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SYNERGETIC BASE-TEN
NUMBER DYNAMICS
"Henri Poincare, for instance, says: "Every whole [natural] number is detached from the others, it possesses its own individuality, so to speak; each one of them forms a kind of exception, for which reason also general theorems of number theory are but seldom forthcoming." Nevertheless, this individual aspect of number appears to contain the mysterious factor that enables it to organize psyche and matter jointly."M.-L. von Franz, Number and Time
"Consequently, it is not only the parallelism of concepts which nowadays draws physics and psychology together, but more significantly the psychic dynamics of the concept of number as an archetypal actuality appearing in its "transgressive" aspect in the realm of matter. It preconsciously orders both psychic thought processes and the manifestations of material reality. As the active ordering factor, it represents the essence of what we generally term 'mind'."M.-L. von Franz, Number and Time
The Synergetic Natural Number Continuum
The continuum of base ten number is generally looked upon as a progressive and linear series of cardinal and ordinal numbers. Iterations signify the simple addition of the initial unit to each resulting member encountered in the continuing series of elements known as numbes. The digits 1 - 9 are known as integers or numerals. Of course, multiples of 10, 100, 1000, etc. are formed simply by adding zeros.
Further analysis discloses that this continuum can be viewed as both progressive and regressive. It is not exclusively linear, but has a cyclic function resulting from the terminal character of the last base digit and the next beginning initiated by zero producing the two-digit range. This doubling of number is for all practical purposes a cyclic function that recycles again and again with each ten-fold group produced.
Besides the cyclic and ambidirectional aspects of the number series, there is also a periodic series of reversals that occur in conjunction with the cyclic aspect.
This ongoing combination of diverse functions can be considered a mixing effect not unlike an egg beater that folds the medium over and over. Remember, from our previous discussion that "OM is said to issue from a process of multifold reflection." That process, exactly, is revealed in the number continuum when we can hold a metaphorical mirror up to nature's primal manifestation--the natural number sequence.
The key to the comprehensive analysis of general number behavior is found in the concept of "circular unity." Circular unity is an idea demonstrated by the harmonious interaction of the first six numbers. SIX is the first perfect number is the sum of its first three digits, or 1 + 2 + 3 = 6.
1 x 2 x 3 = 6; 62 = 36; 36 x 3 = 108; 362 =1296 x 2 = 2592.
The term "unity" (or Universe for that matter) implies something that is composed of parts. Fuller agreed, and stated it as, Unity is plural and at minimum two, or at minimum six.
The sequence of perfect numbers (6, 28, 496, etc.) does not have the structural nor metrological significance of the Holotomic Sequence, which graphically displays an important structural order within prime number distribution.
Note that the first Pythagorean Triple 3:4:5 equals 12, which is Holotome A (which is also twice the first perfect number). 3 + 4 + 5 = 12.
The sheer complexity of the waves and cycles that are generated within the base ten continuum cannot be described nor explained with conventional modes of philosophical description. Number theorists must resort to higher mathematics. But these cycles can be demonstrated graphically so they are visible even to those not mathematically inclined.
Synchrographics has been systematically contrived to formally illustrate behavioral patterns that have successfully led to a general understanding of the fundamental elements of the geometrical nature of the base ten system.
Can we find a cosmic mandate expressing Fuller's assertion that unity is plural and at minimum sixfold? The Holotomes and Holotomic Sequence fulfill this mandate with neat, logical finite sections. They represent circular unity and whatever else remains of the "infinite rest" that swells beyond our immediate focus.
When we refer to the base of a number system, such as base ten, we are also referring to the amount of iterations in a loop or cycle for the FOLDMENT that multiplies the base and the multiples of that base is for all purposes a circle.
The coexistent independency and interdependency of the base digits creates the rational notion that continuity is discontinuous. From this we derive the closed loop logic of the Holotomes as discrete levels of finitude.
The graphic importance of this particular Holotomic Sequence is that circular symmetry is being conserved and may be enlisted as the fundamental reference key in the graphic investigation of number behavior. The primes are deployed in symmetrical interface only within these specific Holotomic domains.
The Synchrographic process of regarding symmetry as a primary analytical aspect of reference makes the Syndex archetypal system of fundamental classes of numbers possible. The foundation of this system is the palindromes and transpalindromes.
These 12 archetypal classes of number are catalogued by the ambidirectional glyphs that discloses the transbinomial nature of any individual number. Only 12 permutations exist in the total foldment of the number field or domain: Retrosquare Primes, Retroprime composites, tranpalindromes squares, etc.
The so-called four fundamental operations of arithmatic are in reality two binomial pairs: addition is reverse subtraction; multiplication is reverse division.
The term transpalindrome is invoked merely to establish a context through which to establish a bilateral system of numerical classification, that is, to create a notational link between any integer of 2-digits or more with its antithetical or reverse companion. For example, number 21 is the transpalindromic companion of number 12.
This simple concept brings into being a host of valid structural relationships that would otherwise be totally ignored. For example, number 16 is the ONLY 2-digit square that is a prime when reversed: 16 is a square, 61 is a prime. We call number 16 a retroprime square, and number 61 is conversely termed a retrosquare prime.
Without this classification system, it is impossible to analyze the number field. Palindromes, or binomial reflection numbers are neither purely accidental nor without significance. Remember, OM issued fom a process of multifold reflection to create the entire universe of phenomena.
It is through the classification process that the enigma of prime number distribution has been solved. By labelling all possible permutations of the ambidirectional system of number dynamics, we find there are twelve discrete members in the domain of number class.
In order to systematize the study of the base ten concept, a graphic format was essential to organize sizable spans of the continuum for in-depth analysis. The enspiralment of number offered itself as the ideal format. The cyclations of the sprial could be referenced to the longer cyclic periods intrinsic to number itself. There are cycles within cycles, more readily seen in graphic format than in a continuous sequence.
This was the general reasoning for adopting the synchrographic methods: to condense, or sample the number continuum, and establish reference to related periods of cyclicity.
