by Paul Pietsch
We can let all values of Y above the equator be positive; below the equator, lets let them be negative. We can do the comparable thing with X, except that we'll use the vertical meridian instead of the equator; then values of X on the right side of our circle will be positive and those on the left will be negative. If we plot a graph of sine or cosine values for angle A versus degrees on the circle, we get a wave. The cosine wave starts out at +1 at 0o (360o), drops to 0 at 90o, plunges down to -1 at 180o and returns to +1 at 360o, the end of the cycle. Meanwhile, the sine wave starts at 0, swells to +1 at 90o, drops back to 0 at 180o, bottoms to -1 at 270o and returns to a value of 0 at the completion of the 360-degree cycle. We really don't need the triangle anymore (so let's chuck it!): the circumference will suffice.
Notice, though, that the degree scale can get to be a real pain in the neck after a single cycle. Roulette wheel, clock hands, meter dials, orbital planets, components of higher frequency, and the like, rarely quit at a single cycle. But there's a simple trick for shifting to a more useful scale. Remember the formula for finding the circumference of a circle? Remember 2pi r? Recall that pi is approximately 3.14, or 22/7. When we make r equal 1, the value for the circumference is simply 2pi. This would convert the 90o mark to 1/2 pi, the 180o mark to 1pi (or just pi)...and so on. When we reach the end of the cycle, we can keep going right on up the pi scale as long as the baker has the dough, so to speak. [8] But we end up with a complete sine or cosine cycle at every 2pi.
With regard to a single sine or cosine wave on the pi scale, what is amplitude? Recall that we said it was maximum displacement from the horizontal plane. Obviously, the amplitude of a sine or cosine wave turns out to be +1. But +1 doesn't tell us where amplitude occurs or even if we have a sine versus cosine wave--or any intermediate wave between a sine and a cosine wave. This is where phase comes in, remember. Phase tells where or when we can find amplitude, or any other point, relative to the reference; i. e., to zero.
Notice that our sine wave reaches +1 at 1/2pi, 2 1/2pi, 4 1/2pi... and so forth. We can actually define the sine wave's phase from this. What about the phase of a cosine wave? Quiz yourself, and I'll put the answer in a footnote.[9]
If someone says, "I have a wave of amplitude +1 with a phase spectrum of 1/2pi, 2 1/2pi, 4 1/2pi," we immediately know that the person is talking about a sine wave. In other words, our ideal system gives us very precise definitions of phase and amplitude. We can also see in the ideal how these two pieces of information, phase and amplitude, actually force us to make what Benjamin Peirce and his son Charles called necessary conclusions! Phase and amplitude spectra completely define our regular waves.
Now let me make a confession. I pulled a philosophical fast one here in order to give us a precise look at phase and amplitude. We know the phase and amplitude of a wave the moment we assert that it is a sine or a cosine wave. Technically, our definition is trivial. To say, "sine wave" is to know automatically where amplitudes occur on our pi scale. But let's invoke Fourier's theorem and apply it to our trivia. If a complicated wave is a series of sine and cosine waves, and those simple waves are their phase and amplitude spectra, then knowing the phase and amplitude spectra for a complicated wave means having a complete definition of it as well. Our trivial definition leads us to a simple explanation of how it is that phase and amplitude completely define even the most complicated waves in existence. It is not easy to explain the inclusiveness of phase and amplitude in the "real" world. But look at how simple the problem becomes in the ideal. First, phase and amplitude define sine and cosine waves. Second, sine and cosine waves define compound waves. It follows quite simply, if perhaps strangely, that phases and amplitudes define compound waves too. But there's a catch.
***
We can define the phase and amplitude of sine and cosine waves because we know where to place zero pi--0-- the origin or reference. We know this location because we put the 0 there ourselves. If we are ignorant of where to begin the pi scale, we don't know whether even a regular wave is a sine or a cosine wave, or something in between. An infinite number of points exist between any two loci on the circumference of a circle, and thus on the pi scale. The pure sine wave stakes out one limit, the cosine wave the other, while in between lie and infinite number of possible waves. Without knowing the origin of our wave, we are ignorant of its phase--infinitely ignorant!
Suppose though that instead of a single regular wave we have two waves that are out of phase by a specific amount of pi? We still can't treat phase in absolute terms. But when we have two waves, we can deal with their phase difference--their relative phase value--just as we did in chapter 3 with the hands of the clock. And even though we may not be able to describe them in absolute terms, we will not be vague in specifying any phase differences in our system: our waves or cycles are out of phase by a definite value of pi. If we transfer our waves onto the circle, we can visualize the phase angle. In Fourier analysis, phase takes on the value of an angle.
What about relative phase in compound waves? Let's approach the problem by considering what happens when we merge simple waves to produce daughter waves. In effect, let's analyze the question of interference, but in the ideal. Consider what happens if we add together two regular waves, both in phase and both of the same amplitude. When and where the two waves rise together, they will push each other up proportionally. Likewise, when the waves move down together, they'll drive values further into the minus zone. If the amplitude is +1 in two colliding waves, the daughter wave will end up with an amplitude of +2, and its trough will bottom out at -2. Except for the increased amplitude, the daughter will look like its parents. This is an example of pure constructive interference; it occurs when the two parents have the same phase, or a relative phase difference of 0. The outcome here depends strictly on the two amplitudes, which, incidentally, do not have to be identical, as in the present example.
