Holographic Images - Part 3
Transforms, transformations and transform space
At the heart
of Fourier's thesis for analyzing waveforms is the contention that any compound, irregular
wave can be shown to be equivalent to the summation of a series of simple, regular waves.
This series is known as a Fourier series. In turn, any physical
phenomenon displaying an oscillatory nature or a periodic character can be expressed as a
Fourier series of sine and cosine waves.
An
irregular, compound wave can be treated as a series of increasingly smaller regular waves.
In fact, as one proceeds along the series, the frequencies of the smaller and smaller
waves becomes increasingly greater. In other words, they complete their cycles at
progressively faster rates.
Fourier's
technique involves selecting some initial regular wave to be used as a working
representation of the period of the compound, irregular wave in which one is interested.
He, then, used his method to establish a set of coefficients to be used in conjunction
with the selected working representation of the initial, compound, irregular waveform.
This process of finding the coefficients is called Fourier analysis.
When
integrated, the series of coefficients and the corresponding increasingly higher
frequencies for the increasingly smaller waves will add up to the value of the fundamental
frequency used as a model for the irregular, compound wave. The coefficients were selected
in order to make the frequencies of these increasingly smaller waves whole number
multiples of the initial regular wave frequency.
Fourier's
method actually uses a kind of dialectic to guide the process of generating the
coefficients to be selected for the Fourier series. By gathering together the values for
all the regular waves derived through Fourier analysis and using these values to make a
compound wave, one has an opportunity to compare this synthesized wave against the
original irregular, compound wave.
When the
synthesized wave can be shown to closely match the original wave, then, one terminates the
analysis. If, on the other hand, the match-up is not sufficiently close, then, one
continues to proceed with further analysis.
The initial
wave in a Fourier series is referred to as the fundamental harmonic.
Each successive wave in the Fourier series is called, in turn: the second, third, fourth,
etc.,harmonic. In most cases, a series consisting of nine coefficients (that is, up to the
ninth harmonic) is able to provide a sufficiently close approximation for even very
complicated, irregular, compound waves.
Once the
series of coefficients has been determined, one is in a position to plot a graph involving
amplitude versus frequency. Graphs can be symbolized in the form of an equation. An
equation consisting of a series of coefficients that represent the amplitude/frequency
properties of a set of regular waves is known as a Fourier transform.
There were
certain technical limitations inherent in the idea originally conceived by Fourier.
However, a number of other theorems have been introduced to permit one to circumvent these
limitations. The most important of these supplementary theorems is the Laplace
transformation.
The term
"transform" can be used in either a verb or noun form. Usually, however, the
term is used in its noun form of transformation- as that which is generated from, or is
the result of, a transforming process. In its noun form, transform refers to either the
graph-figure or the equation which is produced by a specific functional ordering of the
Fourier coefficients.
In essence,
then, a transformation represents both the transition from one mathematical form to
another, as well as the structure produced by that process of transition. Moreover, in
accomplishing this transformation, one also has undergone, in the case of Fourier
analysis, a transition from perceptual space (which is the medium through which the
original irregular, compound wave that is being modeled is given expression) to Fourier
transform space.
In
perceptual space, frequency is a function of time, and, as a result, the 'perceptual
frequency' is expressed in terms of cycles per second or Hertz units (Hz). However,
in transform space, frequency becomes a spatial function. More specifically, frequency is
measured by the density of stripes occurring in a given area of an interference pattern.
The term 'stripes'
refers to the periodic patterns of light and dark which are manifestations of the
junctures of constructive and destructive interference. In fact, the density value of
stripes in a given area depends on the character of the phase difference between the
interfering set of waves.
Therefore,
frequency is fundamentally linked to phase. For example, signals in the nervous system are
sent as waves in which amplitude and frequency are independent of one another, but the
signal is transmitted in transform space as a spectrum of phase differences.
One of the
benefits resulting from the transition to the 'spatial' form of transformation is to help
simplify calculations. In Fourier transform space, one often can accomplish with
multiplication and division what only could be accomplished with the use of calculus in
perceptual space.
Furthermore,
the periodic character of a phenomenon often manifests itself more clearly and markedly in
Fourier transform space (as well as in the still more abstract counterpart of Fourier
transforms known as Laplace transforms) than it does in perceptual space. For example, the
message, signal or interference pattern of a holograph more clearly manifests its
structural character in transform space than it does in perceptual space.
The key to
gaining access to transform space is the Fourier transform. However, the enhanced clarity
of the holographic message in transform space does not mean one visually can see a clearer
signal. The clarity is a manifestation of the way the structural character of the logic of
the relationships involved in, and among, different transforms becomes better resolved in
our understanding. As a result, one can better grasp the structural character of the
latticework of phase relationships that cannot be seen visually.
