Chaotic Attraction - Part 2
The development of fractal methodology
In
statistics and probability, the bell-shaped curve is the usual means of plotting variation
as an expression of the changing relationship between two variables. The bell-shaped curve
represents the normal distribution or standard Gaussian distribution of variation in a
given statistical context.
Basically,
it says there is a tendency for variance to distribute itself around a set of central
values. This means, among other things, that as one becomes further removed from those
central values, one will also encounter an increasingly diminishing number of instances of
the set of values which comprise that variance. However, the character of this decrease as
one moves further away from the central tendencies of a Gaussian distribution is said to
be smooth (i.e., continuous) and regular.
Benoit
Mandelbrot had had an intuition for some time that non-statistical laws governed the
characteristics of so-called random, stochastic processes. This intuition
began to take concrete form in the early 1960s.
Mandelbrot
believed there is an intimate connection between micro-events and macro-events. In fact,
he felt there is a pattern to the manner in which a given phenomenon manifests itself on
different scales. Mandelbrot held there was a symmetry of scale in which the same
structural pattern would manifest itself across whatever range of scales one cared to
examine in relation to that phenomenon. Thus, he believed certain structural invariants
were preserved from one scale to the next, despite the apparent random character of the
phenomenon under consideration.
For
instance, when he analyzed fluctuations in cotton prices, he discovered that the sequence
of price changes were preserved independent of what scale was used as a basis for studying
the changes. One could chart the sequence of price changes for cotton in a day, or a
month, or over a number of years, and the structural character of the sequence of changes
would reassert itself with each change of scale.
Although the
precise value for any single instance of change was unpredictable, a set of changes over
time would manifest the same structural sequence, irrespective of the period one chose as
the basis for a temporal scaling factor. In short, there seemed to be a structural
signature that was being imposed on, or preserved across, scaling factors.
Shortly
after coming to the Yorktown, New York, IBM research center, Mandelbrot became interested
in the problems associated with transmitting computer data over phone lines.
Computer data is transmitted over such lines by means of an electric current.
This current
carries information in the form of discrete packages. However, no matter how strong one
made the current carrying this information, there was always an irreducible amount of
noise arising during the process of transmission. Sometimes this noise would result in the
loss of a portion of the data being transmitted.
There was no
way to predict when and where the noise would arise. Nonetheless, there did seem to be
something of a sequential structure to the appearance of the noise. More specifically,
there seemed to be periods of transmission that were free of errors interspersed with
periods of transmission in which errors were manifested in clusters. Yet, when engineers
tried to grasp the character of these noise-clusters more closely, it seemed to dissolve
into a complex maze that defied analysis.
Mandelbrot
devised a means of characterizing, in a precise fashion, the distribution of the
error-clusters. His methodology produced solutions which reflected what had been observed
empirically. Essentially, Mandelbrot's position maintained that errors were not
distributed continuously throughout the transmission.
If one
examines any given error-cluster, one would find within that cluster regions which were
free from error. Indeed, Mandelbrot contended that irrespective of the temporal scale one
used as a basis for examining a given error-cluster, one would find an invariant
relationship in the ratio of error-free portions of a given cluster to the error-plagued
portions of that cluster.
The
phenomenon of transmitting information through telephone lines, like all phenomena, has a
characteristic structural character or spectrum of ratios of constraints and degrees of
freedom. This spectrum of ratios has an internal dialectic which gives expression to the
sorts of phase relationships and transitions in phase relationships which establish the
envelope of possibilities or parameters of values within which the phenomenon manifests
its structural character.
However, the
ratio (or ratios) governing the relationship of information and noise in the transmission
of a message over telephone lines is nonlinear in character. This means, essentially, that
for any given instant of transmission one cannot predict whether one will encounter noise
or information.
The best one
can do is to capture the general structural character of the ratio between noise and
information for a given interval of time. Moreover, one can show that the ratio will be
independent of the temporal level of scale one uses to demonstrate the ratio.
The reason
one cannot predict whether one will encounter information or noise at any given instant of
transmission is because one has no way of determining the phase state of the system for
any given instant. In other words, for any given instant, one has no way of calculating,
with any precision: (a) which ratios of constraints and degrees of freedom will be given
expression by the electrons making up the transmitted message, nor (b) the extent to which
the spectra of ratios or structural character of the interacting electrons will be
affected by interference phenomena, either from within the transmission process itself or
from outside sources.