Next let's consider the consequences of adding two waves that are out of phase by pi, 180 degrees, but have the same amplitude. They'll end up canceling each other at every point, with the same net consequence as when we add +1 to -1. The value of the daughter's amplitude will be 0. In other words, the daughter here won't really be a wave at all but the original horizontal plane. This is an example of pure destructive interference, which occurs when the colliding waves are of equal amplitude but opposite phase.
Now let's take the case of two waves that have equal amplitude but are out of phase by something less than pi; i.e., something less that 180 degrees. In some instances the point of collision will occur where sections of the two waves rise or fall together, thus constructively interfering with each other--like the interference that occurs when we add together numbers of the same sign. In other instances, the interference will be destructive, like adding + and - numbers. The shape of the resulting daughter becomes quite complicated, even though the two parents may have the same amplitude. Yet any specific shape will be uniquely tied to some specific relative phase value.
Now let's make the problem a little more complicated. Suppose we have two regular ways, out of phase by less than pi; but this time imagine that they have different amplitudes. The phase difference will determine where the constructive and destructive interferences occur. Remember, though, that any daughter resulting from the collision of two regular waves will have a unique shape and size; and the resulting shape and size will be completely determined by the phase and amplitude of the two parents. In addition, if we know the phase and amplitude of just one parent, subtracting those values from the daughter will tell us the phase and amplitude of the other parent.
Suppose we coalesce three waves. The result may be quite complicated, but the basic story won't change: the new waves will be completely determined by the phase and amplitude of three, four, five, six or more interacting waves. Or the new compound wave will bear the phase and amplitude spectra that have determined completely by the interacting waves.
How does the last example differ from Fourier synthesis? For the most
part, it doesn't. Fourier synthesis reverses the sequence of analysis.
The process is an abstract form of a sequence of interferences that produced the original compound wave. But the compound wave, no matter how complicated it is or how many components contributed to its form, is an algebraic sum of a series of phases and amplitudes.
***
A moment ago when we were talking about simple waves, I pointed out that we can figure out the values of an unknown parent wave if we know the phase and amplitude of the other parent and of the daughter. Why couldn't we do the same with an unknown compound wave? Why couldn't we, say, introduce a known simple wave, measure the phase and amplitude spectrum of the new compound wave, and then derive the unknown amplitudes and phases? It might take a long time, but we could do it, in theory. In fact, the holographer's reference wave is a kind of "known." The reference wave is a relative known in that its phase and amplitude spectra are identical to those of the object wave before the latter strikes the scene and acquires warps. The holographer's "known" results from coherency, from a well-defined phase relationship between the interfering waves. But the phase and amplitude spectra in the object wave upon reacting with those of the reference wave will completely determine the outcome of the interference. And the results of that interference, when transferred onto the hologram plate, create the hologram.
***
When dealing with waves, theoretical or physical, it is critically important to remember their continuous nature. True, the physicist tells us that light waves are quantized (come in whole units, not fractions thereof), that filaments emit and detectors absorb light as photons, as discrete particles. We can look upon the quantized transfer of lights as the emission or absorption of complete cycles of energy. But within the particle, the light wave itself is a continuum. And when we do something to a part of the continuum, we do it to all of it. If we increase the radius of a circle, the entire circle increases, if it remains a true circle. And we can see the change just as readily in a wavy plot. Also, the change would affect the outcome of the union of that cycle with another cycle. If we change, say, the Fourier coefficients on the second cosine wave in a series, we would potentially alter the profile of the compound wave. And the effect would be distributed throughout the compound wave. For, again, the components do not influence just one or two parts of the compound wave. They effect it everywhere.
The continuous nature of waves is the soberly scientific reason for the seemingly magical distributive property of Leith and Upatniek's diffuse hologram, wherein every point in the object wave front bore the warping effects of every point in the illuminated scene.
As I've mentioned numerous time, relative phase is the birthmark of all holograms and thus the central issue in hologramic theory of memory. Remember that phase makes a sine wave a sine wave and a cosine wave a cosine wave-- once there's any amplitude to work with. We can come to the very same conclusion for the compound wave: the amplitude spectrum will prescribe how much, but the phase spectrum will determine the distribution of that amplitude spectrum. Thus our profiles, yours and mine, are as recognizable on the surface of a dime as they would be on the face of Mount Rushmore. And your profile is uniquely yours, and mine, because of unique spectra of phase.
***
In this chapter, we have examined waves in the ideal. For only in the ideal can we free our reason from the bondage of experience. We are about to extend our thoughts into the hologram. Using reason, we will ease our minds into an abstract space where phase information lies encoded. This space is most often called Fourier transform space. The entry fee for crossing its boundaries is the Fourier transform (the equation I mentioned earlier in this chapter), the yield of Fourier analysis. We make the journey in the next chapter.