A Fourier
series of coefficients has a corresponding Fourier transform. Therefore, if
the structural properties of superimposing waves (i.e., the operation of convolution)
becomes difficult, if not impossible, to grasp in perceptual space, one may perform the
requisite transform operation to generate a mathematical form which is more accessible to
the understanding, and, therefore, is more open to exploration, manipulation and so on.
The
alterations and transitions occurring in the amplitude and phase of the light waves as a
result of engagement with an object do not constitute an image of the object. These
alterations of the light wave constitute a transform of the object. In order
to restore the image of the object inherent in the information carried in the transform of
the object, one needs to perform a transform of the transform.
The first
Fourier transform translates the object's structural character into an 'object' (which
could be a figure, graph, set, or magnitude of some sort) of transform space. Then, a
second Fourier transform operation occurs when the first transform is run through a lens
system which translates the object of transform space into an object of perceptual space.
The first
Fourier transform operation is comparable to Fourier analysis. This similarity is due to
the way in which the transform translates the irregular, compound wave, constituting the
object, into a set of regular, uniform, simple waveforms in transform space. These latter
waveforms are capable of modeling the original compound wave (i.e., the object).
On the other
hand, the second Fourier transform operation corresponds to Fourier synthesis. This is the
case because the second operation has the effect, like Fourier synthesis, of recombining
the set of waveforms of transform space into an image or figure of perceptual space that
gives synthesized expression to the irregular, compound waveform with which one started.
One of the
essential defining differences between the object and reference wave revolves around
asymmetric alterations in the property of phase variation arising as a result of
differences in the character of the paths undergone by the object and reference waves. For
each aspect of the compound object wave, phase will vary in relation to the corresponding
aspect of the reference wave.
Furthermore,
among all of these phase variations, there will be at least one phase variation which will
remain the same both before and after the point of interference. This fixed-point phase
variation serves as the invariant reference point relative to which all the other phase
variations will take place.
The
foregoing consideration concerning fixed-point phase variation is at the heart of one of
the basic requirements underlying the hologram phenomenon. More specifically, there must
be a spectrum of phase variations in transform space which has the property of being
well-defined. Usually, the meaning of being ‘well-defined’ involves being able
to tie a given variation to some invariant feature. Thus, one of the minimum conditions
that must be satisfied in order for a hologram to be possible is for there to be a
fixed-point relationship between the object and reference waves.
People, like
Karl Pribram and Paul Pietsch, argue that memory is a particular
spectrum of phase variations in transform space. These phase variations exist as a
transform analog of relationships among different sets of neurons in the brain.
As such,
mind is not stored in the form of molecules, action potentials, neuronal cells or any
other aspect of brain functioning or anatomy. Mind is an expression of the variations in
phase relationships that are stored in transform space.
The
physical/material activity of the brain's neural networks may serve as part of the
instrumentality which is necessary to help generate the compound reference and object
waves. However, the storage of the interference patterns of these waveforms is a function
of the spectrum of phase variations arising as a result of the differences between the
reference and object waves. These differences are stored in transform space, not
perceptual/material space, since they involve phase relationships, not actual 'things'.
Seen from
the foregoing perspective, memory is a transform of a transform. This transform of a
transform moves, as well as translates, a structure from transform space into perceptual
space. It is an analog of the reconstruction of a wave-front which occurs when one passes
coherent light through a holographic plate at the appropriate angle of incidence.
Although the
foregoing has a nice theoretical ring to it, one should not lose sight of the fact that
transform space is a mathematical construct which is, at best, an analog for what is
occurring in the dialectic of dimensions (including the material processes of brain
functioning). In other words, the model being put forth by Paul Pietsch, Karl Pribram, and
others presupposes that transform space is primarily mathematical in character, consisting
of the results of operations on sets of points or on magnitudes or on geometric figures in
perceptual space. Nonetheless, actual transform space may not be at all mathematical in
character, although mathematics may provide a means of generating analogs for the
structural character of the ontological counterparts to such a mathematical model.
In the case
of human understanding, transform space may be entirely a function of the hermeneutics of
the phenomenology of the experiential field. This field is generated by the non-linear
dialectic of various dimensions.
The
dialectic of dimensions is, in turn, vectored, oriented, shaped, arranged and organized by
an underlying order-field. Such an 'order-field' establishes the set of constraints and
degrees of freedom governing the flow of the dimensional dialectic that generates the
complex waveforms giving expression to the phenomenology of the experiential field having
the structural character it does on a given occasion.
In the light
of the foregoing possibilities, transform space can be approached in terms of its being a
concrete reality rather than merely a mathematical abstraction. In other words, transform
space is concrete in the sense that it is comprised of a determinate set of constraints
and degrees of freedom as a result of an underlying dimensional dialectic.
However, the
ontological character of this reality is not necessarily physical or material in nature.