On the other
hand, Mandelbrot was able to come up with a description for the general structure of the
ratio of noise to information for a given interval of time. He was able to do this because
the dialectic of electrons, under conditions of telephone transmission, generates a
characteristic range of possibilities. One of these characteristic features is the way in
which the ratio of noise to information persists across all levels of scale of measurement
involving intervals of time.
In effect,
Mandelbrot's ratio is a macro representation (in the sense that it involves an interval
rather than an instant) of the complexities of the dialectic of electrons occurring on a
micro level (i.e., from instant to instant or phase state to phase state). No matter how
small one makes the interval, it will always be an interval and not an instant.
Nonetheless,
his ratio is a macro reflection of what one would find if one had the mathematical and
technological sophistication to see or capture the structural character of the complex
dialectic of the phase states of interacting electrons as it manifests itself at a given
instant of time. In short, the macro reflects the micro because one is dealing with the
same spectrum of ratios of constraints and degrees of freedom irrespective of what level
of scale one is engaging the phenomenon in question.
Although
Mandelbrot had arrived at his position through his geometric intuition, he was providing
an analog for an abstractly derived construction already known to mathematicians: namely,
the Cantor set. To construct a Cantor set, one uses a line segment to represent the
numbers between 0 and 1.
One, then,
proceeds to remove the middle third of that line segment. This leaves two portions of line
segment.
One also
removes the middle third from these two remaining line segment portions, and one continues
to remove the middle third from each subsequent series of remaining line segment portions
which is generated from each preceding removal operation.
Theoretically,
this process can be carried on indefinitely. Ultimately, it yields an infinite set of
clusters of increasingly refined nature as one proceeds from one level of the removal
operation to the next level of the removal operation. Moreover, according to
mathematicians, the total length of this infinite residue is supposedly zero.
In effect,
Mandelbrot was characterizing the ratio of error-free transmission to error-plagued
transmission as a Cantor set plotted against time. Although errors would become
increasingly sparse as one selected smaller and smaller temporal scales through which to
examine the character of the transmission, there would always be a certain irreducible
error component in any transmission. Moreover, this error component was inextricably
linked to error free portions of transmission in the form of a ratio.
Therefore,
in any given temporal period, the ratio of error-free to error-plagued transmission was
invariant. However, this was as close as one could come to pinning down the error
component of transmission. One simply had no means of determining where in that temporal
period one would find error-free transmission as opposed to error-plagued transmission.
Once again,
Mandelbrot had uncovered an invariance which is preserved across differences of temporal
scale. Once again, this invariance was found in the midst of seemingly random
fluctuations.
Mandelbrot
coined a couple of terms to describe certain aspects of the phenomenon he was
investigating. These are referred to as the Noah Effect and the Joseph Effect.
The Noah
Effect referred to the discontinuous character of the way in which many things changed.
Rather than assume, as had been the case traditionally, that a change from point 'A' to
point 'B' necessarily involved a smooth traversing of all intermediate points, Mandelbrot
contended the transition from 'A' to 'B' could occur despite the fact that one or more of
the intermediate points had been by-passed altogether. The movement from 'A' to 'B', in
other words, was in the form of a single discrete jump or in the form of a series of
discrete jumps, depending on the forces which were vectoring the situation in question.
The Joseph
Effect refers to the tendency of the structural properties of a complex, nonlinear system
to persist over time. Although such systems seem to exhibit random-like characteristics,
nonetheless, these characteristics exist side-by-side with certain stable features. These
stable features might disappear from visible sight for a period of time, but they would
reappear at other junctures in time. These features formed a persistent set of themes in
the life of many, complex, nonlinear systems.
In the
terminology of the present article, Mandelbrot's Joseph effect gives expression to the
spectrum of ratios of constraints and degrees of freedom that constitute the structural
identity of any given object, event, process, state, condition and so on. The dialectic of
dimensions establishes complex manifolds out of which emerge point-structures,
neighborhoods and latticeworks which permit various structural properties of nonlinear
systems to persist over time and across different levels of scale.
This
dialectic of dimensions is capable, in turn, of maintaining such stability in the midst of
variability because of the way the underlying order-field has invested the spectra of
ratios of constraints and degrees of freedom of each of the dimensions with their own
structural identity. The phase relationships and phase transitions permitted by a given
dimension's spectrum of ratios serves as the coupling constant through which different
phase state variations of the dimension's structural character are able to maintain an
essential structural integrity.
