case of the four leaf transformations, the attractor is an object that looks very much like the original leaf.
How this part of the process works can best be understood using a simple example on a sheet of squared graph paper. Imagine a rectangle with three of its corners labeled l, 2, and 3. Suppose there are also three transformations. The first shrinks everything toward corner 1, the second shrinks everything toward corner 2, while the third shrinks everything toward corner 3, always by a factor of one-half. Now, select a starting point somewhere on the grid. Randomly apply one of the three transformations. That transformation will designate a new point halfway between the original point and one of the corners, depending on the choice of transformation. Randomly applying a second transformation (it may be any one of the three available) locates another point halfway between the previous point and the appropriate corner. As you chase the point around the sheet of graph paper, marking each landing spot, you are producing a somewhat distorted Sierpifiski gasket.
It turns out that any particular collection of affine transformations, when iterated randomly, produces a unique fractal figure. The trick is to find the right group of transformations to use for generating a particular image. That's done using the "collage" process, for example, a leaf covered by little copies of itself. Furthermore, the probability of using a certain transformation need not be the same as the probability of applying any other transformation in the set. And because some grid squares are likely to be visited more often than others, keeping track of the relative number of visits to each square provides a way to specify color brightness and intensity or to define a gray scale. In this way, a lot of information is packed into a few formulas, and you can compress images by encoding them as a collection of rules. Randomly iterating the rules then recreates the image. Using his technique, Barnsley and his group have been able to create remarkable three-dimensional renderings of natural objects such as ferns. Their graceful model of a black spleenwort fern (see Figure 5.9 and Color Plate 6) is the product of the application of a collage of four affine transformations--each a combination of a translation, a rotation, and a contraction.
Barnsley has also been able to come up with reasonable fractal reproductions of photographs. In one instance, he uses 57 affine transformations, or maps, as they are often called, and four colors-a total of 2,000 bytes of information--to model three chimneys set in a landscape against a cloudy sky. "The idea is that we can fly into this picture," Barnsley says. "You can pan across the image, you can zoom into it, and you can make predictions about what's hidden in the picture."
As you blow up the picture to show more and more detail, parts of the picture degenerate into nonsense, but some features, such as the chimneys, the smoke, and the horizon, remain reasonably realistic, even when the image compression ratio is more than 10,000 to 1. The degree of compression attained depends on how much of the regenerated picture makes sense.
Interestingly, the images created using this scheme aren't self-similar. Instead, they're self-affine because an object such as one of Barnsley's fractal ferns shows slightly different features on different scales. Magnifying the image reveals subtle differences in form and color, and the magnification can be carried on indefinitely, as for any fractal.
If image compression can be made as effective as
Barnsley's work suggests and if the whole process can be automated,
it could provide an efficient means for storing data in a computer's
memory, for transmitting photographs over telephone lines, for
recognizing specific objects in a landscape, and for simulating
natural scenery on a computer. Someday, it may even be possible
to convey a movie from one computer to another simply by sending
a chain of formulas down a telephone line.
When the delicate fragrance of a perfume weaves through the air, individual perfume molecules, Jostled about haphazardly in the hurly-burly of their collisions, follow a Jagged path. Air currents further tangle these tortuous paths into monstrous trails. This type of irregular motion can also be seen in the unceasing, restless dance of tiny, barely visible particles suspended in a liquid. Such jitteriness is known as brownian motion.
Theoretically, a scientist can trace a brownian particle's irregular trajectory by periodically plotting the particle's precise location. The result is a string of points strewn across a sheet of graph paper (see Figure 5.10). But using straight lines to connect the picture's dots fails to capture the motion's true intricacy. Between every pair of plotted points lies another jagged path. That's just the kind of motion whose essential features can be compactly expressed by fractal geometry.
This type of motion is akin to a random walk--a sequence of steps whose size and direction, as we saw earlier, are determined by chance. In a simple, one-dimensional version of a random walk, a person flips a coin and takes one step forward if the result is heads and one step backward if the result is tails. Across a broader stage, a random walk looks even more like the steps of a drunken sailor.
When a brownian trajectory confined to a plane is
Increasingly closely, Its length, like that of a coastline, grows without bound. In fact, the trail itself ends up filling practically the whole plane In which the motion takes place. Although the trajectory Itself is a one-dimensional curve, the path's tendency to fill the plane marks It as a fractal of dimension 2.
Fractal concepts can be used to describe not only an astonishing array of fragmented or branching natural structures but also the dynamic properties of these structures--from the movement of Brownian particles to the drip of scalding water through coffee grounds. In a way, fractals represent a new kind of meter stick that scientists can use to measure natural phenomena. For example, they can use fractals to study the way materials are put together, the way they shatter, the way they branch, and the way they conduct heat or electricity.