The fundamental basis of the compound cycling begins with the circular unity six (ref. Fuller). It is the 2-dimensional circular unity of the spherical 3d model of sacred space which is composed of the nexus of the four cardinal directions with a vertical axis, (T.R.I.), six ambidirectional axes. It leads directly into the full spectrum of SYNDEX discoveries, or the nuts and bolts of general Numeronomy:
#1. The Triaxial Retrograde Interface is the fundamental program for circular and symmetrical retrograde unity and the general basis of the Holotome's profile. 1 + 2 + 3 = 6; 1 x 2 x 3 = 6. 62 =36; 362 = 1296. 60, 602 , 603 , 604 = 12,960,000.
2. Description of the proliferation of the Holotomic Sequence by prime number multiplexing; list of first five holotomes and synchrographics describing them.
12
24
72
360
2520
#3. The twelve Syndex glyphs denoting the archetypal system of ambidirectional number classification; also the general explanation of transpalindromicity. The alphabet of number class includes: palindromic primes (11, 101); retrosquare prime (61); transpalindromic squares (144, 441); retrosquare composite (46); palindromic composite (33, 99); retrocomposite prime (41); transpalindromic primes (13, 31); retroprime square (16); palindromic squares (121); retrocomposite square (64); transpalindromic composite (12, 21); retroprime composite (14).
#4. The exemplary nineleven retrograde octave wavecycle and prime/square/composite triplex diagram, denoting the profile of 9/11 cycle in conjunction with the four 2-digit pairs of transpalindromic primes. Also various descriptions of transpalindromic profile in 2, 3, 4, 5, 6, 7, and 8 digit multiples of 99 sequences. The 9/11 wave cycle was discovered on Synchrograph C, #108.
#5. How retroadditive sums of holotomes produce 1/3, 2/3 or full 99 count or even multiples of 99, thus synchronizing with exemplary wavecycle.
#6. The fourteen 3-digit pairs of transpalindromic primes.
#7. The location of holotomes in exemplary 99 wavecycle path.
#8. Tracks denoting interval symmetry of primes: Holotome D.
#9. Ancient Metrology: The Sumerian knowledge of the Precession of the Equinoxes: 72 x 360 = 25,920; and the Holotomes as circular unity, 12 - 24 72 - 360 - 2520, etc. Here the intrinsic structure of number coincides with nature's scenario. And the Hindus used exactly the same figures as metrological modules, as have all subsequent civilizations. Temple architecture is based on multiples of 36 = 62 .
So, Numeronomy, the laws relating to the essential structure and dynamics of number, is a new word for an extremely ancient science. This science, based on the knowledge that the continuum contains a definite structural order with general laws that describe the nature of that order, has laws that relate to the general behavior of nature itself.
In Synchrographics, the cyclic and reflexive nature of the cardinal/ordinal number series is portrayed through a graphic context which reveals the minimal set of key numbers required to show the coherent nature of prime number order.
Synchrographics suggests some new terms, including a 12 symbol alphabet which is justified by the context. Systematic investigation of the intrinsic structure of the numeric series is purely a matter of selecting a graphic method of mapping numbers in their natural order so that geometrical order is also an integral aspect of that sequence.
The system of multiradial and multiaxial interfacing between number and idea is called spatial formation. The maps that include the relationships of circular unity and the distribution of primes and other classes of number are called mandalogs.
The employment of this multirelational spread sheet permits the number analyst to consider aspects of the numeric continuum that would otherwise not be taken into account and therefore beyond the order-seeking functions of human mentality.
The first important concept of numeronomy is the exemplary base wave. The wave begins both before and after number 10. In fact, it is called the nineleven cycloflex bacause it is the result of the mutual interaction of both nine and eleven.
This wave begins at ten and concludes its first cycle at 99 (9x11 = 99). Then it continues through the multiples of 99 and never ends. This is called exemplary because it sets up a continuous pattern that never ceases and never changes. This pattern is responsible for the continuous integrity of number behavior.
Numeronomy, or the laws governing the behavior of the continuum of quantitative notation is the natural result of numerology, the study of number. Numeronomy is the outcome of Synchrographics. Numbers speak for themselves through structure and behavior.
The single most important discovery of the SYNDEX PROJECT is the Holotome and the Holotomic Sequence, created by prime number multiplexing. It was discovered on Synchrograph C.
The second most important discovery is the Exemplary Basewave Octave or Nineleven Cycloflex. It was also discovered by meditating on the C Graph.
The third important discovery is the four pairs of 2-digit transpalindromic primes which served as major clues to the discovery of the coherent order of prime number distribution.
The fourth discovery is how the Holotomes relate to the exemplary octave wavecycle.
Number/geometry is the fundamental cornerstone of human communication and specifically the alphanumeric principle of descriptive notation. The T.R.I. represents the geonumerical basis of the sequence of minimal pluralities that accomodate the maximum amount of divising factors. The Holotomic mandalogs display the retrograde symmetry of each of the circular unities in the form of a half positive and half negative octave system predicated on the octave nature of the so-called base ten system of number.
The base ten system of number is an octave system, where either one or nine can be seen as a null value event.
Furthermore, this octave can be regarded as a cyclic function. The zero, one, or nine can function as the null event which acts as the null value gap between the beginning and ending of the octave retrograde cyclation, due to its half positive and half negative symmetrical sycle (which negates the numerical value of one or nine hust as if they were of the same nature as zero).
Due to the octave nature of the eight true numbers, no transpalindromic sequence can exceed an octave cycle.
Each Holotome in the sequence of holotomes when represented in a radial series of the numbers is composed of a perfectly symmetrical array of prime numbers diametrically opposed to other prime numbers or numbers composed of primes multiplying other primes. Also, the intervals that separate the primes are diametrically opposed to the same magnitude intervals across the wheel, yielding 100% perfect radial symmetry.