When
different dimensions come into 'contact' with one another through, for example, phase
relationships, they will generate point-structures, neighborhoods and latticeworks which
will manifest various aspects of the structural character of the dimensions involved in
the dialectic.
In this way,
an order-field distributes some of its properties across a variety of levels of scale.
This distributive quality extends all the way from dimensions, to the sorts of
point-structures, neighborhoods and latticeworks that emerge as a result of the dialectic
of dimensions which has been set in motion by the order-field.
Mandelbrot's
fractal geometry is an attempt to reflect the structural character of certain aspects of
reality more accurately than traditional Euclidean geometry is able to do. As Mandelbrot
says: "clouds are not spheres, mountains are not cones...". In
other words, one cannot use the methods and theorems of Euclidean geometry to gain insight
into the complex, irregular structural character of clouds, even though such geometry may
be well suited for describing the structural character of, among other things, regular
spheres.
The face of
nature tends to be more akin to nonlinear dynamics than to regular, linear dynamics. As a
result, one needs a methodological approach which will equip one to handle the
complexities, irregularities and erratic properties manifested in natural phenomena.
Fractal geometry was intended to serve as a means of investigating the discontinuous,
irregular, nonlinear, and erratic character displayed by many aspects of experience.
Similarly,
hermeneutical field theory is intended as a methodological approach that will assist the
individual to grapple with some of the complexities, irregularities and nonlinear
properties that are manifested in many issues concerning understanding, interpretation,
methodology, and knowledge. In other words, hermeneutical field theory is rooted in the
realization that not everything in the phenomenology of the experiential field is capable
of being reduced down to linear configurations.
A milestone
in the development of Mandelbrot's perspective came with the publication of a paper
entitled: "How Long Is the Coast of Britain?". In essence, Mandelbrot contended
one could have a variety of answers to this question.
On the one
hand, he argued there is a sense in which the coastline is infinitely long since with each
succeeding change of scale, one is opened up to a new set of contour irregularities that
were not apparent on the previous level of scale. Conceivably, there may be no end to the
levels of scale one encountered as one's measuring efforts get lost in the mists of the
sub-quantum world beyond the horizon of the Planck length.
On the other
hand, Mandelbrot also maintained the answer to the question being asked in his article
would depend on the length of the ruler one used to measure the coast of Britain.
If, for example, one used a ruler that was one meter in length, one would get a different
answer than if one used a ruler which was one foot in length. The reason for this is that
the smaller, foot ruler is able to have access to more of the irregularities of the
coastline than is the meter ruler.
In other
words, the larger ruler could not be used to measure all those irregularities which were
less than one meter in length. Therefore, measurements based on the one meter ruler would
exclude such irregularities.
The one
meter ruler only could be used to measure those portions of the coastline against which
the entire length of the ruler could be laid. Similarly, although the smaller, foot ruler
could be used to measure many of the irregularities not capable of being measured by the
meter ruler, the foot ruler could not be used to measure the irregularities of the
coastline which were less than one foot.
With each
reduction in the length of the ruler used to measure the coastline, one makes the
transition to another level of scale. As one goes to smaller and smaller modes of
measurement, the length of the coastline is increased because these smaller measuring
units are able to capture all the irregularities which had to be excluded from the
measuring process used on the previous level of scale.
Inherent in
the foregoing analysis concerning the length of the British coastline is a danger that one
may confuse ontology and methodology. In order to better grasp the sense of this concern,
consider the following.
Intuitively,
one might wish to argue that the value of length generated by a measurement process is an
abbreviated index for the way in which a given process of methodology (i.e., measurement)
engages a given aspect of reality or ontology. Part and parcel of this intuition is a
feeling that irrespective of what values are generated by the measurement process, the
object, structure, process, or event being methodologically engaged has a structural
character which is independent of the measurement process. In other words, the structural
character of the object, event, and so on, does not depend on measurement for its
existence as a structure of one kind rather than another.
To be sure,
different aspects of the spectrum of ratios of constraints and degrees of freedom that
constitute an object’s or event's structural character may be tapped by a given
methodological engagement.
As a result,
different modes of measurement may induce different facets of a structure's spectral
character to manifest themselves. Consequently, one will come up with different indices
for the character of the interaction between methodology and ontology.
