“Geometry existed before
Creation. It is co-eternal with
the mind of God. . .
Geometry is God himself.”
Kepler
*
The Mystery of The
Number Symbols
Have you ever really looked at the forms of the symbols of what has truly become the worlds’ first global language? Nobody ever consciously designed them to reflect their roles within the greater system as a whole. Nobody ever consciously thought that the symbol for the two should be the same as the five (but juxtaposed). And that four should be the same symbol as the seven; the six the same as the nine; or that the three should be, for some reason, found hidden in the eight. Yet, they certainly are.
Now the essence of these number forms as they are commonly written is obviously mirror-type imaging of one another. So the mystery is only compounded when we discover the schematic of order and symmetry based solely on the forms of the symbols.
Obviously we could design a system of ten characters which could produce an orderly and symmetric schematic based purely on the forms of its’ symbols if that was our intent. And more importantly, not without our most steadfast purpose would the order amongst the forms of those symbols also be a reflection of the order and inherent structuring within the system of base-ten itself, such as occurs naturally in the set of forms adopted the world over.
It wouldn’t take long after observing the schematic of order and symmetry for one to see that the two actually is related to the five. The base of the system is ten with the one and zero co-equal rulers. They preside over four pairs of identically symbolized quantities. But only the two and five can divide the base evenly. Ten divided by two is five; and ten divided by five is two. The influence of this quality is reflected in the symbols since a horizontal line dividing the ten gives birth to the forms of the two and the five (see illustration on front cover). But the structuring uniting these forms goes much deeper.
The table (above) reveals the number sequences unique to each quantity’s successive powers. Any number to the zeroth power is one, so the powers of two begin 1,2,4,8… 16 is expressed as 7 (1+6=7), 32 becomes a 5 (3+2=5), 64 a 1 (6+4=10 and 1+0=1). Rendered in this fashion this sequence will continue forever. Those describing the powers of the other numbers are derived in the same manner. Now look at the powers of two in the top row of the table. They’re simply the powers of five backwards (second row down)! The powers of four, 1471471, are the powers of seven backwards. The affinity between six and nine is also obvious. Maybe someday the cosmos will reveal to us the reason for the break in the system’s perfection with the powers of the three and eight. But this too has its parallel in nature in the electron populations in the outer shells of the heavier elements.
So the “mystery” is this: the forms of the number symbols seem to be designed in such a manner as to indicate a preconceived intention to create symbols bearing the marks of the logic uniting the greater overall system. But it was not. Historically, these symbols evolved chaotically, haphazardly, and independently. Could it be that nature has taken the initiative, gifting a sign for all to see that the force of the spirit is at work?
So Where Did The Number Symbols Come From?
Confusion abounds regarding the origin of the symbols. We call them “Arabic”, or “Hindu-Arabic” numerals, which is probably fitting since some of the symbols appear independently in India, Egypt, Persia, and Arabia.
It’s probable that traders, who needed a workable number system , carried the symbolic forms used in their homelands along with the goods that they exchanged.
History is the obvious place to look for the forms’ origins. And the historical record shows that the present forms result from a gradual evolution over nearly a thousand year period.
But, as we can see, their evolutionary route was not at all uniform, or consistent.
The Table of Number Powers
And
The Table of Chemical Elements
There is a striking similarity between Nature’s numerical arrangement of the chemical elements and the inherent structuring of Number itself. The Table of Powers pairing the two and the five, the four and the seven, and the six and the nine as obvious kindred relations begs that the remaining quantities three and eight likewise share in some kind of unique relationship identifying them as siblings (such as if the sequence for the three had been 181818…). In this respect it breaks down and fails to deliver on the implied “perfection”.
Study a table of chemical elements arranged to show the distribution of electrons in each unique atom. The K shell or orbit, after hydrogen, is complete with two electrons for every atom. The next shell L fills 1 through 8 and then all atoms after that fill with eight. Shell M follows the order of the previous shell, filling one through eight and then continuing with a couple more eights, then a 9, 10, 11, 12, 13, 14, 15, 16, 18. . . then the rest all hold 18 electrons. The perfect order of the beginning certainly has deteriorated. The next shell N starts off in spurts and then stabilizes one through eight mirroring the beginning. It too then deteriorates in orderliness as it works its way up to completion at 32 electrons. Shell O repeats the previous shell’s less than perfect march to completion at 32 electrons.