In the context of the Holotomes, then, the deployment of prime numbers is an orderly progression. This ends the tradition that the primes do not occure according to any recognizable pattern.
This is the essence of the holotomes and their graphic elegance. Graphical elegance is often found in simplicity of design and complexity of data. Visually attractive graphics also gather their power from content and interpretations beyond the immediate display of some numbers. The best graphics are about the useful and important, about life and death, about the universe. Beautiful graphics do not traffic with the trivial.
On rare occasions, graphical architecture combines with the data content to yield a uniquely spectacular graphic. Such performances can be described and admired, but there are no compositional principles on how to create that one wonderful graphic in a million.
Number is considered so simple and mundane in nature that a popular assumption exists that there is nothing more to know about it that could really be of any valid significance. In a sense, number is self evident and apparently contains no subtle mysteries.
Contrary to this attitude, number is the repository of a highly complex system of very intricate and involved interrelationships that have shaped the cosmological and religious backgrounds of all cultures. They affect us unconsciously at the deepest levels of our belief system, which in turn conditions our thoughts, feeling, and behavior.
The true mechanisms operating in the number chain can be shown in a system of incremental spirals portraying the numeric continuum and the special events which occur in it. R.B. Fuller recognized this when he wrote to Marshall, March 3, 1981: "Your cyclic synchrographing work clarifies and simplifies this whole matter to an epochal degree."
SYNCHROGRAPHICSSynchrographics is an innovative, systematic discipline interfacing the natural base ten integer progression with the fundamental elements of geometry. This institutes a graphic synthesis of the the two basic disciplines which in essence are initially two interdependent concepts that only occur through their mutuality.
The Pythagorean Triples that begin with the 3-4-5 triangle bring to note this initial unity of number and geometry attesting to the scientific validity of the synchrographic method of analysing relationships. They are not at all evident without such an interdisciplinary medium.
Each holotome expressed in synchrographic form is geometrically symmetrical. The base digits of the parallel spirals of iterating squares give direct visual evidence of the factorial degree of any specific integer by the occurrence of squares that have been color-coded for that particular incident of synchronicity.
In that the initial holotome is twelve and all subsequent holotomes are a multiple of that number, the valuable duodecimal interface that encompasses the base digits is reflected in the substratum of all holotomes.
Synchrographics offers a plausible answer to the question of why the Babylonians adopted the 360 degree circular unity. The ancient Hindus chose 108, which is 3 x 36.) The classic answer is that 360 has many divisors. But perhaps some unknown numerist discovered this sequence in the ancient past.
This sequence is generated by doubling the first perfect number six to equal 12. Then doubling 12 for 24. Then multiplying 24 by the first true prime 12 x 3 = 72. Then by multiplying that number by the next true prime 72 x 5 = 360. Then multiplying that by the next true prime: 360 x 7 = 2520, etc.
By beginning with twelve we have already involved 2, 3, 4, and 6. By doubling 12 we have involved #8. or five of the base digits. By multiplying 24 by prime number three, we involve nine, or six base digits. By multiplying 72 by 5, we involve seven base digits or 2, 3, 4, 5, 6, 8 and 9. Finally, by multiplying 360 by 7 we have captured them all: 2520.
If the Babylonian metrologists knew of this, neither they nor modern number theorists make mention of it.
Whatever the case, the Holotomes are ideally adapted as instruments for tranlating intricate geometrical interrelationships into the language of number. Only through the careful study of these special modules does the exquisite order of prime number occurrance become obvious, for the primes are found to be deployed in symmetrical interface only within these specific holotomic domains.
Thus, number stripped of its structural character is reduced to the empty and monotonous iterations used essentially for counting objects and measuring distances. When numbers, on the other hand, are permitted to be deployed in cycles that are in phase with their already inherent rhythms, a clearer picture emerges.
All mandalogs are the product of the systematic generation of the exact sequence of minimax factorization. They have the perfect retrograde feature by which the patterns generated in the first half of the spiral are reversed at midpoint and are reflected as a mirrored image in the second half of the spiral. Remember, OM was formed by multifold reflection!
Also, because of the existence of palindromes and other reflective qualities issuing through each holotome there is an exemplary wave form that begins at the end of the first holotome. This is a dual component wave, resembling the DNA helix. The wave begins amid number ten and is composed of square number nine and palindromic prime number eleven.
This compound cyclic wave is labeled the nineleven cycloflex. It cycles and oscillates through multiples of ninetynine and produces decant or tenfold series of consistent tranpalindromic sequences or numbers. Each number in the sequence has its perfect reversal on the corresponding other side.
The total reversal of number should always have been expected in that the number chain is by its graphic nature a two-way street, refolded again in the four fundamental operations of arithmetic.
The graphic mandalogs contain a rational and logical system of interrelating number and geometry or quality and quantity. They are graphic expressions of identical ideas regarding the descriptions of events in nature.
A critical consideration in expressing the optimum number of interrelated ratios is to do so with a minimum amount of graphic details. That is, to show the most information with the least given axis of reference. The mandalogs, or number wheels, are mathematical entities which express a plurality of interdependent formulae in a simple singular system.
The cornerstone of SYNCHROGRAPHICS is the preliminary Pythagorean Triple:
3 + 4 + 5 = 12:Holotome A times two equals B or 24
times prime number three equals C or 72
times prime number five equals D or 360
times prime number seven equals E or 2520
times prime number eleven equals F or 27720
times prime number thirteen equals G or 360360
In this way, the minimal numbers that accomodate the maximum amount of consecutive factors of division are generated by the multiplication of each resultant sum with the next prime number in its natural order of occurrence.
Each of these Holotomes is a number of special geometry, a circular unity. Expressed as a geometrical entity, a synchrographed Holotome is found to be reflectively symmetrical. At its midpoint, its initial pattern reverses and its second half becomes a reflection of its first half, much as OM created the Universe through its "multifold reflection."