A word of caution is warranted, however. Mathematical fractals have properties that aren't actually found in natural objects. No real structure can be magnified repeatedly an infinite number of times and still look roughly the same. One reason is the finite size of atoms and molecules. Another is that real objects may, at some magnification, abruptly shift from one type of structural pattern to another. Nevertheless, fractal models provide a useful approximation of reality, at least over a finite range of scales.
The microscopically jagged surface of a piece of fractured metal is one example of a material property that lends itself to fractal analysis. A few years ago, Benoit Mandelbrot worked with some metallurgists to come up with a method for specifying the roughness of a given surface. They found that a large number of broken metal surfaces, although not all, have a roughness that can be represented by a fractal dimension. Experiments showed that the measured fractal dimension takes on the same value for different specimens of identically treated samples of the same metal. The investigators also discovered that different heat treatments not only affect the toughness of a metal but also change its fractal dimension. They concluded that a metal surface's fractal dimension may itself be a useful measurement of a metal's toughness or strength, providing metallurgists with a new tool for characterizing metals.
Fractal ideas have also encouraged scientists to look anew at old, seemingly inexplicable experimental results once destined for the wastebasket and to reexamine problems that previously looked so complicated that they were ignored. Physicists and other researchers now realize that many formerly puzzling results actually reflect the dimensions of fractal geometric objects. With this insight, some apparently complex problems become relatively simple.
In analyzing experimental data, scientists usually look for simple relationships between variables; such as the relationship between the intensity of sound waves scattered from a metal surface and the waves' frequency. If a theory predicts that doubling the frequency will quadruple the intensity, the intensity should be proportional to the frequency squared. However, in many experiments, the exponents that express the proportionality turn out to be numbers like 2.79 instead of integers. Scientists who were taught to think of integers as the natural way to represent physical processes are only now beginning to see that noninteger exponents are just as likely, If not more likely, to turn up in nature.
An example from the world of electrochemistry shows the interaction between experimental observation and a mathematical model. For decades, scientists had noticed that the interface between a metal electrode and the electrically conducting liquid, or electrolyte, in which the electrode is immersed has strange electrical properties. Simple electrical theory predicts that the interface should behave like a capacitor, an electrical device that stores then abruptly releases electrical charge. When an alternating current passes between electrode and electrolyte, it ought to meet a resistance that is simply related to the frequency. However, experimental studies show that this resistance is actually inversely proportional to the frequency raised to a fractional power between 0 and 1.
Further study shows that the resistance appears to depend on the roughness of the electrode surface. As the surface is made smoother, the exponent approaches 1. But, under magnification, even well-polished electrode surfaces still show long grooves with jagged cross sections and edges. The grooves themselves have finer scratches, suggesting a degree of self-similarity.
The theorist's job is to come up with a suitable mathematical model that captures an electrode's observed characteristics and its consequent behavior. That mathematical model must be simple and regular enough to be mathematically solvable but not so far removed from an electrode's observed features that any mathematical results from applying the model would be suspect. The idea is not to paint a realistic portrait but to capture the spirit of the phenomenon with a caricature. In that way, researchers may catch a glimpse of what is actually happening at the interface.
Given the nature of the observed electrode grooves, a good place to start is with a fractal. But which fractal? There are many to choose from. Some are easier to handle mathematically than others. One good fractal candidate, related to the Cantor set described earlier, is called the Cantor bar and has a dimension less than 1.
The Cantor bar begins as a thick, solid line segment. Taking a chunk out of the center of the bar breaks it into two pieces. The two resulting fragments are in turn broken according to the same rule, and the process is repeated ad infinitum (see Figure 5.1 I, left). If the length of each broken piece is 1/a times the size of the original piece, where a is greater than 2, then the object has a fractal dimension of log 2/log a, which must be less than 1.
When the increasingly fragmented bars are welded together-starting with the full, original bar on the bottom, then adding the next two pieces, then topping those with the next four broken pieces, and so on--the result is a kind of symmetric urban landscape with stepped towers stretching higher and higher as they get thinner and thinner (see Figure 5.11, right). Although the surface of an electrode isn't nearly this regular, the mathematical model captures some of the features of a rough, grooved surface, especially the idea of grooves within grooves within grooves.
Giving electrical properties to this geometric pattern allows the calculation of various quantities that can also be measured experimentally. Currents, for instance, can pass directly across the interface or out along the grooves. That leads to a simple equation predicting that the interface resistance is inversely proportional to the frequency raised to a certain power, and the exponent depends on the geometry of the surface. Now, it's up to experimentalisis to see how well this fractal theory matches what's seen in the laboratory. The theorist would then be able to refine the model further and to suggest which measurements would more likely reveal the processes occurring at the electrode-electrolyte interface.
