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Optical holograms encode for amplitude as well as for phase variations in the scene. The basic reason for this has to do with the nature of light, with the waves' attainment of their maximum speed. But other kinds of waves also make holograms. And the acoustical holographer Alexander Metherell some years ago had a hunch that phase was really the generic essence of the hologram. Of course we can't have phase all by itself, except in the abstract. But Metherell wondered if he might assume just one amplitude--create a constant background din--and then encode the message strictly with phase variations. It worked. And he went on to demonstrate his point with the phase-only holograrns I referred to earlier. I mention phase-only holograms at this juncture to make a point about holograrnic mind. The frequencies and energy levels in the nervous system do not remotely approach those of light. For this reason we can't make a literal comparison between optical and neural holograms, at least not in using hologramic theory. Also, because of phase-only holograms, amplitude variations would not play a necessary and indispensable role in the storage of information in the brain. Phase is the essence of hologrannic mind. Before I supply more background on holograms, per se, let me raise still another important preliminary question. What do we actually mean when we use the word wave? Let's take an introspective look.
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Many of us first became acquainted viith the word wave when someone hoisted us from the crib or playpen, gently kissed us on the side of the head, and coaxed, "Wave bye-bye to Uncle Herbie!" Later, we may have thought "wave" as we pressed a nose against the cool windowpane and watched little brother Ben's diapers waving on the clothesline in the autumn breeze. Then one Fourth of July or Saint Patrick's Day, our mother perhaps gave us a whole quarter; we ran to the candy store on the corner, and, instead of baseball cards and bubblegum, we bought a little American flag on a stick. We ran over to the park and waved the little flag to the rhythm of the march; then we began to laugh our heads off when the mounted policemen jiggled by in their saddles, out of time with each other and the beat of the drums and the cadence we were keeping with the little waving flag. Still later, perhaps, we learned to wave Morse code with a wigwag flag, dot to the right, dash to the left. Up early to go fishing, when radio station W-whatever-the-heck-itwas signed on, we may have wondered what "kilocycles" or "megahertz" rneant. And it was not until after we began playing rock flute with the Seventh Court that the bearded electronic piano player with the Ph. 1). in astronorny said that "cycle" to an engineer is "wavelet" to a sailor, and that the hertz value means cycles per second--in other words, frequency. If we enrolled in physics in high school, we probably carried out experiments vlith pendulums and tuning forks. An oscillating pendulum scribed a wave on a revolving drum. A vibrating tuning fork also created waves, but of higher frequency: 256 cycles per second when we used the fork with the pitch of middle C on the piano. Moving down an octave, according to the textbook, would give us 128 hertz. Are our usages of wave metaphorical? The word metaphor has become overworked in our times. While I certainly wouldn't want to deny waves to poets, I don't think metaphor is at the nexus of our everyday usage of wave. Analogue is a better choice: something embodying a principle or a logic that we find in something else. (Notice the stem of analogue.) To and fro, rise and fall, up and down, over and under, in and out, tick and tock, round and round, and so on. Cycles. Periodicities. Recurrences. Undulations. Corrugations. Oscillations. Vibrations. Round-trip excursions along a continuum, like the rise, fall, and return of the contour of a wavelet, the revolutions of a wheel, the journey of a piston, the hands of a clock. These are all analogues of waves. Do we really mean that pendular motion is a symbolic expression of the rotations of a clock's hands? No. The motion of one translates into continuous displacements of the other. Is the ride on a roller coaster an allegorical reference to the course of the tracks? Of course not. The conduct of the one issues directly from the character of the other, to borrow a phrase from a John Dewey title. And why would we suppose that a pendulum or a tuning fork could scribe a wave? The answer is that the same logic prevails in all periodic events, pattems, circumstances, conditions, motions, surfaces, and so forth. No, a child's hand isn't the same thing as a fluttering piece of cloth or the ripples on a pond. And yes, there's imprecision and imperfection in our verbal meanings; we wouldn't want it otherwise. Poetry may exist in all of this. Yet by our literal usages of wave we denote what Plato would have called the idea of waviness, the universal logic revealed by all things wavy. And that logic translates, completely, into amplitude and phase. And if the medium stores phase, we have a species of hologram.
o o O Not all physics is about waves, of course. The liveliest endeavor in that science today, the pursuit of the quark, is a search for fundamental purticles--discrete entities--of mass-energy. The photon is a light particle. Light is both particles and waves. The same is true of all mass-energy at the atomic level. The electron microscope, for example, depends on electrons, not as the particles we usually consider them to be but as the electron waves uncovered in the 1920s as the result of de Broglie's theories. And one of the tenets of contemporary physics is that mass-energy is both particulate and wavy. But when we are dealing with particles, the wavy side of mass-energy disappears; and when it is measured as waves, mass-energy doesn't appear as particles. If you want to concentrate on corpuscles of light, or photons, you must witness the transduction of a filament's mass-energy into ]ight, or from ]ight into some other form, as occurs in the quantized chemical reactions in our visual pigment molecules. But if the choice is light on the move between emission and absorption, the techniques must be suitable for waves. Physics would be a side show in our times if the logic of waves had been left out of science. And waves might have been left out of science, were it not for Galileo's discovery of the rules of the pendulum. The pendulum taught us how to build accurate clocks. Without a reliable clock, askonomy would be out of the question. And how could anybody contemplate timing something such as the speed of light without a good clock? It was in 1656 that a twenty-seven-year-old Dutchman named Christiaan Huygens invented the pendular clock. Huygens wasn't just a back-room tinkerer. His work with the pendulum was the result of his preoccupation with waves. The reader may recognize Huygens's name from his famous wave principle. He had studied the question of how waves spread to make an advancing wave front. Have you ever observed ripples diverging in ever-expanding circles from the point where you drop a rock into the water? If not, fill a wash basin halfway, and then let the faucet drip . . . drip . . . drip! Huygens explained how one set of ripples gives rise to the next. He postulated that each point in a ripple acts just like the original disturbance, creating tiny new waves. The new waves then expand and combine to make the next ]ine of ripples, the advancing wave front. A diagram in a treatise Huygens published in 1690 is still the prototype for illustrations of his principle in today's physics textbooks. Nor is it a coincidence that Huygens, "during my sojourn in France in 1678," proposed the wave theory of light.3 (He didn't publish his Treatise on Ligl7t for another twelve years.) We can't see light waves. Even today, the light wave is a theoretical entity. And to scholars in Huygens's times, nothing seemed sillier or more remote from reality than light waves. But on November 24, 1803, Thomas Young, M.D., F.R.S., thought it right "to lay before the Royal Society a short statement of the facts which appear so decisive to me . . .
The hologram is not a phenomenon of light, per se, but of waves--in theory, any waves or wavelike events. I've already mentioned acoustical holograms. X-ray holograms, microwave holograms, and electron holograms also exist, as do 'computer" holograms, which are holograms constructed from mathematical equations and reconstructed by the computer--holograms, in other words, of objects comprised of pure thought. The same kinds of equations can describe holograms of all sorts And the very same phase code can exist simultaneously in several different media. Take acoustical holograms, for instance. The acoustical holographer produces his hologram by transmitting sound waves through an object. (Solids transmit sound as shock vibrations, as, for example, when someone knocks on a door.) He records the interference patterns with a microphone and displays his hologram on a television tube. Sound waves cannot stimulate the light receptors in our retinas. Thus we would not be able to "see" what a sonic wave would reconstruct. But the acoustical holographer can still present the scene to us by making a photograph of the hologram on the TV tube. Then, by shining a laser through the photograph, he reconstructs optically--and therefore visibly--the images he originally holographed by sound. Sound is not light, nor is it the electronic signals in the television set. But infOrmRtiOn carried in the phase and amplitude of sound or electronic waves can be an analogue of the same message or image in a light beam, and vice versa. It is the code, the abstract logic, that the different media must share, not the chemistry. For holograms are encoded, stored inforrnation. They are memories in the most exacting sense of the word--in the mathematical sense. They are abstractable relationships between constituents of the medium, not the constituents themselves. And abstract information is what hologramic theory is about. As l said in the preceding chapter, the inherent logic in waves shows up in many activities, motions, and geometric pattems. For example, the equations of waves can describe a swinging pendulum; a vibrating drum head; flapping butterfly wings; cycling hands of a clock; beating hearts; planets orbiting the sun, or electrons circling an atom's nucleus; the thrust and return of an auto engine's pistons; the spacing of atoms in a crystal; the rise and fall of the tide; the recurrence of the seasons. The terms harmonic motion, periodic motion, and wave motion are interchangeable. The to-and-fro activity of an oscillating crystal, the pulsations in an artery, and the rhythm in a song are analogues of the rise and fall of waves. Under the electron microscope, the fibers of our connective tissues (collagen fibers) show what anatomists call periodicity, meaning a banded, repeated pattem occurring along the fiber's length. The pattern is also an analogue of waves. Of course a periodically pattemed connective-tissue fiber is not literally a wave. It is'a piece of protein. A pendulum is not a wave, either, but brass or wood or ivory. And the head of a tomtom isn't the stormy sea, but the erstwhile hide of an unlucky jackass. Motions, activities, patterns, and waves all obey a common set of abstract rules. And any wavy wave or wavelike event can be defined, described, or, given the engineering wherewithal, reproduced, if one knows amplitude and phase. A crys tallographer who calculates the phase and amplitude spectra ofa crystal's x-ray diffraction pattem knows the intemal anatomy of that crystal in minute detail. An astronomer who knows the phase and amplitude of a planet's moon knows precisely where and when he can take its picture. But let me repeat: the theorist's emphasis is not on the nominalistic fact. It is on logic, which is the basis of the hologram. Nor, in theory, does the hologram necessarily depend on the literal interference of wavy waves. In the acoustical hologram, for example, where is the information? Is it in the interferences of the sonic wave fronts? In the microphone? In the voltage fluctuations initiated by the vibrating microphone? In oscillations among particles within the electronic components of the television set? On the television screen? On the photograph? The answer is that the code--the relationships--and therefore the hologram itself, exists--or once existed--in all these places, sometimes in a form we can readily appreciate as wavy infommation, and in other instances as motion or activity, in forms that don't even remotely resemble what we usualIy think of as waves.