However, the
changing character of the dialectic through which the measurement process engages, and is
engaged by, a given structure does not alter the basic spectrum of ratios of constraints
and degrees of freedom that constitutes the character of the structure being measured. All
that is affected is one's methodological orientation toward that structure's spectrum of
ratios.
All that is
affected is the ratio or ratios which are selected or sampled for examination by the
measurement process. In short, the structure will manifest itself differently according to
the way in which a measurement process engages that structure, but the structure remains
the same as far as its characteristic spectrum of ratios is concerned.
The changing
nature of the answer concerning the length of the British coastline is not a reflection of
the variable character of the coastline's ontology. The changing nature of the answer
concerning length is a reflection of the changing character of the manner in which
methodology or the measurement process engages that coastline.
Traditionally,
Euclidean geometry operated from a perspective that allowed for as many as three
dimensions. These dimensions were conceived of as running at right angles to one another.
A point is
considered to have zero dimensionality. A line is said to be one-dimensional, whereas a
surface consists of two-dimensions and a solid occupies three-dimensions.
However, if
one were to ask: what is the dimensionality of, say, a ball of twine? Mandelbrot
maintained the answer which one gave would depend on one's point of view.
If one were
far enough away from the ball of twine, it could take on the appearance of a point of zero
dimensionality. As one got closer, it would appear three-dimensional, but if one went into
the inner recesses of the ball of twine and examined the twine from different levels of
scale, the twine could appear to be one-dimensional, two-dimensional, or
three-dimensional.
Mandelbrot
also raised the possibility that there might be dimensions in between the normal one-,
two-, or three-dimensions. He referred to these in-between levels of scale as fractal
dimensions.
Fractal
dimensions were intended to serve as a means of describing, through measurement, various
sorts of qualities falling beyond, or outside of, the traditional, Euclidean conceptions
of dimensionality. In other words, Mandelbrot intended fractal dimensions to be an index
for characteristics such as the degree of irregularity which are displayed by whatever is
being measured. Furthermore, Mandelbrot claimed this index of fractal
dimensionality remained invariant across all levels of scale which might be used to
measure or describe the phenomenon or object or structure being examined.
However, in
line with the previous comments concerning the possible dangers of confusing methodology
and ontology in relation to determining the length of the coastline of Britain, one must
exhibit a certain amount of caution with respect to interpreting the significance of
Mandelbrot's idea of a fractal dimension. The sense of dimensionality being given
fractional values in the foregoing cases are not necessarily ontological dimensions.
Mandelbrot's
sense of fractional dimensionality is a reflection of the manner in which a given
methodology is engaging a particular aspect of ontology. In other words, fractional
dimensions are an artifact of methodology, not ontology.
One should
not construe the above caution to mean that fractional dimensions cannot have great
heuristic value. If nothing else, the idea of fractional dimensionality may permit one to
methodologically tap (e.g., measure) different facets of the spectrum of ratios of
constraints and degrees of freedom which constitute a given structure.
As a result,
one may be able to develop a better understanding of the structural character of various
aspects of ontology. However, one should remember that fractional dimensionality is an
expression of methodology, together with the way it engages different aspects of ontology,
rather than an expression of ontology per se.
Keeping in
mind the foregoing, consider the following. A Koch curve (this figure is named after Helge
von Koch who was the first person to describe it- around 1904) supposedly,
represents an infinitely long line surrounding a finite area. To construct a
Koch curve, one takes a triangle that has sides one unit in length. Next, one
places a triangle, which has been reduced by two-thirds, in the middle of each side of the
initial triangle.
One, then,,
repeats this second step an indefinite number of times, making the appropriate reductions
in size for each new level of scale relative to the preceding level of scale. The
perimeter or boundary length of the final figure will be 3 x 4/3 x 4/3 x 4/3... 4/3n, with
'n' being the number of times the second step is repeated.
Of course,
mathematicians argue that, in principle, this process can be repeated an infinite number
of times, thereby generating a curve of infinite length which never intersects itself.
Yet, if one circumscribes the initial triangle with a circle, the infinite Koch curve will
never reach to any point beyond the circle that surrounds the initial triangle.
Consequently,
although the curve itself takes on an infinite complexity, it is enfolded into a finite
space. This space will be larger than the area covered by the original unit triangle but
less than the circle circumscribing that triangle.