The point that is being made here is that just because the powers of three are not the same, or even related to the powers of eight doesn’t render bogus the whole of the previous data. At least no more so than does the Table of Elements’ lack of perfection render bogus it’s relevance to the composition of the elements.
Below is a common version of the Table of Elements: (replace with clear image)
Deriving The Symbols 3 and 8
I hope you find this of some interest. The sketch below relates to my Mystery Of The Number Symbols poster. There I introduce what I’ve come to call “the table of powers” that proves (regardless of their forms) the powers of the number 2 to be the powers of 5 backwards; the powers of 4 are the powers of 7 backwards; and those of 6, identical to the powers of 9. Of course the poster shows the numbers sharing the same power sequences (for some mysterious reason) share identical number symbols, just juxtaposed.
But I point out that the 3 and the 8, which share the same symbolism, deviate from the three previous pairs in that their power sequences don’t seem to have any correlation. I state in the poster text “Maybe someday the cosmos will reveal to us the reason for the break in the system’s perfection with the powers of the three and the eight.”
A few days ago I was entertaining myself musing over this dilemma. I realized that the 2 and 5 could merge and morph “topologically” into the symbol for three. Likewise does the symbol for 8 derive from a merging and morphing of the 6 and 9. If you look at the number line on the poster with its schematic below (based on the shapes of the ten symbols) one can’t help but notice that the 2 and 5 sandwich the 3 on the left side of the schematic in the exact same way as the 6 and 9 sandwich the 8 on the right side. Keep in mind, none of this order would be happening had the symbols assumed other shapes or forms.
HISTORICAL NOTE
In 1977, after three years into what would become a ten-year odyssey throughout much of the eastern Pacific Ocean, far out at sea in a vessel of his own making, a young sailor had a revelation.
Ever since dawn the wind had been slackening and by nightfall a breathless calm had enveloped godot, his sleek 35’ ocean-racing trimaran. As often was the case he had been sailing alone for several days. He was tired and the change in weather gave him a chance to catch-up on some much-needed sleep. Around midnight he awakened to what he described as the “intense sound of absolute silence”. Coming up from below deck his eyes were treated to the most wonderful illusion.
There was no moon, not a breath of wind, nor even a trace of a cloud anywhere in the sky. And the surface of the ocean was without the slightest ripple. The sea was as flat as the face of the finest crystal mirror. And the stars. . . brilliant beyond belief, were doubled deep into the watery depths, making godot appear as the centerpiece of the vast celestial sphere.
He told me he sat there in the cockpit for what seemed like an eternity, spellbound by the sensation of drifting through space on his sailboat, when he sensed he was no longer alone. Attributing his feelings to the uniqueness of those moments, he said he finally got up from where he was sitting and walked to the bow of the boat, eventually easing himself into to the soft folds of the downed genoa-sail piled atop the bow pulpit. The feeling of another presence eerily followed him and had become more intense.
I still get the chills when I recall him telling me this. . . he told me that suddenly, not actually expecting an answer, he shouted out loud “Who are you?”. And then to his terrified surprise, in a voice as clear and as loud as his question, he swears to this day it replied “for now just think of me as The Big O. Fear me not! For you must listen. I have some things I must teach you. In time I will show you the simple keys to the cosmos, the proportional rhythms that guided even my hand. Your years in this watery wilderness have purged you of much of the inculcation of your fellow humankind and have opened your eyes and mind to be receptive. Even so, there will be much work on your part in the years to come. This noble vessel will be your study and the solitude of the sea your safe sanctuary”.
It was on that night when he says he was shown the “Numbers” and the uncanny relationships among their forms. He was told it was a sign. In the beginning it was mostly for him, for strength and courage in the years ahead. Later, they would become a sign for all to see. . . living proof of another force at work in the universe.
Copyright Imagio Savastano 1997
Could it be, just possibly,
that somewhere deep within “Number” itself
is the rulebook which binds both God and the scientist?
Can it be that, just possibly,
“Number” per se embraces the conceptual matrix
upon which Universe was delineated?
pk
AUTHOR’S NOTE FROM
1989 EDITION
Some Thoughts On Universe is a compilation of essays written during the period 1977 through 1989. One of the most difficult tasks during that time was to try to explain to anyone who might ask just what? this book I am writing is all about. Sometimes I would say “numbers”, or “ something to do with numbers and form”. Other times I’d say “the Big Bang”, or “cosmology” from an artist’s point of view; simply a painting (in words and numbers) of the beginning of time. Still at other moments I’d respond with a single word: Philosophy.