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Lashley's experiments can be applied to diffuse holograms. as I have pointed out. His results depended not on where he injured the brain but on how much. Likewise, cropping a comer from a diffuse hologram does not amputate parts from the regenerated scene. Nor does cutting a hole in the center or anywhere else. The remaining hologram still produces an entire scene. In fact, even the amputated pieces reconstruct a whole scene--the same whole scene. What Lashley had inferred about the memory trace is true for the diffuse hologram as well: the code in a diffuse hologram is equipotentially represented throughout the diffuse hologram.
o o o The loss of detail that occurs when we decode a small piece of diffuse hologram is not a property of the code itself. Blurring results mainly from noise, not from the signal. How seriously noise affects the quality of an incoming message depends on the ratio of noise to signal. If the signal is powerful, we may dampen noise by reducing volume or brightness. But with very weak signals, as short-wave radio buffs can testify, a small amount of noise (static) severely impedes reception. In optical holograms, the relative level of noise increases as the size of the hologram decreases. And in a small enough piece of hologram, noise can disperse the image. We have already made the analogy between the survival of memory in a damaged brain and the survival of image in a marred hologram. Signal-to-noise ratio is really an analogue of the decline in efflciency found in Lashley's subjects. In other words, the less brain, the weaker the signal and the greater the deleterious consequence of ''neural noise."
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Loss of detail in an image produced from a small chip of hologram is a function of decoding, not of the code itself. An infinitesimally small code still exists at every point in the diffuse hologram. Like a single geometric point, the individual code is a theoretical, not a physical, entity. As with geometric points, we deal vwith codes physically in groups, not 2S individuals. But the presence of a code at every location is what accounts for the demonstrable fact that any arbitrarily chosen sector of the hologram produces the same scene as any other sector. Granted, this property may not be easy to fathom; for nothing in our everyday experience is like a diffuse hologram. Otherwise, the rnind would have been the subject of scientific inquiry long before Leith and Upatnieks.
o o o lf a single holographic code is so very, very tiny, any physical area should be able to contain many codes--infinitely many, in theory.l Nor would the codes all have to resemble each other. Leith and Upatnieks recognized these properties early in their work. Then, turning theory into practice, they went on to invent the "multiple hologram"--several different holograms actually stacked together within the same film With several holograms in the same film, how could reconstruction proceed without producing utter chaos? How might individual scenes be reconstructed, one at a time? Leith and Upatnieks simply extended the basic operating rules of holography they themselves had developed. During reconstruction, the beam must pass through the film at a critical angle-an angle approximating the one at which the construction beam originally met the film. Thus, during multiple constructions, Leith and Upatnieks set up each hologram at a different angle. Then, during reconstruction, a tilt of the film in the beam was sufficient for one scene to be forgotten and the other remembered. One of Leith and Upatnieks's most famous multiple holograrns is of a little toy chick on wheels. The toy dips over to peck the surface when it's dragged along. Leith and Upatnieks holographed the toy in various positions, tilting the film at each step. Then, during reconstruction, by rotating the film at the correct tempo, they produced images of the little chick. in motion, pecking away at the surface as though going after cracked corn. Some variant of their basic idea could become the cinema and TV of tomorrow.
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' Collier et al. 1971 p. 455, note that certain holograms would be able to store "50 bibles [inl one square inch " Multiple holograms permit us to conceptualize something neither Lashley nor anyone else had ever satisfactorily explained: how one brain can house more than one memory. If the engram is reduplicated and also equally represented throughout the brain, how can room remain for the next thing the animal learns? Multiple holograms illustrate the fact that many codes can be packed together in the same space. Just as important, multiple holograms mimic the actual recalling and forgetting processes: tilt the film in the reconstruction beam, and, instantly, off goes one scene and on comes the next. A few years ago, I met a young man named John Kilpatrick who suggested that a person trying to recollect something may be searching for the equivalent of the correct reconstruction angle. But suppose that instead of using a single reconstruction beam, we use several. And suppose we pass the beams through the multiple hologram at different angles. We may, in this manner, synthesize a composite scene. And the objects in the composite scene may never have been together in objective reality. When the human mind synthesizes memories into unprecedented subjective scenes, we apply terms such as thinking, reasoning, imagining, and even hallucinating. In other words, built right into the hologramic model are analogues of much human mental activity.
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Holography does not require the use of lenses. But lenses may be employed to produce certain special effects. Leith and Upatnieks showed in one of their earliest experiments that when the holographer uses a lens during construction, he must use an identical lens for reconstruction. This fact should (and probably does] interest spies. For not even Gabor or Leith and Upatnieks could read the holographic message directly. It is a code in the most cloak-and-dagger sense of the word. A hologram must be decoded by the appropriate reconstruction beam, under specific conditions. And a lens with an unusual crack in it would create an uncrackable code for all who do not possess that same cracked lens. A combination of different construction angies and flawed lenses might also be used to simulate malfunctions of the mind. Suppose a holographer makes a hologram of, say, a bedroom wall, and onto the same film also encodes the image of an elephant, using a lens at this stage. Given the appropriate conditions, he could synthesize the bizarre scene of a pink elephant emerging from the bedroom wall. Humans hallucinate similar scenes during delirium tremens.
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Leith and Upatnieks also made color part of holography. Physically, a particular hue is the result of a specific energy or wavelength. What we usually think of as light is a range of energies lying in the visible region of the electromagnetic spectrum. Molecules in our rods and cones make the visible region visible. Red ]ight lies on the weaker end of the spectrum, while violet is on the stronger end. Thus, infrared waves have energies just below red and ultraviolet waves are stronger than violet. Physicists often deal with color in conjunction with the subject known as dispersion. For when white light, say a sun ray, passes through a prism, the beam disperses into red, yellow, green, and blue light. (Dispersion also accounts for rainbows.) White light, remember, is a so-called spectral mixture. And fullcolor illumination of a multicolored scene requires white light. It is possible to produce white ]ight by mixing red, green, and blue lights. Thus the latter are called the additive primary colors. Not only will they produce white ]ight but varying combinations of them can yield the half-million or more hues we discriminate. The colors we see depend on which wavelengths reach the retina. The pigment in a swath of red paint looks red in white light because the molecules absorb the other wavelengths and reflect red back to our eyes. The sky looks blue on a clear day because the atmosphere absorbs all but the energetic blue violets. The sea looks black on a moonless night because nearly all the visible wavelengths have been absorbed. Light is energy. Thus tar on a roof heats much more in the sunlight than does a white straw Panama hat; the tar has absorbed more energy than the hat and has therefore reflected less. Photometrists use the word additive to describe red, green, and blue lights because subtractive primaries also exist: magenta, yellow, and green-blue. When magenta, yellow, and green-blue filters are placed in the path of a beam of white light, no visible light can pass through. The result is sometimes called black light. Black light is a potential product of even the three additive primary colors. For red and blue can produce magenta; green and blue can produce yellow, and, if the mix is right, some green-blue as well. And a beam of white light--a mixture of the primaries, red, green, and blue--can color a scene white, black, or anything in between, depending on the relative amounts of each primary color. Leith and Upatnieks described how they would "illuminate a scene with coherent light in each of three primary colors, and the hologram would receive reflected light of each color." Now the hologram plate itself was black and white. For the hologram remembered not color itself but a code for color. Yet when Leith and Upatnieks passed a red-green-blue beam through the hologram, they produced, in their words, "the object in full color."2 Offhand, it might seem as though the reconstruction beam would have to be the same color as the original light source. But Leith's equations said something different: the reconstruction
Before I describe the actual experiments, let's look at the independence principle by way of another imaginary experiment. This time, instead of transparent sheets let's imagine a deck of cards. Let's begin with a conventional nonhologramic message, using a single card as a set for storing one letter. The meaning of our message--let's use DOG--depends on the relationship between our cards: where each card lies in relationship to the others when the deck is at rest, or when a card turns up during the deal. If we shuffle the deck, we obviously run the risk of scrambling the meaning of our message. DOG might become GOD, for instance. Just as with our message, ANATOMY, on the transparent sheets, the message in our conventional deck is made up of inter-dependent elements. The hologramic deck of cards is far different. Here each card contains a whole message. And if the sarne message is on each card, just as the same feeding message is in each part of the salamander's brain, then shuffling will not alter the deal. But each card is an independent carrier of our hologramic code. What's to stop us from slipping in cards with new codes? Certainly not the codes per se. Cards are independent. Therefore, old and new codes can coexist in the same deck without distorting each other's meanings. And nothing in the information itself would prevent us from constructing a compound hologramic deck, or mind, if hologramic theory really does work. The big "if" is the readout: What happens in reconstruc tion? Which independent codes activate and drive an animal's behavior?