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chapter eight
Ideal Mind
IN THE LAST CHAPTER, we learned some of the basic vocabulary of hologramic theory. Now we begin the of constructing a language from those elements, a process of assembling vehicles to convey brand new thoughts about the university within the brain. In this chapter and the next, we will raise questions about the mind that no one could have articulated a relatively short time ago. Let me forewarn the reader, though, that our answers will not assume the form of
the
physiological mechanism,
the
chemical reaction,
the
macromolecule,
the
genes,
the
cellular responses. As hard as it is to swallow, hologramic theory actually denies the assumption implicit in questions that demand the answer as bits of brain.
What is hologramic mind? What is the nature of the phase code? What is remembering? Recalling? Perceiving? Why does hologramic theory assert that physical parts of the brain, as such, do not constitute memory, per se? Why does hologramic theory make no fundamental distinction between learning and instinct? Why does hologramic theory predict the outcome of my experiments on salamanders? We have touched on these issues already, but only in an inferential, analogous and superficial way. Now we are on the brink of deriving the answers directly from hologramic theory itself. We will start by reasoning inductively from waves to the hologram and on to hologramic theory. Then, having done that, we will deduce the principles of hologramic mind.
***
The coordinate system we used in chapter 7 does exist on an ideal plane, true, but one we can easily superimpose on the surfaces we encounter in the realm of our experience. We can draw sine or cosine axes vertically on, say, the bedroom wall, or scribe a pi scale on a roll of toilet paper. If we equate sine values to something such as lumens of moonlight and place 29 1/2-day intervals between each 2pi on our horizontal axis, we can plot the phases of the moon. Alternatively, we could put stock-market quotations on the ordinate (sine or cosine axis) and years on the abscissa (our pi scale), and get rich or go broke applying Fourier analysis and synthesis to the cycles of finance. Ideal though they were, our theoretical waves existed in the space of our intuitive reality. I shall call this space "perceptual space" whether it's "real" or "ideal."
In the last chapter, I mentioned that in analyzing the compound wave (as a Fourier series of components regular waves, remember) the analyst calculates Fourier coefficients--the values required to make each component's frequency an integral part of a continuous, serial progression of frequencies. I also mentioned that the analyst uses coefficients to construct a graph, or write an equation. Recall that such an equation is known as a Fourier transform and that from Fourier transforms the analyst can calculate phase, amplitude and frequency spectra.
In everyday usage, "transform" is verb; it is sometimes a verb in mathematics, too. Usually, though, the mathematician employs "transform" as a noun, as the name for a figure or equation resulting from a transformation. Mathematical dictionaries define transformation as the passage from one figure or expression to another.[1] Although "transform" has specialized implications, its source, transformation, coincides with our general usage. In fact, a few mathematical transformations and their resulting transforms are part of our everyday experience. A good example is the Mercator projection of the earth, in which the apparent size of the United States, relative to Greenland, has mystified more than one school child and where Russia, split down the middle, ends up on opposite edges of the flat transform of the globe. We made use of transformation ourselves when we moved from circles to waves and back again. In executing a Fourier transformation, in creating the Fourier transform of components, the analyst shifts the values from perceptual space to an idealized domain known as Fourier transform space. In the old days (before computers did just about everything but wipe your nose) the analyst, more often than not, was seeking to simplify calculations. Operations that require calculus in perceptual space can be carried out by multiplication and division--simple arithmetic--in transform space. But many events that don't look
wavy in perceptual space show their periodic characteristics when represented as Fourier transforms, and by their more abstract cousins, the Laplace transforms.
But my reason for introducing transform space has to do with the hologram. Transform space is where the hologramic message abides. The Fourier transform is a link to transform space.
We can't directly experience transform space. Is it a construct of pure reason? Alternatively, is it a "place" in the same sense as the glove compartment of a car? I really can't say, one way or the other. But although we cannot visualize transform space on the grand plain of human experience, we can still intuitively establish its existence. We can connect transform space to our awareness, and we can give it an identity among our thoughts.
Have you ever looked directly through the teeth of a comb? If possible, try it in soft candle light. Observe the halos, the diffraction of light at the slits. If you haven't done so, and you can't find a candle or a comb, try this instead. Hold the tips of your thumb and index finger up to your eye and bring them together until they nearly touch. You should be able to see the halos overlap and occlude the slit just a tiny bit before your finger and thumb actually touch. Those halos are physical analogs of Fourier and Laplace transforms. In principle, the edges of your fingers do to the light wave squeezing through the slit what the Fourier analyst does with numerical values: execute a transformation from perceptual to transform space. What is transformed? The answer: the image the light waves would have carried to your eyes if the halos hadn't mutually transformed each other.
If the transform exists, it can be shown the transform space containing it necessarily exists. I say this not to propound a principle but to give the reader an impressionistic awareness of transform space. We have to intuit the ideal domain much as we would surmise that a sea is deep because a gigantic whale suddenly burst upon its surface. At the same time, our own finger tips give us a sense of reality.