Fractal models also play a useful role in percolation processes. The term conjures up an image of brewing coffee or the trickle of a liquid through a gravel bed, but it also applies to an important class of structures known as percolation clusters. Such clusters have properties similar to those of a floor covered with a random mixture of copper and vinyl tiles. Current will flow from one side of the floor to the other if there is a continuous copper path, no matter how roundabout. If most of the tiles are vinyl, current isn't likely to go through. Adding more randomly placed copper tiles increases the likelihood that patches of copper tiles will be linked to form a continuous path. A percolating cluster represents the point at which a conducting path is first created. Percolation clusters are particularly useful for modeling processes such as the seeping of oil through porous rock and the spread of plant species in a meadow.
This construct seems to work as a mathematical metaphor for many materials in which different substances having antagonistic properties are intermingled. Such materials may contain random mixtures of electrical conductors and insulators, magnetic and nonmagnetic components, or elastic and brittle parts. The critical concentration of the two materials in the cluster, not the simple average of the properties of the components, determines which property predominates and how the mixture behaves. That in turn decides physical characteristics, such as the structure of some polymers, the conductivity of alloys, the efficiency of telephone networks, the propagation of forest fires and infectious diseases, and the spread of plant species in an ecosystem. Percolation models also provide useful pictures of abrupt changes in phase, such as the lining up of atomic spins to create a magnet, or the sudden, temperature-dependent onset of superconductivity in a thin metallic film.
Not only are the results of calculations done on an actual, random percolation cluster hard to understand, but the computations use up a great deal of computer time and are only approximate because of the complexity of the process. Researchers therefore look for fractal patterns that are systematic enough to ease mathematical calculations but random enough to be realistic.
Fractals also come up in percolation problems in another way. In a problem first proposed in the sixteenth century, an orchard is laid out in the shape of a checkerboard, with each space occupied by a tree. The spaces are small enough that neighboring trees touch each other. Tragically, a plague invades the orchard and spreads from tree to tree, destroying the entire fruit crop. The question at the time of replanting is how many trees should be excluded from the grid to prevent a disease from spreading from one end of the orchard to the other, while still maximizing the fruit yield.
Over the years, various attempts to answer the question have led to a best estimate that about 41 percent of the checkerboard squares should remain unoccupied. The critical concentration for percolation (when the disease can make its way from tree to tree across the orchard) is 59.2 percent. At this threshold value, the pattern of trees occupying the orchard is self-similar, from the overall pattern spanning the orchard down to the individual tree. Above the percolation threshold, the tree cluster is self-similar only on certain scales. On other scales, the pattern is merely some Euclidean object.
Another intriguing and scientifically rewarding pursuit
is to look at fractals upon fractals. Diffusion, as exemplified
by the peregrinations of perfume molecules, is a fractal process.
When diffusion occurs along a fractal surface, the process is
something like the wanderings of an ant in a labyrinth (see Figure
5.12). An ant constrained to roam along a straight line always
returns to its starting point eventually. On a two-dimensional
plane surface, the ant gets lost. But on fractal paths with dimensions
between 1 and 2, what happens to the ant isn't completely known.
Depending on the nature of its fractal labyrinth, the ant may
keep running into dead ends forever, or it may return to its starting
point very infrequently.
...... STICKY STUFF
One of the wonders of nature is how
simple water molecules can settle into symmetrical structures
that have the limitless variety and intricacy of snowflakes. Very
little is known about how these lacy shapes are created. Why does
a snowflake branch? How does one branch tell another which way
it's going so that the whole flake stays more or less symmetrical?
To try to account for how a snowflake or other growing body flowers into its final form, scientists have developed a variety of simple mathematical models that suggest ways in which growth can occur. By studying these models and comparing them with experimental observations, they can begin to guess what forces and conditions underlie various types of growth.
One of the simpler growth models, out of which a fractal form springs, starts as a tiny cluster, or "seed." Every time a particle wandering nearby happens to bump into the cluster, it sticks and stays put. Once it has arrived, the particle never jumps to another site. This type of process is called aggregation. If particles diffuse to the cluster by means of random walks--ordinary Brownian motion in three dimensions--the process is known as diffusion-limited aggregation.
This growth process can be simulated in two dimensions, step-by-step on a computer. At the center of a grid sits a small, connected set of dots representing the initial aggregate. Each dot lies in a separate square of the grid. Far away from the central cluster of dots, a single particle starts its random walk. When the walker eventually arrives at an unoccupied space neighboring a square already filled, it stops and stays put, and the cluster grows by one unit. The process is continued for perhaps 50,000 or 100,000 wandering particles.
The resulting pattern is a typical fractal object, which looks like a bare tree seen from above, with branches shooting off in all directions (see Figure 5.13, left and Color Plate 7). This particular fractal object also contains tremendous empty regions, often in the form of long, narrow channels, or fiords, that penetrate far into its interior. What accounts for this distinctive structure is the fact that few late arrivals can get deep into a fiord. Random walkers that happen to reach a fiord's outlet and begin their journey down the channel are much more likely to bump into a wall than they are to make it all the way to the end.
