Visualizing and Conceptualizing Higher Dimensions

by Ted Denmark, Ph.D.

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Recently I was listening to a fascinating online conversation between media-star astrophysicist Neil deGrasse Tyson and the amazing cyber-sleuth Edward Snowden, who were talking about the well-known difficulty of visualizing higher dimensions beyond 3D—or at most 4D—now including *time* since Einstein’s Relativistic Space/Time fusion. It was a familiar puzzle that I had also thought about occasionally since first encountering it in school: the familiar notion that the first three spacial dimensions are all mutually perpendicular, which seemingly would also have to be true for higher dimensions; so the question had always been … what could be perpendicular to the 3D spacial triad or 4D Space/Time? My high school math and physics teachers presented 3D as merely obvious, but they were not too sure about the 4D part—the “arrow of time” asymmetry … which was more of a mystery, as it has remained.

This reminded me of a small epiphany I had recently when I remembered the interpretation of the operation of *differentiation,* as the term is used in calculus—I would say, marking the real beginning of higher mathematics. It had been difficult even for me as a so-called “brain” (our generation’s version of “nerd”) to visualize or understand calculus when it was first presented in my advanced college physics class that I was able to take as a senior in high school. And in fact I had never really made much progress with either calculus or this higher-dimensions “thought experiment” in subsequent years either. But some years later, one of my mentors along the way, Arthur Young, an eccentric mathematical physicist (he was also an accomplished technical astrologer) and the inventor of the Bell helicopter rotor (!), used to say that the obvious interpretation of successive differentiations in calculus was 90-degree rotations. I never knew if this was a standard interpretation (which I suspected it was not but yet still do not know), but it certainly was appealing. Why? Well, 90-degree rotations are fundamental to right angles and “normal views,” one of the most significant ways of dividing space, or perhaps “hyper-space” as well, in any kind of intuitive geometric analysis—a most obvious heuristic approach—that is also used, as noted, to comprehend the meaning of the first three ordinary dimensions in geometry: *length*, *width* and *height*.

Arthur Young was also an accomplished aeronautical engineer (like his Russian counterpart Sikorski, who had invented a very similar helicopter rotor at nearly the same time, though his rotated in the opposite direction), who used as his illustration of successive differentiations the main sequence in the analysis of motion: *velocity*, *acceleration* and what was informally called “jerk” by engineers, but which Arthur said should be called “control.” So, *velocity* can be understood as a two component vector number (uniform speed in a particular direction); *acceleration* as a differential ratio (rate of change of velocity with respect to time); and *control* (similarly, rate of change of acceleration with respect to time).

The obvious factor that distinguishes what calculus was invented to analyze (again, nearly simultaneously by Newton and Liebniz) is *motion*, hence the “rate of change with respect to time” ratio. So, in math symbolism the *velocity* differential is written as dx/dt, change of distance (x) with respect to time (t). Similarly, *acceleration* is written as the second derivative, dx^{2}/dt^{2}, change of velocity with respect to time; and *control* would be written as the third derivative, dx^{3}/dt^{3}, rate of change of acceleration with respect to time.

This may be confusing if you are seeing this terminology for the first time, but the basic ideas are probably intuitive enough for anyone familiar with driving a car. “Velocity” we usually call “speed,” and we realize there is a limit for it on the particular road we are traveling on, in some particular direction, often in a straight line, and we have to watch out for other cars going all the many other directions we may encounter as we travel along, so we have to watch out for unexpected trajectories as well as those moving along with us. Acceleration we understand when we push on the “accelerator” or brakes to speed up or slow down, thus changing our velocity. Control is also familiar, even if a special word is not usually used for it: for example, it’s when you “step on it” (rapidly increase the rate of acceleration) or “jam on the brakes” (rapidly decelerate) to control the rate of change of acceleration/ deceleration of the car (in addition to using the steering wheel to change directions). All of this happens in flying airplanes too but in a more complex environment of 3D motions, with special terms used to describe the additional dimensional degrees of freedom. Soon, there will be “flying cars,” and drivers will have to develop intuitions for “3D driving,” rather than moving on a more-or-less 2D flat surface. This should prove to be very interesting (!).

So, we are quite well acquainted with 3D spacial awareness in any event—even if we don’t fly airplanes—it is the full “solid angle” … of all space that can be imagined going as far as we like (to infinity) in any direction—and any generalized location within the 3D space, can be uniquely specified by three component numbers (typically, x, y and z) all understood to be mutually normal (perpendicular) directions, relative to some point of origin (0,0,0). There are other alternative reference systems using angles and distances to uniquely specify 3D points, but they are all comparably interoperable if you stay within the same conventional frame of reference.