J.S. Bach used this numerical phenomena in his Crab Canon or Retrograde Fugue. The breakdown of that notation was 22 x 144 or 3168. This number is cited in the Qabala as the perimeter of Solomon's temple: 3168 - 1008 = 3.1428571 (4 x 252).
The secret traditions seem to have made liberal use of the Holotomes without ever pointing them out.
The number 3168 has special qualities: By adding the palindrome which is the sum of a palindrome times another palindrome:
3168
5445 : 55 x 99
8613We get a reversal of the initial number.
The ninety-nine cycle is the carrier wave of the transpalindromic reflection sequence. This sequence is crucial to the mapping of the natural number scenario because the 99 cycle issues through the Holotomes.
The Holotomes are ideally adapted as an instrument for translating into the language of number the intricate geometrical interrelations between the configurations of cubic space. The Pythagorean Triples are the best examples of the interdependent nature of number and geometry. These triples logically deduced as an "infinite set" all share the 90 degree angle. They show the geonumeric character which describes the same ratios and interrelationships in different styles of notation.
Synchrographics begins with the assumption that since number and geometry are two ways of expressing the same set of ratios or relationships, then it holds true that a graphic device may be generated that faithfully aligns these two methods of notation in a synchronetic diagram. That is, a single notational system may express the geometrical nature of number and visa versa.
The "four progressively additive and four progressively subtractive event octaves with a ninth null event" depicts the primary cycle or finite extent of the initial program parameter.
With the turnaround occuring at ten (between square number nine and prime number eleven), the nineleven wavecycle then begins and proceeds to fortynine and a half (49.5). It turns around and proceeds to ninetynine (99), and thereafter continues through the multiples of 99 to 1089 or the only four-digit transpalindromic square. This is not a guess, since the mandalogs demonstrate it graphically.
The behavior or structure of the baseten system requires the perspective of an integrated complex where number and geometry are interqualifying aspects of an unified system of congruent identities. The character of notation determines whether data is in the form of number or geometry. Each requires the other in order to be expressed.
This interdependency authorizes the synchrograph to represent a number as a geometrical phenomena in which each holotome contains the triquadric core intitiated in number 12.
(Holotome A)
Thus, every subsequent holotome retains a copy of the initial data, plus new more involved data. Each and every holotome is a symmetrical retrograde MANDALOG, representing the four progressively additive then progressively subtractive event octaves with a ninth null event synchronicity. Altogether it represents the octave nine system of R.B. Fuller, or Marshall's nineleven cycloflex.
The exemplary compound wavecycle which proliferates through the multiples of 99 is the carrier wave that both integrates and isolates the Holotomes with accumulative integrity. In this scenario, the primes behave in an orderly manner through their special palindromic members:
13 31 17 71 37 73 79 97
The four pairs of two-digit palindromic primes form the octave bridge in the 99 cycle.
The general laws of number behavior can now be written from the behaviors clarified through synchrographic mapping techniques. Numeronomy is then expressed or emanated through the transpalindromic functions, which remain unseen in classical number theory and structure.
The intellectual separation of geometry from number removed from number the purely geometrical aspects of the numerical continuum that made the holotomes apparent as symmetrical mathematical entities.
Only through the study of these special modules does the exquisite order of prime number occurrence become obvious. These geometric number wheels are unique examples of circular unities.
The primes are deployed in symmetrical interface only within these specific Holotomic domains.
There is a way in which to seek out these entities by intermultiplying the primes from a special base module. Much in the way that the factorials are produced but with the difference that diminishes the huge sums that result from the redundant multiplication of the accumilating composites, out of what would have been primes.
We begin with six, the first perfect number, then double it to produce 12, which we call a holotome. This produces a number wheel that involves all of the base digits plus three of the first two-digit numbers. This number wheel contains all of the needed geometry by which to proliferate the family of related Holotomes.
All mandalogs are the product of the systematic generation of the exact sequence of minimax factorization. They have the perfect retrograde feature which reverses at midpoint, because of the existence of palindromes and other reflective qualities.
The first solid indication of a rational link between prime numbers and square numbers was found in the diagram entitled the prime/square interface which actually includes the composites in that the full overview addresses the holistic interaction of all classes.
The prime/square interface diagram consists of a vertical column of the first one hundred and one numbers with their squares listed on a right hand column. The finite extent of this number map is calculated to encompass the full range of the 99 cycle. The cycle that contains the exemplary basewave that is essential to the structural integity of the Holotomes. This wave in a certain sense is such that it determines the point places in the continuum where discontinuity may or may not occur.
The mandalog, then, is a graphic mathematical entity for the expression of a plurality of interdependent formula in a simplicated singular system, i.e. an information containment mechanism, or book: Holotome.
Such a device contains information and is at the same time a device to convey information clearly and accurately, with a minimal possibility of ambiguity, error, or paradox.
In classical systems of encoding and conveying information, elements of paradox occur through the vehicle of language itself. The fault generally exists in the very foundation of language at the fundamental level of syntax.
It is through the transpalindromic nature of the natural baseten number sequence that the irregular occurrence of prime numbers is recognized as a purely causally-determined pattern of rational explanation.
The classical approach to a study of prime numbers is such that the primes are considered more or less estranged from other classes of numbers in hopes that the primes might manifest some intrinsic rhythm of their own that could be found to account for distribution, density, etc.
No single class can be isolated as an element responsible as a determinant of any specified classes of behavior, since the full compliment of classes that comprize the self-modifying continuum interact congressionally.
Synchrographic analysis has shown that an exemplary wave form is formulated in the structuring of the base digits which when issuing through the sequence of numbers maintains its own structural quality even while it modifies the quality of the numerical event identities it encounters.
This wave form occurs through the mutual interaction of square number nine and palindromic prime number eleven. In that nine times eleven equals ninetynine, the wave proliferates through the multiples of ninetynine.
Fuller did not have the advantage of synchrographs to clearly see and properly describe this basewave. This description of an octave-nine system had the turnaround at fifty. The true nineleven turnaround is a 49.5.