In an effect called growth instability, any cluster that is slightly distorted initially will become even more distorted. Once the initial cluster begins to develop bumps and hollows, the bumps end up growing much faster than the hollows. Because particles are more likely to stick near peaks, the bumps grow even steeper, and the fiords become less and less likely to fill. Eventually, continual growth and splitting of the protruding ends give rise to a heavily branched, globular fractal.
What happens if a particle sometimes bounces off
instead of sticking? Computer simulations show that this leads
to a thickening of the branches and an increase in fractal dimension,
but the figure remains a fractal.
Although diffusion-limited aggregation is easy to describe and simulate, the process is not yet well understood at a deeper level. No one is sure why the process gives rise to fractals rather than shapeless blobs that have no symmetry at all. Why are loops--or open lakes surrounded by matter--so rarely formed? How does the fractal dimension depend on the dimension of the space in which the process occurs? Ordinary mathematical tools seem inadequate for answering these and related questions.
The diffusion-limited aggregation model can be used, nevertheless, to explain a particular natural growth process, such as the deposition of metals during an electrochemical reaction (see Figure 5.14). Not only do many natural forms have fractal characteristics, but fractals can also serve as models for new materials that don't exist naturally. For example, diffusion-limited aggregation leads to materials with an especially large surface area. Experimentalists have created one such material by letting microscopic gold balls diffuse toward a small cluster, which grows into a three-dimensional aggregate that's so branched it's practically all surface.
What about snowflakes? Diffusion-limited aggregation theory has made a good start on reproducing the branched, hexagonal laciness of snowflakes. A sixfold pattern is easily built in when one lets the particles diffuse on a triangular lattice rather than a square grid. A controlled touch of randomness, or "noise," decides which of many equally probable sites should grow at each step. The resulting figures look like snowflakes, although they lack the amazing similarities shown among the branches of different arms in a real snowflake.
Another way to create a growth model is to begin a computer simulation with many particles distributed throughout space and to allow the particles to move about randomly until they meet and stick together. In this case the clusters can also move, which leads to a different type of pattern with a fractal dimension of about 1.4.
In ballistic growth, a given particle moves not on a random walk but along a randomly aimed straight line. If it strikes a predecessor, it sticks where it hits. The process generates patterns in the plane that after a few thousand trials look like somewhat porous, deckle-edged spots roughly circular in shape (see Figure 5.13, right). Simulations show that if growth goes on for long enough over a quarter of a million or so trials the laciness dwindles and the covered area grows with the square of the radius, while the edges become better defined. The fractal dimension approaches the Euclidean value of 2.
But in most fractal growth models, mathematically precise results remain elusive because so many of their properties aren't known or understood. Even the concept of fractal dimension isn't enough to distinguish among the diverse objects that appear to be fractals. To overcome this deficiency, various other types of dimensions and constants have been introduced for special cases. Mathematicians and scientists have also been searching beyond the fractal dimension for other properties that may be universal.
A startling result of current research on fractal-modeled physical processes is that very different phenomena give rise to fractal dimensions that are very close in value. Such events range from the distribution of galaxies in the universe to the nature of turbulence in flowing fluids. It's too early yet to tell whether these phenomena are a collection of separate problems with a different explanation for each case or they depend on some underlying principle that would explain many of them simultaneously.
Meanwhile, the use of fractals as a descriptive tool is diffusing into more and more scientific fields, from cosmology to ecology. To some extent, fractals are in the eye of the beholder. Just as a river system can be described by naming it as a whole or by naming each of its numerous tributaries, researchers can choose various ways to describe the objects they see in nature. They can label the branches individually, or recognize that a branch looks roughly like the whole object and can it a fractal. Time will tell whether or not it's useful to see objects in terms of fractals. A meter stick is a very humble object until it gets into the hands of an Einstein. Einstein's work with meter sticks and clocks led to profound changes in our concepts of space and time. The fractal meter stick has yet to find its Einstein. Can this new meter stick be used to construct a deeper theory of why fractal objects occur, why certain fractal quantities are universal, and why fractals are ubiquitous in nature?
Ever since the earliest applications of fractal geometry,
description has outpaced explanation. Theories that depend on
fractal properties seem to work, but no one really knows why.
Further progress in the field depends on establishing a more substantial
theoretical base in which geometrical form can be deduced from
the mechanisms that produce it. Without a strong theoretical underpinning,
much of the work on fractals seems superficial and perhaps pointless.
It's easy to perform computer simulations on all kinds of models
and to compare the results with each other and with experimental
results. But without organizing principles, the field drifts into
a zoology of interesting specimens and facile classifications.
end of 2nd quote