In general,
if one has a structure consisting of n-dimensions which is, subsequently, divided into
p-equal parts, then, the ratio of similarity, r, between any given part of this structure
and the structure as a whole, will be given by the formula:
r = nth root of 'p'
In the case
of structures like the Koch curve, one is uncertain about the dimensionality of such a
curve. On the other hand, one can determine values for both 'r' and 'p'.
Thus, when
constructing the Koch curve, one takes any given side of the original equilateral
triangle, and replaces, in the prescribed fashion, one line (which constitutes the side of
the triangle) with four lines: the two lines on either side of the section that has been
removed, together with the two sides of the reduced equilateral triangle which are added
on (projecting outwardly from, but affixed) to the side with which one started originally.
If one substitutes these values into the above formula, one gets:
3 = nth root of 4
To solve for 'n', one must use logarithms.
First, one
takes the logarithm for each side of the equation, resulting in the following: log 3 = n
log 4. After one has completed the appropriate calculations or looked up the values in a
set of log tables, one finds that n = 1.2618.62
Consequently,
the dimension of a Koch curve is not a whole number, but fractional. Moreover, this
property of fractional dimensionality also is capable of manifesting itself in the context
of surfaces and solids.
Thus, the
Sierpinski sponge, begins life as a cube. Subsequently, an infinite reiterative process is
applied to it. The end-result of this reiterative process is a structure, supposedly,
consisting of zero volume enclosed within an infinite surface. This structure is said to
have a fractional dimensionality of 2.7268.
Furthermore,
each face of the Sierpinski sponge manifests self-similar properties in relation to the
structure as a whole. In other words, each face has zero area surrounded by a perimeter of
infinite length as well as a fractional dimensionality. These faces are referred to as
Sierpinski carpets, and they each have the same fractional dimensionality as a Koch curve,
namely: 1.2618.
As indicated
previously, structures which possess this property of having a fractional dimensionality
are known, collectively, as fractals. The investigation of fractal properties is known as
fractal geometry.
The Koch
curve, the Sierpinski sponge and Sierpinski carpets are all the result of a conceptual
construction process. The construction processes underlying these structures is an
expression of the way in which methodology, especially in its mode of generating
measurement values, has been applied to an initial seed structure. By recursively altering
such seed structures an infinite number of times, one is alleged to end up with rather
paradoxical figures.
For example,
such construction process permit non-intersecting curves of infinite length to be
circumscribed within a finite area (i.e., the Koch curve). Another paradoxical example
involves 'solid' structures of zero volume and infinite surface (i.e., the Sierpinski
sponge).
Once again,
however, these examples of paradoxical, fractional dimensionality are an artifact of
methodology. In this respect they illustrate precisely the same phenomenon that surfaced
in trying to come up with an answer to the length of the coastline of Britain. As such,
they are heuristic structures which may or may not help one to better understand either
the character of methodology, the measurement process or the character of the dialectic
between methodology and ontology.
While
hermeneutical field theory shares fractal geometry's commitment to, and emphasis on, the
property of invariance across levels of scale as an extremely fundamental theme, the
former differs from the latter in the significance which is attributed to the idea of
fractional dimensionality. Fractional geometry tends to treat the notion of fractional
dimensions as a reflection of the ontological character of spatial dimensionality.
Hermeneutical field theory, on the other hand, considers fractional dimensionality to be
an artifact of the way that methodology engages, and is engaged by, different aspects of
ontology.
Fractal
dimensionality can be a useful tool for probing the spectrum of constraints and degrees of
freedom that constitute the structural character of the dimension of space. However,
fractal dimensionality does not demonstrate that there exists, in any ontological sense,
spatial dimensions that are in between the usual, three spatial dimensions. Fractal
dimensionality represents a particular methodological means of orienting oneself with
respect to the structural character of one's engagement by, and exploration of, different
facets of the spectrum of ratios of constraints and degrees of freedom that give
expression to the dimension of space.
In this
respect, fractional dimensions are not really dimensions at all. They are examples of the
way different ratios of constraints and degrees of freedom inherent in the structural
character of the dimension of space are induced to manifest themselves under different
circumstances of methodological engagement - measurement is our way of attempting to
establish a variety of base lines through which to gauge what the structural character of
that ratio is and may entail.
|