Then, there is the mystery of *time* … in the post-Einstein world, as the fourth dimension that is thought to be integral with space as a unified totality. Time adds the notion of *duration*, so that a fuller measure of existence can be imagined as “persisting” at or through some particular sequence of time … again, though it is not so intuitive in the way we usually think of time, perhaps also from an origin: *Zero Time*. We date our timed historical eras typically relative to some well-known event, such as what is believed to have been the birth of Jesus Christ (B.C. and A.D.) or more neutrally as the “common era” (C.E.). A real Zero Time that is absolute rather than relative is unfamiliar but follows logically—at least there is no obvious reason why there could not be, or have been, a Zero Time … unless we believe that time has “no beginning and no end” (like space?) as many people probably do, as an easy solution (no one could have been around to know about Zero Time, etc.). At least I found it a fascinating concept when I first encountered it in the teachings of Semjase, the Pleiadian star woman.

This is usually as far as we can go with our ordinary understanding of dimensionality: 4D—even for very practical or theoretical technical people like scientists or more deeply thoughtful people like philosophers. It is considered to be like, the edge of the known universe. There have been some attempts to think of various 5D imaginings and beyond but mostly without persuasive results. Mathematical formalisms are easily notated, but venturing what they might mean is not. Mathematicians usually point to the contradictory notion of the square root of -1 as the best example of a mismatched meaning of mathematical formalism, and may think higher dimensions are similarly contradictory … so there might only ever be three (or four).

What I am going to suggest now, as an original interpretation, as far as I know, goes beyond this, what I believe to be the misunderstood boundary of dimensionality, that in some ways should have always been obvious—taking calculus into account as the appropriate conceptual tool and using the preliminary workup we have already seen—will serve as the completion for a fundamental ‘octave metaphor’ or analyzing our astonishingly Magical Universe (variant for Multiverse) with its level upon level of energy octaves, likely all of resonant holographic fractal nature, operating from sub-quantum to astronomical levels of scale—thought to be practically infinite.

The basic notion is that the first three geometric dimensions are scalar numbers (x, y, and z), that is, described as simple quantities (compared to complex numbers, as we have seen, like velocity with both speed and direction). Time is also understandable analogically as a scalar quantity, like spacial directions, as mathematicians routinely say, from “zero to infinity.” Whether time is really a scalar (no direction) or a vector (similar to velocity with perhaps something like directional “time lines”), we can’t really be sure, but maybe it’s both—or the fractal-like *transition* from one to the other (!)—as I will soon suggest.

This octave metaphor as it occurs throughout the natural world of science (periodic table), music (well, the ubiquitous “musical octave”), the single octave of visual color perception and elsewhere (days of the week, toroidal geometry, cross-cultural mythic imagery, etc.) in a sense contains either seven elements or eight, depending on how we think of it: the analogue octave is seven (a duality that can be grouped into 3+4 or 1-3-5-7 & 2-4-6, or various combinations such as or 3+1+3 elements, etc). The computational binary or digital octave, as an approximation, is 8 (2^{3}). The basic idea of the octave is that the frequency of the first element doubles in the eighth element in the ascending octave or halves in the descending octave. So, are there a total of seven unique elements in the octave or eight (the point at which the frequency doubles)? It seems to be either or both, suggesting a simultaneous analogue “real” world and a parallel virtual one of digital approximation (this is only one way of thinking about this phenomenon, of course, using the two most obvious logical modeling modes … or even the basis for the brain’s dual hemispheric complementary for reality error-checking).

As it occurred to me, the completed notion of an “Octave Module” (OM) emerges from adding the triad of “higher dimensions” beyond the 4^{th}, as the 5^{th}, 6^{th}, and 7^{th}, corresponding to the triad of lower dimensions (1^{st}, 2^{nd} & 3^{rd}), as a way of filling out the remaining three spaces being sought, giving a total of seven to satisfy the octaval metric. If you are wondering why I might have introduced the differential calculus at the beginning or have already guessed this is where it fits, you are right: these three higher dimensions would therefore be characterized by the three phases of *motion: *velocity, acceleration and control while the first three dimensions are characterized by *stasis* or “being at rest,” to use the old Newtonian term of choice. The first three-dimensional mutually perpendicular triad of dimensions are *scalars* and the second set are *vectors*, with the 4^{th} dimension (time) being the transition. Bingo!

So, we really have seven dimensions—and we only thought we had three … or four. Thus, *motion* is a very big deal (!), probably much bigger than we could have imagined (like time!) since it is so familiar. The corollary is that we are really at least seven dimensional beings since we are (get ready) … “animals”—beings that *move* as well as *have being,* with *velocity*, *acceleration* and *control* built into their neural operating systems. I should probably say, “move … under their own guidance,” hence evoking the highest 7^{th} dimensional quality in this scheme: *control*. In an amusing way it reminds me of the marvelous French story called T*he Bourgeois Gentilhomme* by Moliere in which the protagonist, after successfully getting through his whole life and reflecting back, finally realizes that he has been speaking prose the whole time. Likewise, we thought we were living in three dimensions, or maybe four, but have been holding forth and moving through at least seven dimensions of existence all the while (unlike, say rocks and plants).