Lately, when asked the same question I am likely to respond that this book brings to light the inherent mathematical affinity between one sphere dividing into two spheres and a seven point system transforming into an eight point system. The primal singularity of modern sciences Big Bang cosmology is analogous to the single sphere, and the two spheres the result of its first transformation. Why one sphere becoming two and a seven point system becoming an eight point system are related; and, how this geometric relationship might artfully be composed into a model for viewing our physical reality is really what this book is all about.
I ask the reader to approach these concepts with an open mind. And for the sake of the total art piece this picture is intended to convey to allow the last brushstrokes to dry before appraising its value as either new insights into the geometry of form, or, their application (if any) at helping to better understand the composition of our physical reality.
FOREVER THEREAFTER,
FROM THE COMMENSMENT
OF THE BIG BANG,
UNIVERSE WAS THE WORLD
OF “BECOMING”.
IN ITS SIMPLIEST EXPLANATION
IT IS THAT OF “ONE”
STRIVING FOR THE IMPOSSIBLE:
THAT OF BECOMING “TWO”.
pk
o
_______________________________________________________________________________________________________________
FOREWORD TO
1989 EDITION
The inherent potential of Unity is the introduction of seven “Others” identical to itself into universe. Together, as eight, they form all other units of Universe. These relations are primal and manifest as the system of number logic uniting the eternal ideal forms of geometry and Unity. Never before have mathematicians been aware that a system of seven spheres, and a system of eight spheres, and the transformation of one sphere into two spheres are all related through Unity. Some Thoughts On Universe shows the logical derivation of these relationships. In the process the “transit-tetrahedron” is discovered revealing itself as the fundamental unit in a hierarchy of geometrical forms. The mathematics of these contentions can be easily verified by the reader. An abundance of evidence indicates that this geometry might even extend to the actual quanta-sization of physical reality.
One thing is for certain. Some Thoughts On Universe is sure to incite heated argument. Established scientists and mathematicians by nature will not embrace any work coming as it does from someone outside their specific fields of expertise. However, they will find it difficult arguing with the numbers. The lay person might find much to identify with, and not being threatened professionally will by nature be more open minded and receptive to this free-thinking addendum to many established thoughts.
It is obvious that we must regard the universe as extending infinitely,
forever, in every direction;
or we must regard it as not so extending.
Both possibilities go beyond us.
Scientific American Editorial
March 13, 1921
^
_______________________________________________________________________________________________________________
SOME THOUGHTS ON UNIVERSE
In the beginning there was one sphere, a singularity which “exploded” into the Universe as we know it today. This is where scientific cosmology begins, and this is ultimately where the mathematical concepts in Some Thoughts On Universe are rooted.
Why should one sphere dividing into two spheres be related to a system of seven spheres transforming into a system of eight spheres? What or who are the Decahedron, the Star-tetrahedron, and the transit-tet? What is the maximum potential of a sphere, the cones of maximum volume, and the natural division of unity process?
The cornerstone of the physical sciences is mathematics. And the mathematics presented here with is for the most part original and unknown to the storehouse of mathematical knowledge. I call this a transformational geometry, and it seems to be a fundamental logic governing certain relationships among the forms of geometry. That these mathematical relationships extend to those governing the composition of physical reality might someday be concluded from the experimentation of physicists and chemists.
An example of possible applications to interpreting the rules forming our cosmos might be seen in the mystery regarding the masses of the proton/neutron and electron. For ever since these were measured with precision, scientists and natural philosophers have wondered why nature divided all of the “stuff” of the three-dimensionality into just two sized packets in a ratio of 1/1837. Actual experimentation provides us with the ratio, yet there is still today no theory as to its’ logical derivation. Transformational geometry might just reveal some of the logic which guided nature’s hand. For it shows the ratio 1/1837 to be at the very heart and soul of primal geometric forms which have been designed to illustrate the concept of the most economical packaging of stuff, with the most economical being nature’s path without exception.