The ontological character might involve other dimensions such as consciousness,
understanding (expressed as hermeneutical operations), will, and so on.
All of these
other dimensions are capable of interacting with the physical/material realms, but the
former cannot be reduced to being functional expressions of these latter dimensions.
Indeed, the structures or waveforms generated through, for example, neural activity may
have to be subjected to a set of non-material/non-physical operations in order for the
neural waveform activity to be translated into hermeneutical transform space.
Once
translated in this fashion, the neural activity may act as vectors which are capable of
helping shape and orient the events of hermeneutical transform space. However, one need
not suppose that transformed neural waveform structures are the sole vectoral determinants
of that space.
Logical relationships: an expression of focal/horizonal phase differences
In one sense
logical relationships are really a study in phase differences either within one
latticework or between latticeworks or among latticeworks. However, rather than being
linked with issues of frequency or temporal/spatial functions as is the case with
frequency modulation or neural activity, respectively, logical relationships concern phase
differences involving focal/horizonal orientation and engagement.
These phase
differences can be relative since one can choose either horizon or focus or any one
latticework as the point of reference against which one explores and measures differences
in phase orientation and engagement in relation to whatever other structures, foci or
horizons one is studying. Nevertheless, these phase differences can exhibit greater and
lesser degrees of relativity depending on which dimensions and latticeworks, or which foci
or horizons, one selects as a basis for reference and exploration.
Some
reference points are more accurately and objectively reflective of the structural
character of certain aspects of reality than are other such reference points. As a result,
the former sorts of reference points are more capable than the latter sort of reference
points of permitting one to properly orient oneself in relation to the study of logical
relationships among different latticeworks or within a latticework or among various
dimensions.
In any
event, when one treats logical relationships as a species of phase differences, one is
drawing attention to the way latticework orientation and engagement properties have
vectoring and structural characteristics that manifest themselves in the form of various
kinds of connections, linkages and relationships under different circumstances. These
orientation and engagement properties are capable of being mapped as a set of complex
dialectical interactions.
These
interactions, in turn, are characterized by shifting ratios of constraints and degrees of
freedom. Such shifting ratios reflect transitions in logical relationships as a function
of alterations in the structural character, orientation and mode of ontological engagement
of latticeworks and dimensions, one with another, as well as within themselves.
Memory and holographic theory
When two or
more wave systems interact to generate a memory, one cannot stipulate that memory is
attached to any particular structural feature of the interacting systems. In holographic
theory, any given memory is stored in transform space as a set of phase relationships.
These phase relationships describe periodicy in terms of its essential characteristics.
Such
relationships or characteristics do not, in and of themselves, give expression to any
specific size, proportion or concrete form. They indicate relationships in the form of
phase differences which do not have size, nor do they occupy space, nor do they have any
particular concrete form of a physical or material nature.
As a result,
in the holographic theory of mind, the mind cannot be reduced to the activity or anatomy
or chemistry or electrical activity of the brain. This cannot be done since, in essence,
the mind exists in transform space while the brain exists in perceptual space.
A question
facing anyone who would propose a holographic theory of memory involves the problem of
going from perceptual space to transform space. More specifically, what makes possible the
translation or transduction process that converts perceptual space structures into
transform space structures in view of the unlike nature of the two kinds of 'spaces'?
Seemingly,
this is just another version of the mind-body problem of Descartes, for one would like to
know how a physical/material process produces a non-physical and non-material structure.
Perhaps even more importantly, how is transform space able to maintain or sustain or
preserve relationships, given that it is non-physical and non-material in nature?
Similarly, how does an element of transform space get re-converted into a perceptual space
structure?
A
holographic plate stores interference patterns in a form that can be re-accessed through
wave-front reconstruction. The mathematical description of this process talks of the
movement between perceptual and transform space.
This sort of
description is useful because it permits one to understand, within certain limits, some of
the structural character of what is going on. One can, then, exploit that understanding to
produce tangible results of a determinate, predictable sort. However, as previously
suggested, the mathematical description or model may be, at best, only an analog for what
actually occurs.
Even if one
assumes that the physical plate only intercepts, somehow, the interference pattern
existing in transform space and that the interference pattern is completely separate from
the physical system used to intercept it, one still needs to know how such a process of
interception works. How does a physical/material plate get affected and shaped by a
non-physical and non-material set of relationships in transform space? Where and how do
perceptual space and transform space interact? What serves as the mediator between these
two realms?
The
mathematical model can be shown to work because of the existence of a physical
medium-namely the plate. In other words, theory maintains that the holographic plate
stores the interference pattern in a form that is accessible by physical means.