As a scalable modular unit the Octave Module can be imagined as the fundamental resonant topological element in a holographic universe in which everything is related to everything else in the way conveyed by the Medieval Latin figure of speech called *pars pro toto*, that is, the “part stands for the whole.” This is what is exhibited in a holographic image in which a small part of the image can be seen to contain the whole image, with somewhat less resolution but still recognizably so—the smaller the part, the less accurate the resolution—and this suggests the appropriately fractal nature of the Universe and its modular representation. So, in this sense the Octave Module in presenting a seven-fold internal structure, reflects all the larger and smaller scales of the universe simultaneously at a resolution that corresponds to the scale at which it exists in relation to everything else. So, it reflects a world or Universe that is one of *infinite dimensonality* rather than one having a particular finite number of dimensions (like three, four or seven), though at the highest level, the structure there would also be seven-fold, just as it is at every other level of scale, from the very large (galaxy) to the very small (red blood cell). Thus modularity, by itself, is clearly a very big (and very small) idea, but having the correct common internal metric structure is a huge breakthrough!

Why haven’t we heard about this before? Many of us have, but it has still not been systematically incorporated into scientific or mathematical basics even after the revolutionary implications of both holographic photonics and fractal dimensionality began to be understood late in the 20^{th} Century. Of course, it was there all along, as noted, particularly in the Periodic Table of Elements, in music (harmonic overtones) and in the physics of interacting coherent energy phenomena (lasers, leading to holographic imagery). It’s also in many esoteric studies of a more philosophical nature such as the works of the Tibetan DK in the metaphysical multivolume works of Alice A. Bailey and many others. Arthur Young was also very interested in the seven-fold mathematical surface of the torus, the “two pi plane.” Perhaps the time has finally come in which this insight might become available in a more generalizable way of reasoning as it once was in the historical era of Aristotelian Latin scholarship (“Great Chain of Being”) among others.

As an example of interpreting the Octave Module in a Newtonian context at the universal scale of matter and energy, we would associate the first dimensional triad (the three classical geometric dimensions) with matter, which, unless perturbed by an external force, tends to remain at rest. The external force, or dynamic element, would be analyzable as radiant or photonic energy of various wavelengths and would correspond to the higher triad of dimensions (5^{th}, 6^{th} and 7^{th}). Energy is inherently a substance in motion, even “potential energy” in which the dynamic element is seen as being stored in some structured way, and the calculus was clearly required to analyze its more complex dynamic aspect. The middle dimension, the 4^{th}, would apply again as time, the mysterious intermediary (speed of light, the “quantum of action,” delimited in time units) between mass and energy.

So the OM can be seen as a general model, not only of “seven dimensions” but as an asymmetric triune structure (mass, time and energy) in which time is an explicit element unifying mass and energy or additionally as a dualistic model in which mass and energy are abstracted as a timeless interactive unity. If we move to an Einsteinian context, the OM model becomes the visualization for his most famous law itself: E=MC^{2} … in which the transformation between matter and energy goes as the square of the speed of light, the speed of light being the absolute limit for the propagation of energy coalescing into matter and back again. This is a bit of a conundrum but acceleration is believed by many theoretical physicists to be indistinguishable from the “force of Gravity;” and if so, then gravity would be the penultimate factor in this Octave Module hierarchy with Control as the top level factor (I will opt for consciousness as the full equivalency for Control). This Octave Module would then be the solution for one of the most transcendental searches for infinity in the Transfinite Mathematics of Cantor, Russel and Whitehead: the Set of all Sets—the infinite structure of infinities.

The OM can be applied in many other ways in both conventional and esoteric sciences since “all that is” is either matter or energy (bracketing the most mysterious spiritual aspect for a future discussion), with resonant properties within the universal holographic matrix of existence. One of the most interesting places is esoteric psychology and its primary analytical tool, esoteric astrology in which the teaching of the “Seven Rays” is seen to be the basic system of types and developmental stages for human beings, just as the Periodic Table of Elements provided the basic understanding for the varieties of chemical elements, and the octaval structure also provided the basic theoretic understanding for the interaction of radiant energies since they appear infinitely interactive. A further refinement of the OM metric would carry it into a more finely divided “well-tempered” structure, having 12 internal elements (what J.S. Bach did for multi-instrumental musical performance and the East Indians did with quarter tones for a metric or “frequency” of 24).

At a more practical level, the highly engaging feedback process of making, analyzing or listening to, music, remains accessible to all whether formally (scribbles on pages or computed matrices) or intuitively for those who have ears to hear—the Octave Module reigns supreme here. The progressions of various harmonic and dissonant patterns provide for endless hours of fascinating combinations of riffs at various paces, rhythms and syncopations, and it is symbolically and holographically representational of all the other higher and/or lower resonances in the universe—that’s why it’s so compelling! And you can even dance to it, but that’s a whole ‘nuther story too (talk about motion …!). A similar world of color is used by visual artists to produce corresponding effects of a similarly engaging infinite variety.

And finally, the most simplified Zero/ Infinity/ Middle Earth image of the Octave Module (look like anyone you know?).

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