The graphic mandalogs allow us to monitor the exemplary basewave that is guided through the continuum of natural number by the cyclic and reflexive qualities inherent in the special or noble numbers.
In the Prime/Square Interface Diagram, the basewave is seen to contain itself throught the palindromic mechanism that is sustained through the four pairs of transpalindromic primes that act as transnumeric relay stations.
The tapestry of number is literally woven with the four warps and four woofs, or octave, of the transpalindromic bridge between the fist and only two-digit pralindromic prime number 11 and the first, but not only, 3-digit prime number 101--primes that are known to proliferate palindromes in being multiplied by themselves.
The full importance of the basewave continuity observed in the multiples of 99 is only realized when investigating its involvement in the structure of the Holotomes. The initial holotomes contain only a rational section of a complete cycle; that portion necessary to insure a quality of infinity, (the number repeating itself indefinitely).
The number structure or number behavior mapping technique makes number theory visibly coherent. Synchrographic techniques are scientifically systematic. The general scheme of Numeronomy involves a more complete system of classification which takes special note of both the palindromic and transpalindromic nature of number. It is possible (but remains to be calculated), that the holotomes contain a consistent ratio between primes and non-primes with the holotomes that precede and follow.
TRANSPALINDROMESTranspalindromicity of number is merely a term by which we include the reversal of any particular number exceeding a single digit. For example, 16 and 61.
Another example involving a transpalindromic distinction is when a reversed number remains in its class, like 13 and 31 which are both primes, i.e. transpalindromic primes as compared to a simple palindromic prime such as eleven. These are only twelve permutations possible within the whole continuum of number.
The complete analysis of number behavior is not possible without taking into account the palindromic and transpalindromic characters of number. This is the crucial and paramount reason that the behvaior of prime numbers have remained an enigma for so long.
Number theory has ignored the ambidirectional reflexivity inherent in the number continuum. Regaining an impartial view of how reflexivity is totally conserved within the continuum leaves no gaps in the concept of numerical continuity.
The synchrographs and mandalogs used in this study are to support the continuity that has been disrupted by the belief that prime numbers are without connective order. Toward this end, we introduce approximately 16 new words that fill in the missing blanks to form a coherent picture or concept of true number dynamics.
The term "synchrostat" designates an event synchronicity in the numeric continuum. This term and its subsequent Tables were valuable tools in indentifying a cyclic series of numbers that embody features that were common to all members of the series. They embody and exemplify a practice that was used to explore many other cycles until the main base cycle was finally discovered.
(Synchrostat illus. here.)
Comprehend the universal nature of the transpalindromic function of number behavior is not easy. We tend to see the number chain as a unidirectional continuum, which is too linear for a synergetic perspective. Revisioning it with the concept of simultaneous counterflow yields a more accurate picture. With large spans of number, the complex interrelationships become difficult to visualize.
Fortunately, because of the octave nature of the base cycle there cannot be more than four consecutive transpalindromic pairs in a single symmetrical sequence, regardless of the amount of digits in each individual number.
However, we are only looking at the multiples of nine. But, in fact, all the numbers in between are also involved in transpalindromic transactions. To address such a complex interchange by graphic means can only be accomplished in a series of static cross-sections each involving no more than the eight required transactions.
The exemplary 99 wavecycle affords us a context that gives a graphic expression of total transpalindromic symmetry which it is possible to contemplate with clarity.
All the Holotomes contain the same general mechanism seen in the 99 cycle, but even the Holotomes must be contemplated in various graphic modes to capture their full integrity.
Transpalindromicity functions through the ambidirectional nature of number. It is therefore, the initial or primary function of number behavior in general. All details of number behavior derive primarily from a transpalindromic function operating through an ambidirectional chain of ambidirectional numerical events.
If any aspect of direction is left unconsidered in the behavior of any specified event then behavior observed in connection with that event will have been compromised. In the past, there has been a general disregard for the retrocity of number in general. Yet, no single function is more important or interesting than this transpalindromic nature in determining the basic waveforms inherent in baseten numeration.
Transpalindromic symmetry refers to a circular module of numerical relationships and interoperations which form a totally harmonious retrograde and symmetrical octave unity. The emphasis is not on palindromes, but on the relationship existing between palindromes and transpalindromes:
1089
Palindromes: 55 x 99 = 5445
9018We can take any number, reverse it, add the two numbers together, continue to reverse and add, and eventually end up with a palindrome--a number whose digits appear in the same order whether they are read from left to right or from right to left.
Transpalindromic symmetry is not isolated to the few graphic expressions submitted here as examples to clarify the meaning of this neologism (transpalindromicity) on the subject of number behavior.
Transpalindromic behavior itself is a totally general condition of the number chain and is operative throughout the continuum. It is also a primary factor in the graphic disclosure of many subtle but important aspects of number behavior; an intrinsic structural condition effecting every and all numbers in their collective interactions.
Transpalindromic symmetry is a condition relating to specific isolated groups of number or special numerical quata where symmetry manifests as a collective mutual interaction.
A Holotome is such a collective interactive group where symmetry is herein claimed to be absolute.
A Holotome is distinguished by an integrity of retrograde octave symmetry, discrete levels of finitude and circular unity. Its expression in synchrographic form is a holistic synthesis of graphic syntax.
A contemplative instrument of inquiry into the relationships between words and the idea-pictures they mean to represent.
Transpalindromic symmetry means cycloreflexive synchronicity: what comes around goes around. Palindromes are symmetrically flanked by up to four transpalindromic pairs.
SUMMARYTo summarize our main premises, note that the multiples of nine produce a transpalindromic loop, or wavecycle, which turns around at midpoint to produce a reverse series of companion numbers:
9
18
27
36
45
54
63
72
81
90The first eight multiples of nine disclose an octave cycle of four forward and then four backward number events. This retrograde function progresses to a point between 45 and 54, where it reverses and then continues on in retrograde manner to 81, which is the reverse of 18. Thus a cycloflex or wave-cycle is apparent.