IN THE LAST CHAPTER, we learned some of the basic vocabulary of hologramic theory. Now we begin the process of constructing a language, of assembling vehicles to convey brand new thoughts about the universe within the brain. In this chapter and the next, we will raise questions about the mind that no one could have articulated just a few years ago. Let me forewarn the reader, though, that our answers will not take the form of the physiological mechanism, the chemical reaction, the molecule, or the cellular response. Hologramic theory 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 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 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. o o o
The coordinate system we used in chapter 7 exists on an ideal plane, but one we can easily superimpose on the surfaces we experience We can draw sine or cosine axes vertically on, say, the bedroom wall, or scribe a pi scale on a paper towel. If we equate sine values with something such as lumens of moonlight, and place 29 l/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 dwelt in the space of our intuitive reality. I shall call this space 'perceptual" space, whether it be "real" or "ideal." I mentioned in the last chapter that in analyzing the compound wave as a Fourier series of component regular waves, the analyst calculates Fourier coefficients--the values required to make each component's frequency part of a continuous, serial progression of frequencies. I also mentioned that the analyst uses the coefficients to construct a graph, or write its equation. This is known as the Fourier transform. From Fourier transforms, the analyst calculates phase, amplitude, and frequency spectra. In general usage, "transform" is a verb; and it sometirnes is in mathematics. 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.l While "transform" has specialized implications, its source, transformation, coincides with general usage. In fact, a few mathematical transformations and their resulting transforms are a part of everyday experience. A good example is the
lSee James and James, ]968; also, TJ?e Interrnational Dictilmary of Applied Mathematics. -
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 in which the Soviet Union, split down the middle, ends up on opposite edges of the flat transform of the globe. We have made use of kansformation ourselves, by moving from circles to waves and back again. In executing a Fourier transformation, in creating the Fourier transforrn of components, the analyst shifts the values from perceptual space to an idealized domain known as a Fourier transform space. Sometimes the analyst's objective is to simplify calculations. Operations that would require calculus in perceptual space employ multiplication and division in transform space. Also, many events that don't look wavy in perceptual space show their periodic characteristics when represented as Fourier transforms, and as 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 hologram's message abides. The Fourier transform is our link to transform space. We cannot directly experience transform space. Is it pure construct of reasoning? Or alternatively, is it a "place" in the same sense as the glove compartment of a car? I cannot say, one way or the other. But though we may not visualize transform space on the planes of experience, we can nonetheless establish its existence within our intuition; we can connect it to our awareness; and we give it an identity among our thoughts. Have you ever looked through the teeth of a comb, in soft candle light, and observed the halos, the diffractions of light, at the slits? If not, you might try this: hold the tips of your thumb and index finger close to your eye; bring them together until they almost touch. You should notice that the halos overlap and occlude the slit before your fingers actually touch. Those halos are physical analogues of Fourier and Laplace transforms. In principle, the edges of your fingers do to the light waves what the Fourier analyst does with numerical values: they execute a transformation from perceptual to transform space. What is transformed? The image the light waves would have carried to your eyes, if the halos hadn't transformed each other. If the transform exists, the transform space containing it also 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 bursts from its surface. Now I will put forth a principle. Although we cannot literally visualize the interior of a transform space, we can grasp the logical interplay of transformed entities. And very often, with the correct choice of a specific transform space, we can greatly simplify the meaning of an otherwise arcane idea. Let me demonstrate this point by introducing a process called convolution, whose ramifications and underlying theorem--convolution theorem--we will soon call upon. Convolution refers to the superimposition of independent sets, planes, or magnitudes. Consider two initially separated sets of dots. A and B, in perceptual space, as depicted in the figure on page 146. (My description is a highly modified version of one in Holmes and Blow, 1966.) The dots of A line up on the horizontal axis at intervals of 5 units, say, inches. Those of B lie 2 inches apart and run obliquely upward from left to right. If we "convolute" A and B, we create a two-dimensional lattice. And if we use an asterisk to indicate the convolution operation, we may define the lattice as A*B, which we would read "A convoluted on B." But what do we mean by convolute? And just how does the operation produce a lattice? The answers are quite complicated in perceptual space, but are very simple in what is called a reciprocal space (1/ space). We create 1/ space from Fourier transforms. The Fourier transform of a line of dots is a grating; that is, a series of uniform lines. In 1/ space the transform of A, let's call it T(A), isa grating made up of vertical lines whose spacings are the reciprocal of 5 inches _ 1/5 or 0.2 inches. The transform of B, T(B), is oblique lines running downward from left to right, with /2 or 0.5 inch spacings--the reciprocal of 2 inches. We can actually superimpose the planes represented in T(A) and T(B). Because the transforms are lines, we can see that the superimposition of the two creates a grid, something we could not observe with dots. The abstract operation corresponding to our superimposition is the same as the uniting of height and width to produce the area of a rectangle. This operation is multiplication. And we can define the grid simply as T(A) x T(B). Now let's take stock. First, perceptual and 1/ space are reciprocal transformations of each other. Second, A and T(A) are transforms of each other, as are B and T(B). Therefore, the mysterious A*B--the lattice produced by the convolution of A and B in perceptual space--is simply the reciprocal of T(A) x T(B). In other words, the asterisk in perceptual space is the equivalent of the multiplication sign in 1/ space. We can't say that convolution is multiplication. But we can see for ourselves that multiplication is the transform of convolution. Now multiplication is so much a part of our everyday lives that we hardly think of it as an act of pure reason. But that's precisely what it is. And in opting for transform space, we made a small sacrifice in terms of intuition, for a substantial gain in what we could make available to pure reason: simple arithmetic. The convolution theorem, which we are about to employ, is the mathematical proof that indeed convolution is what we said it is--an arithmetic operation on transforms. The critical lesson in our exercise is that while transform space is as remote as it ever was, it is far from incomprehensible. Fourier series have their corresponding Fourier transforms. And when attributes of waves do become incomprehensible to the intuition, in perceptual space, the appropriate transformation will put those attributes within the reach of reason.
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When objects distort the phase and amplitude of light, the resulting warps add up not to an image of the object but to the object's transform. The comea and lens--the eye's optical system--transform the transform into the object's image. The convolution theorem shows that the Fourier transform of the Fourier transform of an object yields the image. I'm sorry about the double talk. But the convolution theorem explains how the eye, a projector, or a microscope can turn the wave's warps into an image: the objects transform the carrier waves, and the lens system transforms the transforms from transform to perceptual space. Now I must apologize for triple talk. But think back for a moment to the halos. They are transforms of transforms right at your finger tips. Because your eye performed a third transformation, you saw transforms instead of images of the tips of your fingers. Just as the Mercator projection and the globe are different ways of representing the same thing, so Fourier transforms and Fourier series provide different perspectives on periodic phenomena. We can use what we learned in chapter 7 to understand the ideas we're formulating now. Think of the visible features of a face, a dewdrop, or a stand of pine trees as a potential compound wave in three dimensions. The interaction between carrier waves and objects is comparable to Fourier analysis--to the dissection of the compound wave into a series of its components, except in transform space instead of perceptual space. In other words, the first transformation is much like producing a Fourier series. The second transformation--the one that occurs at the eye and shifts the components from transform space to perceptual space--is comparable to Fourier synthesis, to synthesis of the series of components into a compound wave. The hologram captures the transform of an object, not the object's image. The interference of object and reference waves shifts their components into transform space. To conceptualize the reaction, let's form our imagery around waves, in perceptual space, but let's use reasoning alone for events occurring in transform space. Visualize the components of the compound object wave as strung out in a row, as we might draw them on a piece of graph paper in perceptual space. Along come reference waves that collide and interfere with each component. Each collision produces a daughter wave whose phase and amplitude are the algebraic sums of the phase and amplitude of the particular component plus those of the reference wave; or the reference wave and each component superimpose on each other. But the reactions take place in transform space, remember, and not in the perceptual space we use to aid our imagery. Therefore, we would observe not the image carried in the object wave, but an interference pattern the transform--the hologram. By imagining the components of a transform to be a Fourier series, we provide ourselves with something to "picture." This analogy could create the false impression that the object wave loses its continuity, that each component congeals into an isolated little unit in transform space. A continuum is a system in which the parts aren't separated; and the hologram is a continuum. We can appreciate its continuous nature by observing diffuse-light holograms, any arbitrarily selected piece of which will reconstruct a whole image of the scene. Although the reference must act upon each of the object wave's components, as in our conceptualization, the interference pattern represents the whole. Of course, no interference patterns or holograms can develop unless the reference and object waves have a "well defined" phase relationship. Recall that the optical holographer, using Young's and Fresnel's old tricks, produces orderly phase relationships by deriving object and reference waves from the same coherent source. What do we meam though, by "welldefined?" Let me invoke a strange but powerful theorem of topology that will take us into the general meaning of "well-defined''--a theorem known as Brouwer's fixed-point theorem. Brouwer's theorem is at the foundations of several mathematical ideas, and it is implicit in a great many more. The interested reader will enjoy M. Shinbrot's excellent article on the fixed-point theorem, which I list in the bibliography. Here, I will simply state the theorem without probing its simple but tricky proof. Brouwer's theorem guarantees that in a continuous distortion of a system--as in stretching without tearing a rubber sheet, or stirring without splashing a bowl of clam chowder-at least one point must remain unchanged. This point is the fixed point. Shinbrot describes how variants of the theorem have actually been used to predict contours on the ocean's floor from characteristics of the water's surface. The absence of a fixed point is enough to deny a truly continuous relationship between two entities or magnitudes. What is an object wave? In terms of frequency, it is the reference wave plus the changes imposed by the object. Before the object wave arrives at the object, its frequency is identical to that of the reference wave. The object imposes a spectrum of new frequencies on the object wave. But, invoking Brouwer's theorem, we note that one point in the object wave is not changed as a result of the collision, and the frequency at that point is the same before and after the object. In other words, the frequencies in the object wave will vary, but relative to the invariant frequency at the fixed point. Because the reference and object waves once had identical frequencies, the fixed point in the object wave must have a counterpart in the reference wave. Through the fixed point, the frequency spectrum in the object wave varies--but relative to the frequency of the reference. We must take note of an important difference between object and reference waves, namely the phase variation resulting from their different paths. Let's call this phase variation D. D will vary for each object component vis-a-vis the reference. But because of the fixed point, one of those D's will have the same value before and after the reference and object waves interfere; and all the other D's will vary relative to the invariant D. Variation relative to some invariant quantity is the general meaning of "well-defined," including "well-defined" phase relationships in interference phenomena. And a well-defined spectrum of D's in transform space is the minimum condition of the hologram. The minimum requirement is a fixed-point relationship between object and reference waves. A specific hologram is, at minimum, a particular spectrum of well-defined D's in transform space.