Natural philosophy and theoretical physics tend to merge with considerations of the beginning. However, physicists-cosmologists jump from the singularity state of universe immediately to a universe of immense complexity . . . without any intervening stages. One moment it is a singularity containing in that one point all that we now call matter, energy (and space, if we are to accept their notion of “inflation”, about which more will be said later). And the very next moment it is a fiery cloud of gaseous particles of unimaginable temperature and unpredictable random and chaotic trajectories. To modern science this point exploded, assumedly into untold bigillions of pieces, or particles. And it again must be emphasized, all without any intervening transformations. The in between state is missing, the goings on from the first moment the One Thing began to transform into the many.
The cosmological theory which is presented here views the transformation of the primal singularity as having first multiplied by dividing one-into-two (two-into-four, and then four-into-eight) enroute to its composition as the “many” of today’s view. The explosion that science deduced occurs far after the beginning and only after the most formative rules of nature have emerged as a consequence of the earliest transformations. That the primal seed or parent cell of the organism “Universe” should divide and grow under the dictates of the powers-of-two number matrix is to liken the growth process of macro-universe to the same process governing the development of all other living organisms.
But what of this number logic within these first most formative transformations? From where does it derive and how does its influence manifest in physical universe today? For there were certainly dictates prevailing upon that singularity which influenced what it could or could not do.
SOME THOUGHTS ON
THE BEGINNING
One of my underlying postulates arises from the scientifically deduced assertion that at some distant time in the past our universe was set in motion by the magnificent explosion of what the scientists call the primal atom. This is what has come to be called the “Big Bang” of modern cosmology. If we agree on that general conceptual model, which is one of a single point-like object exploding to form universe, then any cosmological investigation should begin by exploring the state of that object immediately prior to the big bang.
But what can be said for certain about that, let’s call it from here on out that “atom” (since the true meaning of “atom” that comes to us from the Greek philosopher Democritus, is indivisible or indestructible). Is it free floating, suspended, in motion or at rest? For that matter, can it even be said that it possesses dimension at that time, let alone three or four dimensions? The answers to these questions involve careful consideration of knowledge bridging the disciplines of science and philosophy. For if it is as most scientists contend, then all matter and all that exists in universe was contained in that primal atom encompassing the full spectrum of that one continuum which we have come to call matter and energy.
It must be emphasized that the primal singularity was not an atom in the sense we are accustomed to think of an atom today. It did not contain protons, neutrons, and electrons. For at this point prior to the beginning it was but one thing, without any parts. All that was in existence was that one atom.
Now the scientist have come to believe that even space itself was contained (somehow) within that singularity. They don’t believe the singularity exploded into an already existing spatial surround. Instead, they postulate that space (the void between both particles and galaxies) arises, or is born at the commencement of the big bang. They have adopted what has come to be called the inflationary theory which arose due to the awkwardness impose on the scientific community after Edwin Hubble discovered what is referred to as the recession of the galaxies and the expansion of the universe. Hubble found that the light spectrum of distant galaxies proved they were all moving away from Earth as if flung far and wide by some kind of explosion. It was this discovery that compelled scientist to embrace the “Big Bang Theory”. But it came with an uncomfortable caveat. Everything “out there”, regardless of what direction they looked, was moving away from Earth giving the impression that our tiny insignificant planet occupied a special place in the Universe. By extension one could conclude that Earth itself must be special.
To the scientific community this was a problem. Earth is so tiny. Our sun is unremarkable when compared to other suns. And the vast distances between stars and the sheer magnitude of celestial otherness compared to Earth convinced scientists they needed to come up with some other explanation for what they were observing.
Surely no intelligent, scientific person would believe the Earth is special. After all, they are scientists, not theologians.
Their answer to this dilemma is the inflationary theory which might just as well be a theology. For beyond their believing in it, there is absolutely no evidence or experimental results that support this conclusion. The inflationary theory, explaining the expansion of the universe, must be accepted on faith alone.
Generally, the inflation theory states that prior to the big bang there was no space for the explosion to expand into. There was only the singularity. Space comes into existence simultaneously with the particulates of the explosion. And this space between the particles expands as the explosion unfolds driving each particle away from every other particle. A common analogy often cited in the popular literature on the subject is to liken inflation to a loaf of raisin bread. Each raisin moves apart from every other raisin as the bread dough rises. If we could shrink ourselves down in size and stand on one of the raisins, indeed we would observe raisins in all directions moving away from us. Obviously the bread dough is analogous to space; the raisins are the individual particles.