In addition to this +4, -4 cycle, the multiples of eleven produce a consistent series of two-digit palindromes: The first eight multiples of 11 are palindromes which then represent an octave of 8 forward ambidirectional number events, ending in 99 or 9 x 11:
11
22
33
44
55
66
77
88
99Thus, the number 99 is composed of 2 discrete octave sets that are synchronous:
The number 99, then, is a crucial module in the description of number behavior. It contains a full octave of eight sequential palindromes in the multiples of 11 and eight sequential transpalindromes in the multiples of nine. This module is then the archetypal prototype of the transpalindromic sequence which issues through the multiples of 99.9 11
18 22
27 33
36 44
45 55
54 66
63 77
72 88
81 99
9099
198
297
396
495
594
693
792
891
990
1089Four forward then four backward event octaves with a ninth null event, altogether represent a
NINELEVEN OCTAVE CYCLE.It is through the transpalindromic nature of the natural number sequence that the irregular occurence of prime numbers is recognized. It is a purely causally determined pattern of rational whole numbers.
In the classical approach to the study of prime numbers, the primes are considered more or less estranged from the other classes of number, in hopes that the primes might manifest some intrinsic rhythm of their own that could be found to account for distribution, density, etc.
Since the full compliment of classes that comprize the self-modifying numerical continuum interact congressionally, no single class can be isolated as an element responsible as a determinant of any specified classes behavior.
Synchrographic analysis has shown that an exemplary waveform is inherent in the structuring of the base digits. When issuing through the sequence of numbers it maintains its own structural quality while modifying the quality of the numerical event identities it encounters. This waveform occurs through the mutual interaction of square number nine and palindromic prime number eleven. In that nine times eleven equals ninetynine, the wave proliferates through the multiples of ninetynine.
Together, this pair interact to produce the exemplary transpalindromic wavecycle sequence which integrates the full band of number classes.
Buckminster Fuller, not having the advantage of the synchrograph did not clearly see and properly describe this basewave. His description of an octave-nine system had the wae turnaround at 50. The diagram shows the true nineleven turnaround is at 49.5.
His description of "four progressively additive, then, four progressively subtractive event octaves, with a ninth null event" is a good description and perfectly adaptable to the actual scenario. The full ninetynine cycle divided by two equals 49.5.
The involvement of prime number eleven in the palindromes has always been known.
11 x 11 = 121 x 11 = 1331 x 14641
And prime number 101:
101 x 101 = 10201 x 101 = 1030301
But the exact nature of the exemplary nineleven sequence has not been described until the present document.
The relation of the palindromic primes to the transpalindromic primes are found in the nintynine cycle. The four pairs of two-digit transpalindromic primes:
13 31 17 71 37 73 79 97
form a relay bridge between ten and one hundred within the nintynine cycle. See diagram entitled the PRIME/SQUARE INTERFACE.
The continuity in and the discontinuity between the holotomes is effected by the integrity of the exemplary nineleven wavecycle. Each holotome contains a rational section or an even multiple of nintynine:
12 / 99 = 0.121212
24 / 99 = 0.242424
72 / 99 = 0.727272
360 / 99 = 3.636363
2520 / 99 = 25.454545
27720 / 99 = 280.
360360 / 99 = 3640
6126120 / 99 = 61880
etc.There is a class of number not generally acknowledged in traditional number theory, found through the systematic exercise of synchrographics. It is mechanically instrumental to the general operations of number behavior through interactions with other familiar classes of number.
That any number of two or more digits, with the exception of numbers ending with a zero, has a reverse mate is obvious. The reverse of number twelve is twenty one, the reverse of thirteen is thirtyone, etc. This is the transpalindromic aspect of real numbers.
If the reverse mate of any specific number was of the same class as its forerunner then there would be no reason to regard the transpalindromic aspect as a designation of any structural distinction. It would simply be an "artifact" of the so-called Arabic numeral system. But, since for example, a square may be a prime when turned around, a structural distinction is obvious.
The square number sixteen becomes a prime when reversed. Sixteen, then, may be referred to as a retroprime square. As a matter of special interest, it is the ONLY two-digit square that is this class.
Any number which remains in its same class when reveresed is a transpalindrome of that class. For example, thirteen and thirtyone are both primes, i.e. transpalindromic primes.
There are only four pairs of transpalindromic primes:
13 31 17 71 37 73 79 97
The following reference guide shows the structural relationships that influence the interbehaviors of number due to the transpalindromic distinctions.
(Insert the archetypal alphabet of number class here)
144 and 169 are the ONLY THREE DIGIT EXAMPLES of a transpalindromic square number.
122 = 144
212 = 441
132 = 169
312 = 961The 12 -144 loop shows composites multiplying into squares of transpalindromic composites, while the 13 - 169 loop shows primes multiplying to squares of transpalindromic primes. Their roots are the transpalindromic composite number 12 and the transpalindromic prime number 13.
The structural distinction of these numerical identities is made evident. Their special characters are identified as active members of the mechanisms of numerical interaction. The graphic paraphenalia is available to monitor the exemplary basewave that is guided through the continuum of natural number by the cyclic and reflexive qualities inherent in the special members.
The flow or cascade of number is simulated in the synchrographic of Holotome E. It is characterized by the congressional interactions of a complete system, i.e. all classes of number are accounted for. This is not the case in classical number theory, which omitted the transpalindromic classes of number. Yet these classes are most responsible for the behavior of number, per se.
Again, the crucial double helix of the nineleven wavecycle is based on two odd numbers, nine and eleven. 9 + 11 = 20: two odds made an even. Twenty is also the base of the Mayan number system, which operates with the same numerical modules as the ancient Hindu system: all multiples of 108.
The prime/square interface diagram shows the basewave, graphically. The palindromic mechanism is sustained through the four pairs of transpalindromic primes that act as transnumeric relay stations. The tapestry of number is literally woven with the four warps and four woofs, or octave, of the transpalindromic bridge. This bridge links the first and only two-digit palindromic prime number eleven and the first, but not the only, three digit prime number 101. Primes are known to proliferate palindromes in being multiplied by themselves.