Reconstruction of the images from the hologram involves transforming the transform, synthesizing the original compound wave, and transferring the visible features of the scene back to perceptual space. This statement is a veritable reiteration of how the object originally communicated its image. In theory, the hologram regenerates what the object generates. In order for a wave to serve reconstruction, it must interact with all the components and must also satisfy the fixed-point requirement.
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What then is a memory? Transferring the principles we have developed to hologramic theory, and using the language we have developed thus far, we can define a specific memory as a particular spectrum of D's in transform space. What are D's? They are phase differences--relative values, relationships between and among constituents of the storage medium, the brain. Thus, in hologramic theory, the brain stores mind not as cells, chemicals, electrical currents, or any other entities of perceptual space, but as relationships at least as abstract as any information housed in the transform space of a physical hologram. The parts and mechanisms of the brain do count; but the D's they establish in transform space are what make memory what it is. If we try to visualize stored mind by literal comparisons with experience, we surrender the chance of forming any valid concept at all of hologramic mind, and quite possibly yield all hope of ever establishing the existence of the noumenon where brain stores thought.
In hologramic theory, the utilization of series of D's during covert or overt behavior, in recall or thoughts or feelings or whatever, is transforming the transform into perceptual space. This is a good point at which to examine the theoretical meaning of a percept. A percept, the dictionary tells us, is what we're aware of through our senses or by apprehension and understanding with the mind. In hologramic theory, a percept is a phase spectrum, a series of D's, in perceptual space. An activated memory, a reminiscence, is a back-transformed series of D's that have been moved from transform to perceptual space. In terms of the phase code, then, perception and reminiscence involve the same basic information, the difference being the sources of the D's: the percept is analogous to image generation by the object, while the activated memory is analogous to the reconstruction from the hologram. But both synthesize the message in the same theoretical way: the percept is a transformed transform of signals from the sense organs: the activated memory is the back-transformed series of D's stored in transform space. Just as the hologram regenerates what lhe object generates, memory regenerates what perception generates, as Karl Pribram asserted in a lecture some years ago. The specific character of the activated memory depends on the particular readout. A useful analogy here would be to the holographer's use of light instead of sound to decode an acoustical hologram, except that the nervous system has many more options than does the engineer. The activahng signal would determine the special features of the transformed transform, but the phase spectrum--the basic series of D's--would be the same whether the imagination or the fist punched somebody in the nose. In other words, through the code in transform space, behavior is a transduced version of perception. I will expand upon this idea more fully in a later chapter, after we've extended hologramic theory beyond where we are now. But we've already come far enough for me to say that hologramic theory provides a unified view of the subjective cosmos. There's not a box over here labeled 'perception" and one over there marked "behavior," with fundamentally different laws of Nature governing each. One abstract set of rules works ubiquitously.
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Frequency in perceptual space depends on time. Thus the Hz (hertz) value specifies cycles per second. In an interference pattern, however, frequency refers to how many stripes or beats occur in a given area. And whereas frequency assumes a temporal character in perceptual space, it takes on spatial meaning in transform space. We can draw an insightful corollary from the preceding paragraph. The phase difference between two interfering sets of waves determines the frequency of beats or stripes within the interference pattern. An intimate relationship exists, then, between phase and frequency. In FM (frequency modulated) radio, a specific message is a particular spectrum of phase variations. In waves with frequency independent of amplitude, as is true in the nervous system, the phase-difference spectrum in transform space is the stored memory. Earlier we used Metherell's phase-only acoustical holograms to postulate phase codes as the character of hologramic mind. But we have just deduced this conclusion directly from the theory itself. Karl Pribram suggested in the 1960s that visual perception may be analogous to Fourier transforms. He had in mind the hologram. Rather recently, a couple named De Valois and their collaborators developed a computerized system for calculating Fourier transforms of checkerboard and plaid pattems. They used the system to analyze neurons of the visual cortexes of monkeys and cats. Are these cells coded to perceive structural elements of the patterns? No. The neurons in the visual cortex responded to Fourier transforms of the patterns, rather than to the patterns' structural elements. The De Valois's results "were not just approximately correct, but were exact" within the precision of their system.2 As Pribram had predicted, the retina sends the brain not a literal rendition of the image, but a transform of the image. To phrase this in the language of hologramic theory, the phase codes stored in cells in the visual cortex transform the transform into images in the perceptual space of the conscious mind.
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2 De Valois et al . 1979. p 50 1 Mathematicians discover the properties of their theorems by manipulating equations, by bringing different terms together in novel ways. We will manipulate phase codes. Instead of equations, though, we will do our "calculations" in the "real" world, with pictures. Notice the identical sets of rings in figures 1 and 2. They're optical analogues of the crests of ripples on a pond. I superimposed these two sets of rings out of phase to produce the moire patterns in figures 3 and 4. The moires are interference patterns, and the phase difference is how off-center the two sets are relative to each other. If we compare figures 3 and 4, it is apparent that the frequency of beats, as well as their widths and spacings, vary with the phase shift. When the two centers lie closer together, as in figure 3, the stripes are coarse, widely spaced and of low frequency. Where the phase difference is great, as in figure 4, frequency is high, spacings are narrow, and the stripes are thin. The stripes are precisely determined by the phase shift. And these stripes represent the phase code in transform space. Are the stripes memories of the phase spectrum in the rings? The answer is yes, when the stripes reach high frequency. But let's not take my word for it. Let's turn the statement into a hypothesis and test it. If stripes encode for rings, we should be able to back-transform to rings from stripes alone. We ought to be able to overlay stripes on a set of rings and make new rings in perceptual space. Figure 5 is the putative memory, figure 6 its posited transform, and figure 7 the actual result of the test. New rings do indeed back-transform into perceptual space when we superimpose only stripes on them. What do we mean by stripes? They are beats, yes. But stripes are periodic patterns of light and dark, a harmonic array of alternating densities. Given this, the memory of rings shouldn't literally be confined to stripes. The memory is a periodicity, a wavy logic. We should be able to back-transform rings from, say, dots. Figure 8 shows that dots on rings will indeed create new rings. I am presenting this dot experiment for another reason as well. Perhaps the rings are just a matter of luck. Maybe the various dots are spaced fortuitously to interact with the correct arcs. lf we look at figure 8 carefully, we see that not all rings are the same. In fact, those on opposite ends of the vertical and horizontal axes, thus Iying on perspective arcs of the same circles, are mirror images of each other: where one ring has a dark center, the corresponding ring is light. If the "maybe" speculations above were right, these rings would be identical. But they are not. The dots also show something I hadn't foreseen but can't resist pointing out, for it illustrates perception and memory so well. I mentioned in chapter 4 that we often have difficulty remembering something exactly as we originally experienced it. Have you ever fumbled around with several vivid recollections that are similar, and wondered which is the correct one? Even our simple optical patterns seem to have this difficulty. Of course, taking a look at one's notes or using a fresh percept for comparison usually solves the problem much better than does memory alone. Likewise, with our dot system we merely have to compare the ůvarious readouts with the rings in the center. Figure 8 shows the cross-correlation of two series, thus indicating the apprehension part of perception. In hologramic theory, reasoning, thinking, associating, or any equivalent of correlating is matching the newly transformed transform with the back-transform. (The technical term for such matching is autocorrelation.)