But we don’t really know that space is expanding like bread dough. Who has ever stood on another star in a distant galaxy? How do we know observations made there wouldn’t be different than those here on Earth? The fact is, we don’t know. We must assume on faith alone that it is true. For some reason it is easier for people to believe that Earth is not unique, or special, and that we must surely be commonplace, unremarkable, and without purpose (as far as the rest of universe is concerned).
This is certainly a flawed view for a number of reasons. First and foremost is the contradiction that arises when this primal atom is said to have “exploded” into bigillions of gaseous particles. By the scientists’ own definition that primal atom was one thing without any parts or sub-units. That it may have contained the spatial component of universe within along with particles or any other components is simply contradictory. And secondly, It seems that even cosmologist have to admit they need two different operational laws governing the behavior of spatial inflation. Neither of these has actually been found. One is needed to account for the observations showing the recession of distant galaxies due (as they say) to the inflation of space; and another to explain the lack of similar observations in our local region of the universe. Simply put, our moon and companion planets seem to be unaffected by this alleged inflation phenomenon.
What if the scientists’ are wrong? What if the singularity or primal atom actually had a space to “explode” into after a series of preliminary transformations (which will be explained shortly)? If nothing else, the conceptual view of a single spherical unit amidst a seemingly infinite surround of spatial void seems highly superior to one of the same unit containing space itself. The first view is actually capable of imagining or modeling. In your mind think of a billiard ball suspended in what appears to be an infinite surround of space. We see the ball and its spatial surround in our minds. Try to imagine the ball alone. It can’t be done.
For the purposes of modeling imagine this singularity the moment before the beginning. Since we don’t know with any certainty whether Earth is special, or not, we need not be bothered with uncomfortable observational results and thus have no need for an inflation of space theory. We are free to visualize a spherical singularity immersed in a spatial surround.
Not only is it impossible to say that this atom is in motion, but equally impossible to speak of it in terms of being either large or small. “It” is all that “is”, except for its surround of void. And since all motion must be relative to some otherness, as too is the concept of size, the primal singularity must be viewed as both motionless and size-less*.
* When the beginning of the universe is explained to those of us untrained in the sciences, they often refer to the primal atom as being “smaller than a grain of sand”. This is a ridiculous statement since at that moment “size” is a meaningless and an impossible attribute.
Where did this primal atom come from? It either came into being from nothingness at some prior time; or, it had endured eternally in that or some other form without ever having come into existence. These are the two choices for describing the history of the singularity.
_________________________________________________________________________________________________
SOME THOUGHTS ON NOTHINGNESS
When we look outward at the night sky we see two things: stars, and the space between the stars. We see some-things, and no-thing. Though we cannot see or verify for ourselves firsthand, we are told the no-thingness between the stars is really filled with some-things (photons, neutrinos, gravitons, etc.) all traveling at the speed of light. Therefore, our notion about the no-thingness we observe today being an empty void is misleading at best.
Undoubtedly this is the case. One can certainly conceptualize space and its included star points as a continuum of sub-atomic particles with the stars representing more densely populated regions. But we really haven’t changed our two ingredients of some-things and no-thing, although we have made their differentiation less distinct. The singularity prior to the Big Bang provides a much clearer picture of true no-thingness. In this view, there are no other parts in the nothingness surrounding the singularity. And this no-thingness is the nothingness into which the singularity exploded and into which our Universe is still expanding.
“Nothing” very often seems to be a more difficult concept to accept than is that of “something”. Things are. They exist. But “nothing” is something else altogether. Generally, we know where some-things begin and end. And we can even say that “nothing” begins wherever “something” ends. But where does “nothingness” end? What is that extent of that black we see beyond the stars?
Moving forward in time from the Big Bang brings us to the view we see at night. Run it backwards to the singularity state and we arrive at one-thing and no-thingness. If at this point we do not worry about the ultimate questions regarding from where it came and whether it came into being or is eternal, etc., then we can conceive this one thing in its entirety. It begins and ends at its finite surface (where the nothingness begins). There are no photons, neutrinos, gravitons, or any others to muddle the picture so that nothingness begins at the one-things surface and travels outward . . . to what end? Infinity? Infinity might be a condition, an adjective rather than a noun. Infinity is not a place.