The full importance of the basewave continuity observed in the multiples of nintynine is only realized when investigating its involvement in the structure of the holotomes. The initial holotomes contain only a rational section of a complete cycle--that portion necessary to insure a quality of infinity (the number repeating itself indefinitely).
Holotome A 12 / 99 = 0.121212
B 24 / 99 = 0.242424
C 72 /99 = 0.727272
D 360 / 99 = 3.636363Whereas Holotome E is 45 more than 25 x 99, and 54 less than 26 x 99, (45 + 54 = 99), or a rational transpalindromic and transholotomic link.
Holtome E: 2520 / 99 = 25.454545
25 x 99 = 2475 + 45 = 2520 + 54 = 2574 (26 x 99)
The next holotomes are even multiples of nintynine:
27720 / 99 = 280
360360 / 99 = 3640
6126120 / 99 = 61880Note also that the preliminary holotomes themselves are either palindromes or cyclic repeating numbers.
(Insert holotomes, cont. Basewave involvement)
The Syndex system of the classification of number takes special note of both the palindromic and transpalindromic nature of number, i.e. the directionality of multi-digit groupings as they determine a local and intrinsic geometry to the sequence.
This mainly geometrical analysis of number concerns itself especially with the character of the index of factorial synchronicity and the specific locations in the continuum at which they occur. It is a study which would be impossible without the aid of a structured graphic medium by which to note the relating numerical events.
Equipped with the archetypal alphabet of number class as a reference guide, Marshall began exploring the relative positions of these identities on serial listings of the natural number chain. The first significant discovery he noted was in regard to the transpalindromic primes, squares, and composites. Not only were they directly related, but the squares were separated by either a palindrome or a minus or plus one palindrome.
(12 and 13 syndex pretzels, insert here)
The transpalindromic composites produced transpalindromic squares and so did the transpalindromic primes. In the two-digit range, there is only a single pair of transpalindromic squares: 144 and 169. Both have transpalindromic roots. In the two-digit range there is but one retroprime square which is number sixteen (16)
61 divided by 16 = 45 5 x 9; 61 + 16 = 77 = 7 x 11.This exceptional example of transpalindromicity is labeled a retroprime square, a class of special numbers. It is preceded by fifteen which is a transpalindromic composite, and followed by seventeen, one of only four pairs of 2-digit transpalindromic prime numbers.
The nineleven cycloflex or exemplary basewave is by nature a palindromic and transpalindromic wavecycle. The palindromic multiples of eleven (11, 22, 33, etc.) and the transpalindromic multiples of nine (9, 18, 27, 36, 45, 54, etc.) synchronize at ninetynine (9 x 11 = 99). This establishes a base cycle that involves the intermediate numbers between the only two-digit palindromic prime number 11 and the first 3-digit palindromic prime number 101. This basewave then repeats itself indefinitely through the multiples of 99 (99, 198, etc.).
99
198
297
396
495
594
693
792
891
990(insert poemgraf #101)
The next observation Marshall made was that there are exactly four pairs of transpalindromic primes in the 2-digit range. These four pairs are commonly separated by even multiples of square number nine.
13 17 37 79
+18 +54 +36 +1831 71 73 97
And added together produce even multiples of prime number eleven, the first palindromic prime:
13 17 37 79
31 71 73 974x11 44 8x11 88 10x11 110 16x11 176
These four pairs of transpalindromic primes prove to function as a wave guide for the exemplary nineleven base wave that terminates at one hundred, in between palindromic 99 and palindromic 101. These fours pairs, four forward and four reverse, also represent the octave which equates with the exemplary basewave cycle.
(Insert Transpalindromic Prime/Square/Composite Interface)
The Holotomes owe their integrity of continuity and holistic discontinuity (circular unity) to the structural polarity of this cycloscillating and octave-containing basewave. Each holotome is itself a totally symmetrical retrograde mathematical entity of four progressively additive, then four progressively subtractive event octaves which contain a "one half 99" turnaround point.
The mandalogs clearly show this turnaround at 49.5, but in light of our general theory we must conjecture why this is so, and it still remains somewhat obscured in our thinking. If nature works in rational whole number increments, how do we account for the fact that the 9/11 turnaround is at 49.5? Fuller thought it was at 50 perhaps because of his prejudice toward fractions. Since the graphics clearly show otherwise, how do we reconcile this with synergetics? We know it is 99 divided by 2, yet the verbal "excuse" for the fraction constitutes a "hitch in the giddyup" of rational whole number increments as displayed by the octave wavecycle itself. The hitch seems to be the exception to the rule.
CONCLUSIONSThe Goldback Postulate alleges that 'every even number is the sum of two primes.' The case has not yet been reported where this does not hold true. Even though this postulate is unprovable in the sense that all even numbers can't be checked out, it may be provable through a true understanding of how the sequence of number operates.
No even number may be a prime because of the prior occurence of number 2, which acts as a divisor for all even numbers. The first three odd numbers are primes because no number preceding them has occurred that could act as a divisor. Number 9 is the first odd number that is not a prime, because of the prior occurrence of number 3.
Therefore, the occurrence of the first odd number as a composite is the result of the previous occurrence of an odd number that had no preceding idvisors except unity, which by definition is indivisible. So, the composite nature of 9 is causally determined in that 9 would have been a prime like all previous odd numbers, but for the prior occurrence of number 3.
From this it may be postulated that all even numbers will be composites, and all odd numbers are potentially primes unless some previous odd number has occurred that acts as a factor, rendering them composites.
This is also a way of saying that every odd number that is not a prime is the product of at least one previous prime and one previous composite, (3 + 6 = 9).
But the serial equitability of the progressively divisible number chain produces a surplus of combinations to produce primes from even numbers. In fact, as numbers progress composites will occur that accomodate multiple sets of primes and visa versa. Does this not render the Goldbach Postulate only a probability?