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Let's shift our focus back to the nature of the phase code, which in our ring system is the preservation of ring information by penodic patterns. The ring memory is not limited to dots and stripes When we react rings with too great a distance between their centers, we do not produce stripes. Instead, something interesting happens. Inspect figure 9, and notice that rings are forming in the regions of overlap. Built into the higher frequency rings is a memory of rings closer to the center. Let me explain. First of all, as I have pointed out, the phase code is not literally stripes or dots, but rather a certain periodicity, a logic. Our rings are much like ripples on a pond; they expand from the center just as any wave front advances from the origin. Recall from Huygens's principle that each point in a wave contributes to the advancing wave front. The waves at the periphery contain a memory of their enbre ancestry. When we superimpose sets of rings in the manner shown in figure 9, we back-transform those hidden, unsuspected "ancestral" memories into perceptual space Figure 9 shows that no necessary relationship exists between the nature of a phase code and how the code came into being. The "calculation" represented by figure 9 shows why hologramic theory fits the prescriptions of neither empirical nor rational schools of thought. In figures 3 and 4, the system had to "learn" the code; the two systems had to "experience" each other within a certain boundary, in order to transfer their phase variations into transform space. But the very same code also grew spontaneously out of the ''innate" advance of the wave front. These are the reasons I would not define memory on the basis of either learning or instinct. Memory is phase codes: whether it's "learned" or "instinctive" has no bearing on its mathematical, and therefore necessary, features.
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Consider something else our stripes, dots, and rings reveal about the phase code. We can't assign memory to specific structural attributes of the system. In hologramic theory, memory is without fixed size, absolute proportions, or particular architecture. Memory is stored as abstract periodicity in transformspace. This abstract property is the theoretical basis for the predictions my shufflebrain experiments vindicated, and for why shuffling a salamander's brain doesn't scramble its stored mind. My instruments cannot reach into the ideal transform space where mind is stored. Hologramic mind will not reduce directly to constituents of the brain.
Now let's make a preliminary first fitting of hologramic mind to the world of Riemann. In terms of our search, a periodic event in perceptual space is a transformation, as a least curvature, to any other coordinates within the mental continuum. The same would be true of a series of periodic events. Since phase variation must be part of those events, memory (phase codes) becomes transformable to any coordinates in the mental continuum. A specific phase spectrum--a particular memory--becomes a definite path of least curvature in transformations from sensations to perceptions to stored memories to covert or overt behaviors, thoughts, feelings, or whatever else exists in the mental continuum. Calling upon our e's for imagery, the least pearly path will not change merely because the coordinates change. Behavior, then, is an informational transform of, for instance, perception. Transformation within the Riemann-style mental continuum is the means by which hologramic mind stores itself and manifests its existence in different ways. But we need some way of carrying out these transformations. For this purpose, I must introduce the reader to other abstract entities, quite implicit in Riemann's work but not fully developed until some years after his death. These entities are known as tensors.
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The mathematician Leon Brillouin credits the crystal physicist W. Voigt with the discovery of tensors.7 When placed under stress or strain, a crystal's anatomy will deform. But certain relative values within the crystal remain invariant before and after deformation. Like Riemann's invariant curvature relationships, the relative values that tensors represent will survive transformation anywhere, at any time. Just in time for Einstein, mathematicians worked out theorems for tensors. In the process they found tensors to be the most splendid abstract entities yet discovered for investigating ideal as well as physical change. Tensors provided a whole new concept of the coordinate. And they furnished Einstein with the language to phrase relativity, as well as the means to deal with invariance in an ever-varying universe- And with regard to their power and generality, Brillouin tells us: "An equation can have meaning only if the two members are of the same tensor character."8 Alleged equations without tensor characteristics turn out to be empirical formulas, and lack the necessity Benjamin Peirce talked about.
7 Brillouin. 1964, p. 3.
Tensors depict change, changing changes, changes in changing changes, and even higher order variations. Conceptually, they are very closely related to relative phase. Tensor relationships transform in the same way relative phase does. This feature affords us an impressionistic look at their meaning. Examine figure 8 on page 157, and keep a finger there for ready reference. Notice the three similar rings on the diagonal running from lower left to upper right. If you inspect the dots carefully outside the zone of overlap you'll see that we can draw a similar diagonal line between them--from lower left to upper right, as on the corresponding rings. Now, while looking at figure 8, rotate the page until the similar rings on the diagonal lie horizontal. Also watch what happens to the arrays of dots. Notice that the dots come to lie in horizontal rows like the rings. The rings and dots are transforms of each other. The fundamental direction of change remains constant in the transform as well as its back-transform. And in rotation, the basic orientation of rings and dots remains invariant. Absolute values differ greatly. But as we ourselves can observe, relative values tansform in the same way. This is the essence of the tensor: it preserves an abstract ratio independent of the coordinate system. If we stop to think about this, we realize that if tensors carry their meaning wherever they go, they should be able to define the coordinate system itself. And they do.
8 Brillouin, 1964, p. 3
Ordinary mathematical operations begin with a definition of the coordinate system. Let's ask, as Riemann did, what's the basis for such definitions? With what omniscience do we survey the totality of the real and ideal--from outside, no less--and decide a priori just what a universe must be like? The user of tensors begins with a humble attitude. He begins, as Riemann did, ignorant of the universe--and aware that he is ignorant. He is obliged to calculate the coordinate system only after he arrives there, and is not free to proclaim it in advance. Tensors work in the Cartesian systems of ordinary graphs. They also work in Euclid's world. But it is almost as though they were created for travel in Riemann's abstract universes. There are two senses of change: covariation, where changes proceed in the same direction, as when a beagle chases a jackrabbit; and contravariation, as occurs in the ends of a stretching rubber band. Tensors depict either type of change, or both simultaneously. If the beagle gains on the jackrabbit, when the quarry fatigues, for instance. or if the rubber offers more and more resistance as tension is increased, then the changes assume a higher order of complexity. Thus not only do covariant, contravariant, and mixed tensors exist, they can also attain higher ranks. Thus, packaged into one entity, there can be an incredible amount of information about how things change. Ordinary mathematics become cumbersome and eventually fail when dealing with such degrees of complexity. You may think that it would take pages to write representations of tensors. But the mathematician has invented very simple means of representing them: subscripts denote covariance and superscripts indicate contravariance. Thus Rt means a covariant tensor, or rank one. Rkt is a rank two contravariant tensor, and Rih/jk is a mixed tensor, once contravariant, thrice covariant--a simple statement of a very complicated situation In tensor transformations the mathematician applies formal rules to make, for instance, Ri in one coordinate equal, say, Rm in another. If, having applied the rules and performed the calculation, Ri doesn't equal Rm, then the changes aren't really a tensor and represent local fluctuations peculiar to the coordinate system; they have an empirical, not an analytical, meaning. If the mathematician even bothers with such parochial factors at all, he'll call them "local constants." Many features of holograms cannot be explained by ordinary transformations. In acoustical holograms, for instance, the sound waves in the air around the microphone don't linearly transform to all the changes in the receiver or on the television screen. Tensors do. The complete construction of any hologram can be looked upon as tensor transformations, the reference wave doing to the object wave what transformational rules do to make, say, Rij equal Rnm. Decoding, likewise, becomes much easier to explain with tensors. We can drop the double and triple talk, especially if we work within Riemann's continuum. The back-transformation of phase codes from transform to perceptual space becomes, simply, the shift of the same relative values from a spatial to a temporal coordinate of the same continuum.
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We imagined mind as a version of Riemann's theoretical world, a continuous universe of phase codes. Now we add the concept of tensors to the picture. Tensors represent phase relationships that will transform messages, independent of any coordinates within the system. Indeed, phase relationships, as tensors, will define the coordinates--the percept, memory. or whatever.