THE NATURAL DIVISION OF UNITY AND IT’S EFFECT ON
THE LOGICAL STRUCTURING
OF UNIVERSE
The Division of The One Into Two
Since we are starting with one spherical singularity amidst infinite nothingness, one unit of everything that exists, we can make a not so outlandish assumption about he nature of that explosion science calls the “Big Bang”. We can assume that the primal atom, which today is divided into so many bigillion parts, at the commencement of the Big Bang divided first from one unit into two.
Even a stick of dynamite, which to us “appears” to explode, would (if time were slowed enough to permit observation) show a continuous trail of burning beginning at the fuse right down to the end of the stick. Like any other fire, the burning powder gives off radiant energy. But in this case, it burns off so fast due to its state of compression that it “appears” to explode.
It is quite possible that the ultimate explosion resulting in the creation of the universe, if slowed sufficiently for minute observation, would show at the beginning the primal singularity in its earliest transformation. That is transforming from one sphere of everything into two.
There are several reasons not to dismiss this notion without consideration. First, as previously mentioned, the very definition of a singularity having no sub-structuring should preclude us from allowing it to explode. Also, viewing the singularity one moment as one minimal “thing” with no sub-parts, and then the next moment (like the next frame on a movie film) it is a soup of super heated particles; and, all without any intervening stages? There is something not only aesthetically lacking, but logically as well.
But if the Universe were viewed as a living creature and its singularity stage likened to its form at the time of its birth, then there should be no problem imagining this creature growing and changing form in the same way as we know all other living things grow and transform . . . i.e., from multiplying by dividing one-into-two. In this manuscript, this is referred to as the natural division of unity process.
ACCELERATED MOTION
And The Velocity Threshold
The division-by-multiplication of the original One into two only began with the division of the One’s “center”, which is the singularity. This first transformation could by no means be considered “complete” at that very first instant when the now two spherical points ceased to be tangent and were separated from one another. For just as infinite nothingness, void of any other, is the nature of the original’s surround, so to it must be for each and every “other” patterned on the original. Therefore the propensity is for the points to move apart from one another as this division or multiplication process of the one becoming two continues to unfold. Each spherical point attempts to seek out its own surround of empty void in an effort to replicate the original.
These points move apart from the original’s state of absolute rest and motion is necessarily introduced into the view. This motion is specifically accelerated motion having started out from zero speed. They would continue to accelerate since at this point in time there are no other forces present which can effect motion such as friction or the gravity of other bodies.
Their movement apart from one another must be visualized as accelerating either to an infinite rate or some velocity threshold. An infinite rate in itself seems illogical since the concept of acceleration implies a continual increase in rate. When the rate ceases to increase then acceleration ceases. And because the rate cannot be infinite and still be increasing there must be some plateau or threshold which forms the very maximum rate of acceleration that the two points can attain. At that instant some sort of change must have occurred.
Apparently inertial forces arise as a consequence of accelerated motion. And the effects of inertia are the same as gravity. It is possible that at the velocity threshold inertial forces have built up such pressure that each spherical point reacts by dividing again into two. This time they cling together. This is because of the gravitational/inertial effect overpowering their natural tendency to move apart in search of their own nothingness surrounds, void of all others.
When this multiplication of the original continues to unfold with the division of the now four sphere points into eight, they too cling together. At this time the eight sphere points are arranged closest packed in two separate assemblages. Each forms a tetrahedronal array of four “balls”. Each of these tetrahedronal systems is in motion relative to the other.
SUBSEQUENT DIVISIONS
The “Original Unit” divided from 1 into 2, 4, and 8, at which point is formed the first stable power of the Unit. It now consists of two tetrahedrons, each the product of four sphere points gravitationally held together. These tetrahedrons are vast distances from one another and in motion. All together, the tetrahedrons and the vast empty space between them, along with their motion, constitute this first power of the unit. As a consequence of this structuring, with the next division, rather than regarding the 8 points dividing into 16 points, which they do, we find the one system (comprised of the eight points in two tetrahedronal sub-systems) dividing into two such systems.
Since the constituent parts of these two systems are gravitationally attractive, so too are these systems themselves. They also cling to each other. As this natural division of unity process continues to play out, what in one sense can be seen as the 16 points becoming 32, and then 64, is equally but more accurately interpreted as these first two systems dividing into 4, and then 8 such systems.