Thus, if one
wishes to retrieve the stored information, all one has to do is to engage the photographic
plate with coherent light at the appropriate angle of orientation in order to reproduce
the image of the object. What constitutes an 'appropriate angle' will be a function of the
angle at which the interference pattern interacted with the plate when the transform of
the object's image was originally stored. Without the plate, the mathematical model would
be just an empty theory without any counterpart in the perceptual world.
Consequently,
one wonders what will serve as the mind's counterpart for the physical plate of the
holographic process. If the mind in holographic theory cannot be reduced to the brain, and
if memories are not stored in the brain but in transform space, then, how does wave-front
reconstruction take place so that one can have a memory-correlate in perceptual space? How
does the brain manage to intercept the interference pattern of transform space to produce
an image in perceptual space?
In addition,
none of the foregoing mentions the problems surrounding the identity of the coherent light
(or its source) that is to be used to help reconstruct the wave-front which exists in
transform space. One also would like to know how such coherent light is to be sent through
transform space at the appropriate angle. After all, transform space has no size or
proportion or structure that would seem to permit one to have angles of any sort.
One possible
approach to some of the foregoing issues and questions is outlined briefly in the
following considerations. To begin with, the idea of transform space can be construed as
an analog representation of the possibilities inherent in the dimensional dialectic which
underwrites or makes possible the holographic process. In other words, transform space is
a description of certain aspects of the structural character of the complex latticework
generated by the dialectic of dimensions such as energy, temporality, space, materiality
and intelligence (the latter introduced through the efforts of the scientists and
mathematicians who devise and set up the holographic process).
More
specifically, transform space is an analog representation or model of a subset of the
phase relationships that are generated by the aforementioned dimensional dialectic.
Transform space involves an inferential mapping which attempts to capture, or give
expression to, the character of some of the linkages tying together the different
dimensions under a given set of experimental or applied circumstances.
Therefore,
in the case of a transform of a transform, such as occurs in wave-front reconstruction, a
description is being given. This description is an analog representation of the sorts of
phase transitions that are necessary to induce the dimensional dialectic to give
expression to certain aspects of the phase relationships which were created when the
original holographic interference waveform was generated.
Nothing is
stored in transform space except a conceptual description. Indeed, transform space is just
a label given to a certain kind of hermeneutical construction. This construction makes
identifying reference to, as well as establishes inferential mapping relations and
congruence functions with, those aspects of ontology involving holographic phenomena.
Information
concerning the latter sort of phenomena is stored in the phase relationships which have
been generated, and which are being maintained, by a specific arrangement of dimensional
dialectics created through the holographic set-up. Viewed from this perspective, a
holographic plate doesn't store information, so much as it is part of the dimensional
dialectic which collectively underwrites the holographic phenomenon. As such, the plate is
really a passageway through which one gains access, under appropriate circumstances of
reconstruction, to those phase relationships that arose when the original pattern of
interference was generated.
Thus,
irrespective of whether one is talking about mental or material holographic plates, the
principle may be the same. In each case, reconstructed images might be translations or
reflections or transductions of certain aspects of the phase relationships that arose as a
result of dimensional dialectics concerning the initial holographic process.
Although the
plate and/or brain play a role in this dialectic, the role of the plate/brain may be that
of a transducer rather than a storage medium. In other words, certain aspects of the plate
or brain may serve as the physical/material pole of a complex latticework of phase
relationships which links the plate/brain to other dimensional poles by means of the
temporal dimension. As such, the plate/brain is capable of serving as a transducer that:
translates, interprets, and generates, as well as, is shaped by, shifts in phase
relationships concerning a wide variety of themes involving emotion, motivation,
spirituality, intelligence, sensation, and so on.
A second
point to keep in mind is this. In the hermeneutical/phenomenological context, phase gives
expression to the individual's mode of engagement of, or orientation toward, the spectrum
of ratios of constraints and degrees of freedom constituting the range of possibilities
inherent in the structural character of the dialectic between individual and ontology.
While the
attractor basins giving expression to the foregoing dialectic circumscribe all the
possibilities inherent in the spectrum of ratios, under normal circumstances, not all of
these possibilities can be engaged at any one time. When one of these possibilities is
manifested- whether through inducement or spontaneous activity, the individual becomes
oriented toward the on-going dialectic in a particular way. Consequently, the individual's
mode of engagement or orientation becomes the hermeneutical angle of dialectical
interaction at a given moment in time.
The term
"hermeneutical angle" is used in the foregoing because the point of engagement
or the point of orientation represents a phenomenological encounter of one ratio from
among the spectrum of ratios of constraints and degrees of freedom that are possible to
experience. Therefore, hermeneutical engagement establishes an experiential asymmetry
which stands in focal relief to the horizon of remaining possibilities of the spectrum of
ratios of constraints and degrees and freedom. This relationship between focus and horizon
constitutes the hermeneutical analog counterpart to the notion of angle in geometry.
|