A universe constructed from Riemann's guidelines is an exceedingly abstract entity. Diagrams, because of their Euclidean features, undermine the very abstractions they attempt to depict. But just as we did with perceptual and transform space, let's let our imaginations operate in a Euclidean world and, with reason, cautiously, step by step7 think our way beyond our intuitive reality. Imagine two points, A and B, on our Euclidean world. But instead of joining them with a flat line, let's connect them with two curves. And let's imagine that the path describes a circle, with A and B Iying 180 degrees apart. Now let's begin a clockwise joumey from A. But when we get to B, instead of continuing on around to 360ř, and without reversing our forward motion, let's extend our journey into another circular dimension, this time a dimension of slightly greater size. We end up with a figure 8. Although the bottom cycle is larger than the top one, the relative values remain the same. Notice that we "define" our universe by how we travel in it. On a curve, remember, we move continuously over each point. When we get to B, we have to travel out onto that second dimension, if it is there. If we don't, and elect not to count it, it might as well be "what ain't." But if the dimensions join, then a full cycle from point A out and back again is quite different from a journey around a single dimension. Notice that we would have to go through two 360-degree cycles to return to A. The point I want to stress is that while curvature is our elemental rule, and while the relative values remain unchanged, an increase in dimension fundamentally changes the nature of our system. Suppose we add another dimension, at the bottom of the lower circle, for instance, increasing its absolute size even further to produce a snowman. Again our fundamental rules hold, and again relative values transform unchanged, but again the course and nature of our excursion is fundamentally unlike what we experienced with one and two dimensions. For although we have a single curved genus of figures, each universe is a new species. We can grow larger and larger circles along the bottom of the figure. But why restrict ourselves to point B and its counterparts? We chose B arbitrarily. We can pick any point around our original circle, and extend off in any direction. Nor are we limited to a plane. A new cycle may extend up or down or obliquely or in any direction. And what limits us to larger cycles? We can make them smaller, or proceed in the negative direction. The point is that we can evolve incredible variety from a very simple rule. I used circles for the sake of imagery. But let's get rid of literal circles and replace them with the concept of a pair of curvatures. In whatever direction we grow our universe, it's not a matter of circles or ellipses, but an abstract relationship. The relative values of the curved, mutually referencing pairs will remain invariant no matter where we extend our universe. To handle this idea, we must resort to reason.
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Let me summarize hologramic theory at this point, and connect it to what we've been discussing in this chapter. First, we conceive of mind as information, phase information. And we put that information into an ideal, Riemann-like universe, a continuum of unspecified dimensions, whose fundamental rule is curvature. A relative phase value--a "piece of mind"--is the ratio of curvatures. We give phase ratios expression in the form of tensors. And the modalities and operations of mind then become tensor transformations of the same relative values between all coordinates within the continuum. We dispose of our need to distinguish perceptual space from Fourier transforms and kindred transform spaces because we do not set forth the coordinate systems in advance. Coordinates come after, not before, the transformation. For illustrative purposes, we can consider hologramic mind as analogous to operations on a pocket calculator. The buttons, display, battery, and circuitries--counterparts of brain--can produce the result, say, of taking the square root of 9. The calculator and its components are very much a part of the real world. But the operations, the energy relationships within it, belong to the ideal world. One of those ideal coordinates coincides with experience.
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I appreciate the demands Riemann's world must make on the reader. Therefore, allow me to construct a metaphor to assist our imagination. Let's imagine a system whose rules apparently violate Riemann's curvature, a system that seems to be governed by straight lines, sharp corners, and apparent discontinuities everywhere. Imagine a giant checkerboard, on which each square is square like the whole. Now imagine that one square is subdivided into smaller red and black squares. Take one of those small squares and imagine that it is made up of still smaller squares. The pattem repeats itself again and again and again, throughout all levels. The various levels can be seen as the equivalents of dimensions within our curved continuum. Thus we can model the same red-black phase spectrum at any level. And, because we can subdivide any square as much as we please, we can make two checkerboards carry vastly different specific internal patterns. Now that we have created the imagery we want, let's get rid of the metaphor, but with reasoning rather than by fiat. We said that the red and black squares were infinitely divisible. Assume that we approach the dimensions of a single point (as we did with the hypotenuse and our pearly e's). lf the system is infinitely divisible, there ought to be a single unreachable point-sized square, at infinity. If there are two squares down there, then our system is not infinitely divisible; and if it's not infinitely divisible, where do we get the license to divide it up at all? Thus we must place a single square at infinity, so that we can keep on subdividing to create our metaphor. But what kind of square occupies infinity? Red or black? The answer is, both red and black. At infinity our apparently discontinuous system becomes continuous: red and black squares superimpose and, as in Riemann's universe, become part of each other. This is a hidden continuity that underlies the true nature of the repetitive pattern, and it is the reason why we can deal with the pattern systematically at all. Our checkerboard metaphor of hologramic mind tums out to be a disguised version of Riemann's universe.
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Now that we have a general system, let's use it. Let's answer a few questions with it. How do we really account for the results of shufflebrain experiments? How could Buster's fish codes blend in so smoothly with his own? How was it that Punky's salamander medulla could receive the tadpole messages from the rest of his brain? The same questions exist for "looking up" and the rest of my experiments. Why weren't the experiments like trying to pound a square peg into a round hole? Continuity had to exist. And phase transformations had to define the coordinate system, rather than the other way around. Consider another question, now that I've mentioned salamanders and the mixing of species. How can we explain the similarities and the differences between them and us? Hologramic mind, constructed as a version of Riemann's universe, supplies the answer in two words: curvature and dimension. We share the rule of curvature, but we and they are totally different universes by virtue of dimension. (I will return to dimension in the next chapter when discussing the cerebral cortex.) And how can we sum together phase codes of learned and instinctive origins, if fundamentally different abstract rules govern, say, a reflex kick of a leg and a 6/8-time tarantella? We'd move like jerk-jointed robots if our inner universe were a series of discontinuous pieces. How could we condition a reflex if we couldn't blend the new information smoothly with what's already there? Speaking of robots, we are vastly different from the digital computer. The computer's mind is a creature of the linear, Euclidean world of its origin. Its memory reduces to discrete bits. A bit is a binary choice--a clean, crisp, clear, yes-no, on-off, either-or, efficient choice. The computer's memories are clean, crisp, clear, linear arrays of efficient choices. The hologramic continuum is not linear; it is not either-or; it is not efficient. Hologramic mind acts flat and Euclidean and imitates the computer only when the items of discrete, discontinuous data are few. We're swamped when we try to remember or manipulate an array of twenty-six individual digits, a simple task for the computer. Yet ask the digital computer to distinguish your face from a dozen randomly sampled other faces--with and without eyeglasses, lipstick, and mustaches, and from various distances and angles--and it fails. Brains and computers operate on fundamentally different principles, and they mimic each other only when the task is trivial. Consider also the problem of perceptions of time and space, as opposed to physical time and space. People, the author included, have dreamed ten-year scenes within the span of a tenminute dream. The reverse also can happen: a horror lived during a second of physical time can protract to many minutes, during a nightmare. To the scuba diver out of air, a minute seems very long. But time seems to compress during a race to the airport, when we are a few minutes behind schedule. Space may do strange things, too. A character in a recent Neil Simon play tells how, during a bout of depression, he couldn't cross the street because the other side was too far away.9
What do we do about subjective phenomena? Discount them from Nature because they are subjective? In Fourier (and kindred) transforms, the time-dependent features of relative phase become space-dependent. But the relationships in transform space obey what time-dependent ones do in perceptual space: the axes don't contract and expand. Tensors, on the other hand, aren't constrained by presumptions about coordinate axes. In the curved continuum, time-dependent ratios may turn up on an elastic axis. And because the hologramic universe is a continuum, we lose the distinction between perceptual space and other kinds of space; or we may have the conscious impression that time is expanding or that distances will not close. Yes, it's ideal, subjective, illusory. Subjective time and space are informational transforms of what the clock gauges and the meter stick measures. The constraints on the clock and meter stick are physical. Constraints on the transformations of the mind are ideal. But both belong to Nature.
9The clinical literature abounds with reports of this sort, some related to obvious brain damage, others not. For instance, N. J. David (1964, p. 150) - presents the case history of a young man whom the police arrested for being drunk when he asked them why They were only two feet tall. The young man was not drunk. He was suffenng an attack of psychomotor epilepsy. o o O
Nonetheless, hologramic theory suffers from a major deficit, and we will have to correct it. Our construct is too perfect, too ball-bearing smooth, too devoid of the chance for error, for the twig missing from a nest, or the freckles on a face. We can't see ourselves in such a picture. We must add to the theory what doesn't transform, what won't remain invariant in all other coordinates. Our picture needs precisely what the mathematician often goes to great pains to get out of his way--parochial conditions, particular features, local constants. Physiologist E. Roy John identified some local constants (see chapter 2) in the form of noise.
Hologramic theory does not predict that microbes, beheaded bugs, or decerebrated animals necessarily perceive, remember, and behave. Experiments furnish the underlying evidence; and some of it, particularly in the case of bacteria, has been far more rigorously gathered than any evidence we might cite in support of memory in rats or monkeys or human beings. But the relative nature of the phase code explains how an organism 2 micrometers long--or a thousand times smaller than that, if need be--can house complete sets of instructions; and transformations within the continuum give us a theory of how biochemical and physiological mechanisms quite different from those in the intact brains and bodies of vertebrates may neverdleless perform the same overall informational activities. Yet hologramic theory does not force us to abandon everything else we know. As we will see fram the theory's own internal logic, we do not dispense with the brain. The theory does not explode the foundations of our reality. Instead, hologramic theory gives new meaning to old evidence; it allows us to reassemble the original facts, come back to where our quest began, and, in the words of T. S. Eliot, "know the place for the first time." In the last chapter, I pointed out that two universes developed according to Riemann's general plan would obey a single unifying principle, curvature, and yet differ from each other totally if they varied with respect to dimension. Thus the hologramic continuum of both a salamander and a human being would depend on the phase code, and tensor transformations therein; but our worlds are far different from theirs by virtue of dimension. Now 1 would like to take this statement out of the abstract.