These eight (spherical) systems gravitationally closest pack in the form of a star-tetrahedron. Imagine a four ball tetrahedron and then place another ball on each of its four faces. This is the star-tetrahedron. Each ball represents an individual system domain of eight point-like units in two moving tetrahedronal sub-systems. As the multiplication by division process continues with 64 points becoming 128, the formal correspondence is that of the 1 star-tetrahedron multiplying into 2 star-tets.
It’s easy to see that as subsequent systems continue to come into being gravitational attraction amasses them into one growing pile. Pressures will become increasingly more pronounced around those systems at the center of the pile. Eventually the force becomes too great and the systems at the center of the pile run out room in which to squeeze. They are crushed together triggering the cataclysmic explosion known as the Big Bang of modern cosmology.
Star Tetrahedron
THE WORLD OF BECOMING
There’s a fundamental truth regarding the division of the Original One into two (from which commenced the beginning of the Big Bang). For when the singularity divided from 1 into 2 it is impossible to conceive of a point (position or time) at which it can truly be said that the division/multiplication process has been “completed”. This is because the primal egg, atom, unit, singularity, or by whatever name . . . in this cosmology, presupposes the inseparableness of that surround of nothingness, void of any other. The nothingness surround is part of the primal unit’s essence. The singularity itself is merely the center of the unit. It is no more separable from its surround at that time than is Earth separable from the rest of Universe.
This initial multiplication from one into two can be thought of as having “divided” in a limited sense only. Its center divided into two points initiating the division process. But because this Magnapunta (as I first called it in the late seventies) is also the spatial surround, void of any other, the process only began with the division of its center. This results in the propensity for the points to move apart from one another as each attempt to seek out its own surround, void of any other. Thus it is this very presence of the “other sphere” which drives them apart.
The problem is that no matter how far they move away from one another “the other” will always be out there somewhere relative to “the other”. A true division or multiplication from 1 into 2 would result in each being exactly like the original: alone in an otherwise empty void. This is impossible. Thus, forever thereafter, from the commencement of the Big Bang, Universe was as Plato once described: “the world of becoming”. In its simplest expression it is that of the One striving for the impossible, that of becoming two.
THE NUMERICAL CONSEQUENCE FROM
THE WORLD OF BECOMING
It will be helpful at this point to assign number symbols to the Unit and its first, most formative transformation. This will prove to be a powerful tool toward both understanding and illustrating the nature of this event.
QUANTA-SIZING THE VIEW
When the primal unit is isolated in the beginning, the view is of one sphere amidst a surround of (what appears to be*) infinite void. Since this is “one unit” of everything that exists, it can be symbolized by the number one:
* More is said about this appearance. Refer to “Walking Into The Sphere” and “Thought As The Observer”.
1
But the view is slightly more complicated than depicted by the lone symbol above. It fails to illustrate the inherent dual nature of this cosmological unit. This is a more descriptive numerical depiction:
1.0
This is because what is being viewed is one unit of everything that exists (1) and (.) one unit of nothingness (0). Though both numerical expressions represent “one unit”, the latter is a far more detailed description of that original unit’s composition.
The transformation itself can likewise be depicted by simple numeric expression. Normally one unit divided into two units, or
1.0/2
means one divided into two, and the result equaling what is represented by
0.5
But in this cosmology, the Original One could never fully divide into two. Thus, as a form of a simple equation, cosmologically
1.0/2 ≠ 0.5
since the division of the Original One is never completed. “Completion” is only approached and varies in degree in a direct relation to the distance of separation. This is saying that the farther they move away from one another, the greater the approach will be to the Original’s ideal condition of the single sphere. This unattainable ideal condition is represented by the 0.5 in the above equation. And since cosmologically it was impossible for the One to ever have truly divided into two completely, the numerical consequence becomes
0.49999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999 . . . as a result.
Obviously, the greater the number of “9”s after the four, the farther the points have moved away from one another in an attempt to reach the perfection of the ideal. The fewer the “9”s, the closer the spheres are together and the closer we are in time to the actual beginning. This descriptive number expression can be reduced to its lowest form, 0.49; but no further without losing its essence.
Now the question is whether or not there is any evidence that connects both this cosmology based on the natural division of unity progression (1 dividing into 2, 4, 8, 16, 32, 64, etc.) and this number sequence rooted (at minimum) in the number “49”?
The answer is obvious once one begins to look. It’s fair to say that, at minimum, “49” shares a special relationship to unity. For when they are compared to one another, we find that the natural division of unity sequence is at the very heart of their relationship!