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Given the capabilities of single cells, it is not surprising that a monkey will sit in front of a display panel and win peanuts by choosing a triangle instead of a square. By doing essentially the same thing, rats and pigeons follow a trend Edward Thorndike first called attention to during the late 1890s. Even a goldfish, when presented with an apparatus much like a gum machine, quickly learns that bumping its snout against a green button will earn it a juicy tubifex worm, whereas the red button brings forth nothing. It is not surprising that those who were aware of such choice-learning experiments began to think about the evolution of intelligence in terms of arithmetic: thus if we added enough bacteria, we'd get a fish or a man. In the late 1950s, the behavioral psychologist M. E. Bitterman began wondering if it was all this simple. Something about the choice method of assessing learning didn't smell quite right to him. Bitterman decided to add a new dimension to the experiment. His results showed that behaviors cannot be explained by simple linear addihon and subtraction. Bitterman began by training various species in the traditional choice method: his animals had to discriminate A from B in order to win a prize. Then after his goldfish, turtles, pigeons, rats, and monkeys associated A with reward and B with no reward, Bitterman played a dirty trick. He switched the reward button Chaos broke out in the laboratory. Even the monkey became confused. But as time went by, the monkey began making fewer and fewer mistakes. Then the rat and pigeon began to get the idea. They were reversing the habit. The longer they played the game, the fewer mistakes they made and the more conspicuous the habit reversal became Meanwhile, over in the aquarium, the goldfish was still hammering away hopelessly, trying to win a tubifex worm by playing the same old choice. The fish could not kick the old habit and learn the new one. What about Bitterman's turtle? It was the most interesting of all his subjects. Confronted with a choice involving spatial discrimination, the turtle easily reversed the habit. But when the task involved visual discrimination, the turtle was no better off than the goldfish. In other words, it was as though the turtle's behavior lay somewhere between that of the fish and the bird. Turtles are reptiles, remember. And during vertebrate evolution, reptiles appeared after fishes and amphibians but before birds and mammals. Now an interesting event takes place in the evolution of the vertebrate brain. In reptiles, the cerebrum begins to acquire a cortex Was the cerebral cortex at the basis of his results? Bitterman decided to test this hypothesis by partially damaging the rat's cerebral cortex. What did he observe? He found that a rat with a damaged cerebral cortex made the habit reversal when given a spatial problem, but failed to do so when the choice involved visual discrimination. Bitterman's rats acted like the turtle
o o o Bitterman's experiments tell us that with the evolution of a cerebral cortex something different began to emerge in the vertebrate character. Simple arithmetic won't take us from a bacterium to a human being. As embryos, each of us once reenacted evolution, in appearance as well as behavior. Up to about the fourth intrauterine month, a human embryo is quite salamanderlike. We evolve a primate cerebrum between the fourth and sixth months. When the process fails, we see in a tragic way how essential the human cerebral cortex is to us. Mesencepha Cia? is one term applied to an infant whose cerebrum fails to differentiate a cortex.9
Fractals and consciousness. jour98a # 47 Backwards time or self as information space of coherent source orthogonal to biological time would be an iterated process where "sleep" was the reset to the beginning. Thus I can ask where to reset: music, learning JAVA, VB fractal music, or whatever. The cycle seems to be on a single scale maybe for 5 or 6 or 8 or 7 or 12 times and angles until entering the next "scale" or level of resolution for more detail. I am seeing a star of David or other large diagram!
2/22/1998 jour98a # 48 Action is connected to name and in animals, smell. It closes off eternity and throws us [humans and animals] into biological time. Self is in orthogonal information space of no-time and no action and maps of biotime we call thought. This thought is a blueprint for new biostorage in DNA. jour98a # 49 The phase space in n.nets is the "Shape" of the energy surface! Attractor basins are entire complexes whose boarder/boundary is "Word"!
I hope that revealing the oneness of humankind on the same footing as the laws of gravity will allow an awakening to human nature, which human abilities have primarily here-to-fore been used by and for the training of our animal nature [as if it is only the "Dumb" animal in us].
jour99 # 12 The evolution of animal consciousness is fully coherent contained within the DNA structures of millions of years and does not respond to current planet positions in the here-and-now. This old intelligence is responsive to the positions of the planets, but doesn't reorder itself on the basis of realtime positions. This is with strong contrast with human personality which reforms itself at birth within a ONE MIND context. The entire human species is coherently ordered by structures like the Stars as multicoherent sources, but it is whole groups and civilizations that is ordered with ezch individual taking a individual phase angle within that coherence as if it is a single organism with many holographic self organizing positions. jour99 # 13 The development of this personality arose from the ability of humans to be entranced by animal totem consciousness. Humans learned to go into a trance that involved sharing that awareness as if we are seeing thru the animal's eyes, hearing and feeling thru their mind. This is validated by the real experiences of access to information unavailable in "normal" consciousness as in far-seeing when becoming a bird. When this new skill was open up to in humans it required stepping out of the old unity human/monkey consciousness to experience a larger frame / level of resolution. In that frame could take that larger frame and focus it on a subset as if one packs many pixels into a small space so that the human experiences the animal and themselves at the same time. [more?] So when humans developed this skill, it is applied to allow the person to use this larger frame to respond by structuring themselves as "personalities/egos" which was not possible before. jour99 # 14 Each Human has the entire structure present that is selected by the positions of the cosmos. This is as if we have the 4 arms and many legs of some Hindu gods but only allow ourselves to employ a subset empowered by trance. jour99 # 15 This visions human personality/ego as a new power of mind that we do not even know exists. [except NLP].
This multiple coherence of over layering is like learning another language or learning to dance: during the beginning stages we still think in the old language or must think of the way to move our feet, but later we can think in the new language or make up steps and directions within the framework of the dance. Here the music becomes the coherent source that tells us where the steps go.
This is very important illustration of how holographic mind works. It also unifies how we moved and developed from totem animal programs loaded in a trance to birth days loading a totem animal projected onto the cosmos to "totem" individuals from "God" whose birthday we shared to scientific truth taking the place of "God" and free national holidays becoming a universal "birthday".
The laptop computer battery is a model of individual separate existence whereas the plugged into the wall electric network is the holomind as everywhere having the same functionality. This shows the coherence of function that is added to by the "information" of whatever is "using". This is source as well as other places of source like gas or gasoline! Each coherence can "source" many "holograms"! So the animal brain is like the car and the human brain is like all the occupants and driver [ego]. Humans can go to any virtual position as if they were loading the virtual DNA code of any animal [other cars or trucks]. This virtual space is of the same cosmic direction as putting the earth in other positions in our chart from other persons earth or teachers / concepts. Rational mind in this case does not see the functionality of holomind but sees only the organizing map and picture called "knowledge / wisdom". A functional space in mathematics is different than a space of coordinates or parameters [location and value worth]. Thus what seems like "mind" is created from virtual coherent positions, not from functional "DNA" coherence and has this freedom that we mistake for consciousness. Consciousness is the re-membering as a function of a hologram restored to action. In plants it is the seasons re-membering different holograms into action. Any functional action is an action of consciousness as a reconstructed hologram mind/body.
Science and religion is about Us, but is structured and focused on what is labeled "The Universe" or "God - Spiritual" or "Truth - scientific Laws". I see the roots of our experience of understanding as very ancient: millions of years back to our roots as tree - forest dwellers. In that environment proto humans had their logics and intelligence built in over millions of years of evolution. These logics of action - process and behavior had to deal with a 3D environment of the tree tops, and are still "built into" modern man, but not directly accessible by our language consciousness of our newly developed human brain. People who allow these and other older logics to emerge into consciousness have been labeled "mystics" or religious figures by our scientific culture, but this intuitive thinking has played a major part in the development of modern science. The dream of Crick that resulted in understanding the shape of DNA is the most well known.
The difficulty in accessing our human mind comes from those successes in human evolution which resulted in our skills at controlling our environment. This is a beautiful contradiction: the very development which allows humans to think and produce what we call "knowledge of ourselves and the universe" are the very skills that restrict our access to our older [by millions of years] intelligences. Let me review the process I am referring to. When proto humans emerged from the forests onto the plain, we had to compete and contend with larger, faster and stronger animals, yet we developed cooperative hunting societies by using our 3D logics as hierarchical trees tops whose branches connected humans and allowed them to develop procedures and maps - models of our environment. A model of this mapping process can be made using the analogy of many spiders on the top of tree that spin webs that connect to the leaves and branches below them. Each spider also connects to the webs of the other spiders at their same level. This allows the information contained in the specific connections to the leaves to be reflected everywhere among the all the spiders. Then the off-springs of each spider can migrate to other trees and spin almost identical webs given the fact that each tree in unique! This is a model of language in holographic mind where the spiders are the tokens we label "words" and the connections with the lower "tree" is the meanings and organizational structure is provided by the lateral "webs" or networks.