1.0/49 = 0.020408163264. . .
1.0/499 = 0.002004008016032064. . .
1.0/4999 = 0.000200040008001600320064. . .
1.0/49999 = 0.000020000400008000160003200064.
This number pattern will appear regardless of where one places a decimal point in this unique “4999…” sequence (for example, 1.0/.49 = 2.0408163264…). But the essence of it all, again at minimum, is the relationship embodied in the ratio “1.0/.49”, and can be alternatively expressed as
(1.0/0.7)2
giving up even more secrets into the true nature of Universe’s first and most formative unit, the beast with ten heads and seven horns.
DISCOVERING
THE DECAHEDRON
The Beast That Was, and Is Not, and Yet Is
And the beast that was, and is not,
even he is the eighth,
and is of the seven,
and goeth into perdition.
(Revelation 17)
The natural division of unity process is the “multiplication” of the 1 by “division” into 2, 4, 8, etc.* It was explained earlier that if the singularity transformed under the dictates of this process, then at the stage corresponding to 8 spheres a stable plateau of whole unit aggregation is reached. In this cosmology it can be described as consisting of two four ball assemblages (tetrahedrons) in motion having gravitational characteristics resulting in each exhibiting attraction for the other.
*And in that first most formative cosmological transformation, when the 1 becomes 2, it is certainly impossible and meaningless to try to choose whether the event is an actual multiplication or division. For when the 1 sphere transforms into 2 spheres the concept of “size” still is not present. Previously, the single sphere was size-less because there was no “other” to compare. With the first two spheres there is an “other”, but there is no difference. There’s still no meaningful concept of size, and consequently no measurements an observer could perform to distinguish between a dividing and multiplying.
It is regarded as a closed system which becomes two such systems as the process unfolds and the 8 become 16. The important distinction is between two systems each consisting of 8 sub-units; and that of one system of 16 sub-units.
In this cosmology, it is not simply a subjective choice to view an initial eight natured system as the basic building block at the reductive heart of all other future systems in universe. Instead it is the conclusion deduced from geometric modeling of this natural division of unity process.
T
he cube provides the simplest model for illustrating this idea of stable plateaus of whole unit aggregation being associated with the repeated process of 1 dividing into 2. Imagine the form of a cube as model of a volume-unit transforming according to the natural division of unity process. The first transformation into 2 cubes shows that when they are reassembled in one system the result is an asymmetrical form resembling a squared-off loaf of bread. Division into 4 sub-cubes results in a form reassembling a squared-off cake or box. But with division into 8 sub-cubes, and their reassembly into one form, we arrive once again with the form of the original unit. . . in this case a cube.
Continued division into 16 sub-cubes will result in two cubic systems, each with eight sub-units. In fact, 16 cubes are impossible to arrange in a symmetrical closest-packed single system. The diagram at right illustrates the first three transformations of this cubic form of the volume unit multiplying by repeated division into two.
Clearly stage number 4 is the next power of the unit after the 1st. Continued divisions manifest powers of the original unit at the 7th, 10th, 13th, 16th, 19th, and so on. When the number of each plateau of whole unit aggregation is reduced to a single digit, it is found that these stages correspond to a 1, 4, 7, 1, 4, 7, etc. pattern*. These very same patterns also emerge when the spherical form of the volume unit transforms through the natural division of unity process. (*Maybe not just coincidentally does this pattern also describe the powers of the numbers 4 and 7 as was shown in the earlier chapter titled The Mystery of the Number Symbols.)
The diagram below shows the cross-sections of the spherical volume unit as it transforms like the cube above from 1 into 2, 4, and 8. Notice that at the stage corresponding to the eight spherical volumes their diameters are equal to the original’s radius.
The forms at this stage are commensurate to the original unlike the sphere’s first two divisions.
Another indication that the eight sphere stage represents the next power of the original spherical unit is that they closest pack in a stable plateau of whole unit aggregation as a star tetrahedron (refer to illustration at end of previous chapter, Subsequent Divisions).
Even the form of the regular tetrahedron, when transforming by repeated division into two, shows stable stages of whole unit aggregation at the same intervals as the cube and sphere. For example, eight sub-tetrahedrons in one system touch tips at the center and edge-bond to their neighbors. The resulting form is a cuboctahedron.