This development in the human species allowed humans to avoid the biological impossibility of external supervision by anything or anyone exterior to neural networks. This results from the recently developed mathematical "fact" that the only structural type of neural networks that can function in any biological system must be self organizing. So whatever responses to any event or anyone in the exterior environment is totally determined by structures and programs already present within the neural network and to self organizing changes made only within those networks. Within these limits those procedures necessary to the formation of human culture like understanding "NO" or instructions, commands and "knowledge" or passing on traditions or supervised teaching are biologically impossible!
So language and maps provide a way around this limit by allowing humans to create a virtual One Mind that everyone can share as if we are all within a single virtual skull. This solution has been pointed to for centuries by the creation of Cathedrals, Mosques, Temples, Churches and government buildings built in the shape of a large skull!
More on this new definition of language and human nature later, but now back to the emerging procedural use of language. As humans learned to simulate control and supervision of others as if they are one unified individual, humans developed control and supervision of other species called domestication both in plants and animals. This of course opened a whole new map of being human that was organized around a central rational principle governed - driven by the Sun and seasons and the need for discipline in the emerging agricultural societies. The idea of discipline and control needed to be invented because the self organizing neural networks were [ARE] still inventing new way to do tasks as hunters and in competitive ways. Competition is built into the biology of holographic neural networks where the same input [problem] appears laterally at many locations in a network where each location operates with self organization which results in many solutions which compete with each other, and the winners laterally totally inhibit [or turn off] the losers. In most common situations requiring decisive action this is totally necessary, called being of one mind or single intention, or an individual would be trying to go in many directions at the same time.
This natural behavior of neural networks is projected into the ONE MIND called culture where it can result in so called "evils" as cultural ethnic cleansing to totalitarian governments. But this also results in the so-called "goods" of singular reductivistic provable scientific truth and theory. That leaves out those areas which are chaotic and self organizing: where each instance is individual.
The search for self understanding is thus blocked by the very success that makes being human in a shared human culture possible: we look for cause and effect and the laws or "gODs" that govern.
Thus humans hope for more control by discovering the laws of the universe: but what if the laws that govern life and self organization are just that: organized by "choosing" from or trying an infinite number of possibilities.
Thus survival of the fittest is not a "Law of evolution" but a reflection of the behavior of lateral inhibition in neural networks.
Yet it is those procedures that allow organization to take place which will survive and the holographic paradigm is just such a way to organize. This resulted in the holographic phase space code of the DNA and the over-layered linked phase codes resulting from using the cosmos as sources of coherence!
So where do we look for self understanding? In places that can describe natural process, structure and connection.
My model of connection to the cosmos is that life systems use the cycles of time of the earth's rotation, nodal orientation and orbital position as well as the Moon's orbit and nodal orientation [height dimension] as the first coherent sources in the oldest structures of single cells up to the reptilian brain. That further evolution of the reptilian brain up to the mammal brain included the orbital positions of the planets as well as the passage of the Sun thru the galaxy and started to use multi-coherent sources as pairs of planets [and triples?]. These structures are locked into the DNA much as computer chips are hard wired with the code burned into ROM as 586 in all series up to RISC chips. The new evolution of the neocortex in the mammal brain leading up to Being Human allows orientation in time of the day on the day of birth to be used to differentiate between humans. Thus the same code can be loaded in different ways much as various operational codes are loaded into RISC chips which can then run many codes of different previous chips as if they are IBM and MAC at the same time. This virtual behavior is what is pointed to by the human types first introduced in Astrology maps such as LEO, Taurus, Scorpio or snake, Ox, tiger, rat, dog etc. in various systems. Then the human mind becomes a virtual ecological mapping onto a single species! The only problem is that very little code is written in the "real" or native code space of human being. This code space is beginning to be explored by mathematics, science, and with the new developments in psychology of NLP and cognitive therapies. But what else can the idea of "gOD" be but a view from a virtual animal of the human mind!!!! The ratio of the lower brains as fully engaged 1 to 20,000 in the neocortex is like [compared to] a solid to a gas [air] or mist which is similar to what is called "spirit". Thus the fuzzy, creative and enormous information and coding space of the neocortex which creates many, many maps from all angles of reality and even maps of the maps as minuets are maps of hours and seconds of minuets.
Then as we humans started to become self aware or self referent or find self similarities to map our inner processes [we invented "understanding"] we came up with very global transcendent concepts like spirit and God, because mathematical logic shows us that the map maker cannot make a map of itself. This is NOT a definition of God or god but of a process by which humans created self reflection and self understanding. But I am not avoiding definitions using science of those areas referred to as God. Nor am I saying that science has all the answers. But if we fail to see that many of the issues raised by thousands of years of religious belief are correctly answered by modern science, then we cannot move on the still other unsolved and uncharted areas. My assertions about the connection of the cosmos as sources of coherence with all life on earth is one such issue for me.
Group [religious or other] as protection from individual older brain reactions of reptilian strength: "being in the body of Christ" allows one to use the weak neocortex language brain to control the individually much stronger lower centers. Thus one can refer to group action such as law and police or moral rules.
The perception of unchanging eternal "God" is the view of DNA by the virtual mind which experiences the stability of DNA as the ancient of ancients. Thus the very first beginners procedures become embedded in our present process as if we had to use a team of horses to pull a modern car. Thus "gOD" does change and I AM god means I have access to the newest creativity - self organization of the Universe. Starting new is very important.
The angle of focus of coherent sources also is the angle of retrieval and information access! These personal angles of access is really all that can be different in humans! If the DNA and the biosphere is in phase space and embedded phase codes, then "consciousness" as coherent sources that "re-member" states and process are accessed differently from different phase angles. This is what the evolution from fixed phase angle for coherent source of the day-night cycle to a mapping of this day-night onto a virtual cycle based as if dawn was the Eastern horizon at the moment of ones birth such that any moment of a day can be a virtual dawn is the mapping of what in traditional astrology is called the house system. In this same direction of evolution, we developed individual orientation to the moons nodes, the Sun's nodes and recently the planet nodes. The Sun's nodes are the Qi of the back called constellations and the earth's node the signs as the Qi of the front. I am pursuing the connection between the moon's nodes and the cycle length of DNA as 19 turns.
These orientation processes are the procedures used to create individual consciousness and access to personal memories and procedures.
4/16/97 2pm 12.2 The signs are like 12 primitive video cassette recorders that are hooked up to a modern super video recorder that is equipped with video camera, sound and radio etc. and can mix / input from all the 12 sources. This modern recorder - player can also control the inputs, but it is instructed in only recording from the primitive recorders as a play thru unit. This is our modern "human" neo-cortex which learns to be an animal trainer - snake charmer to our lower brains: It is like the shepherd of the flock with the sheep like our mid-brain and we train the sheep to look out and avoid snakes [our reptilian brain] and other "predators". So many humans take the role of "sheep" and "cows" [Taurus] and other wild animals that need guidance or taming - training. This is perpetuated by Christianity who after discovering this instead of freeing us from it and allowing us to connect with our human being [brain] kept us in-slaved by programming us to accept only one person is / was and ever will be able to be human: Jesus Christ. This self organizing creativity called "gOD" is then never available to any human to be human! 12.3 The attitude of science is to find out what problems exist, what are their causes and how to fix the problem, not to use what they know to enslave others! This was the attitude of the Roman empire and for me not what a real spiritual person would do or think: "You are the sheep and I am the shepherd!" That is the attitude of a beginner in the spiritual path who ends up starting a cult! When we see that we are human we see that we can create our own videos by using the cameras and sound recorders and computer that is our neo-cortex! But we can also share the videos made by other humans! In this metaphor of mind the planets are like many different speeds of the video recorders and dubbing - mixing procedures.
12.2The signs are like 12 primitive video cassette recorders that are hooked up to a modern super video recorder that is equipped with video camera, sound and radio etc. and can mix / input from all the 12 sources. This modern recorder - player can also control the inputs, but it is instructed in only recording from the primitive recorders as a play thru unit. This is our modern "human" neo-cortex which learns to be an animal trainer - snake charmer to our lower brains: It is like the shepherd of the flock with the sheep like our mid-brain and we train the sheep to look out and avoid snakes [our reptilian brain] and other "predators". So many humans take the role of "sheep" and "cows" [Taurus] and other wild animals that need guidance or taming - training. This is perpetuated by Christianity who after discovering this instead of freeing us from it and allowing us to connect with our human being [brain] kept us in-slaved by programming us to accept only one person is / was and ever will be able to be human: Jesus Christ. This self organizing creativity called "gOD" is then never available to any human to be human!
12.3The attitude of science is to find out what problems exist, what are their causes and how to fix the problem, not to use what they know to enslave others! This was the attitude of the Roman empire and for me not what a real spiritual person would do or think: "You are the sheep and I am the shepherd!" That is the attitude of a beginner in the spiritual path who ends up starting a cult! When we see that we are human we see that we can create our own videos by using the cameras and sound recorders and computer that is our neo-cortex! But we can also share the videos made by other humans! In this metaphor of mind the planets are like many different speeds of the video recorders and dubbing - mixing procedures.
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