The assumption that all odd numbers would be primes if no previous number had occurred that would act as a divisible factor perfectly fits the casse. If the prior number had bnot occurred, or could be stricken our, as in the case of number three, 9 would be a prime number, but since three is three and not something else, 9 is a composite.
The fact that nine is also a square number tells us also that the generation of squares, cubes, etc. is also the result of the occurrence of previous odd numbers, as two odds make an even and two of the same numbers multiply into a square number.
Continuity is a provisional term that requires a context describing a series of entity events that connect or disconnect to or from each other.
The primes have always been regarded as a single class with no breakdown into subclasses, no different kinds of primes.
Secondly, the primes have been viewed apart from the composites, thereby ignoring any relationships that they certainly must share with the other classes of number.
The first indications that other classes of primes exist was found in the nature of square numbers which range from 2 to 20 with one exception: none differ from each other by 16, reflecting our exceptional 2-digit prime/square reversal.
The first three digits--1, 2, and 3--are geneerally regarded as primes, but the initial members are in a class by themselves. One is not even a number, in the serial sense. It doesn't multiply like real numbers do. Two is also not a number, but is the source of the doubling of unity--duality. 1 x 2 is 2 x 1. It is the source of duality, eveness or balanced symmetry, and essentially static state. Two is the symmetry aspect of the one continuum.
Three is also not a number, but the source of imbalance and asymmetry, a source of dynamic flow. It represents the unitary aspect of the bipolar continuum. The source of oddity or difference with these three qualities produces number four, the first real number or the idea of quantity/quality and dimensionality of the tetrahedron.
"This intellectual step from two to three is a retrograde one, a reflection leading from two back to the primal one," according to von Franz. "In principle this procedure can be repeated with all subsequent numbers. The retrograde counting step leading from the number three to four has even been made historically famous by Maria Prophetessa's alchemical axiom." (see Jung's Transformative Process, SYNDEX I). It means that the number three, taken as a unity related back to the primal one, becomes the fourth. The four is understood not so much to have 'originated' progressively, but to have retrospectively existed from the very beginning.
Even and odd are established in the ideas of two and three. They are essentially spatial or geometrical ideas relating to shape and form. They are the prototypes that establish the classes of number yet to come.
All even numbers are non-primes because they are matched pairs or symmetrical sets of two.
All odd numbers are potentially primes, being uneven non-symmetrical amounts. The first odd number to become a composite is nine. It is not a prime because of the initial occurrence of 3, which acts as a factor making 9 geometrically symmetrical. Two odds make an even. 9 is axially odd, but radially even.
The SYNDEX description of number behavior employs the overview of the cycloflex. It represents the reality of function that is both cyclic and oscillatory, i.e. both rotational and reciprocating.
In describing number behavior and dynamics, it becomes necessary to show the interaction existing between palindromic numbers and transpalindromic numbers--an interaction that is in essence a cyclic behavior.
The retrograde functions progress to a point and then reverse in octave cycles of four forward and four backward number events. This is a wave-cycle or cycloflex.
Continuity in the description of the exemplary basewave cannot mean in the sense of something uninterrupted because it must be curtailed on the upper limit of the Holotome for the sake of holistic unity.
Continuity has given trouble to the number theorist because of the mind's insistance that any specific integer is an isolated idea entity and connot be effected by another integer, several or even many times removed. But this is, in reality, the case when number 9 is not a prime because of the usurpation of its "primeness" by 3.
This transinteger dynamic is a passive dynamic. It happens as acausal determinate which means the retrograde loops on the SYNDEX number maps are left to show the truth.
The number reversals that purport to carry the exemplary wave are gestures of quasi-disconnective continuity because the relations must be shown that we may see how numbers interact as they flow towards some "nothing" called infinity.
The prior reference to approach of holistic continuity at the end of a Holotome is the final non-event of such a specific system. The octave is best deactivated at these discrete points. It is predominantly the Holotomes that clarify the distribution of prime numbers, for the primes are symmetrically arrayed within the context of each holotome.
The radial symmetry of the holotomes is, in itself, the geometriccal proof of numerical coherency. That each subsequent holotome admits one, and only one, additional divisor establishes a rational medium between prime numbers and the sequence of specific compositry.
Though the structure of the basseten continuum is a highly complex order, there are graphic methods by which the more important aspects of that order may be aprreciably simplified.
For example, there are very rare, often unique examples, of certain classes of number that occur seldom or even just once in the two-digit numbers. It was through the discovery of these rare or noble examples that the exemplary basewave cycle was discovered. Once that cycle was isolated it was a simple matter to extrapolate into the higher orders to secure the multiple digit cousins of these nobel examples for comparisons to confirm the sustained functions of cycles being investigated.
A basewave was long ago suspected by the classical number theorists, but its elusive nature was due to the fact that it was not a singular wave form but a compound variety. A wave composed of the mutual interaction of square number nine and prime number eleven, whose essential palindromic nature is connected with the transpalindromic aspects of the total numeric profile.
The synchrographs act as maps by which we may follow the progress of this exemplary wave cycle. It is a feature that would not in any way be apparent without such a systematic graphic mechanism.
The nineleven wave is further caged by such features of the continuum as the lone pair of 4-digit transpalindromic squares--those being the square of 33 and 99: 1089 and 9801. These two interreflecting squares neatly bracket an octave sequence of four forward and four reverse multiples of the first square which signficantly includes a center or nave number which is a palindrome which results from two 2-digit palindromes.
By and large, it is predominantly the holotomes that clarify the distribution of prime numbers because the primes are symmetrically arrayed within the context of each holotome. And the number of primes in each holotome is determined by the total modular amount of the previous holotome. For example, Holotome E or #2520 contains 367 primes or just seven more than Holotome D which is #360 (360 x 7 = 2520), and Holotome D contains 72 primes where #72 is the modular sum of the previous holotome.
The fundamental intent of the initial idea was to create a context which would geometrically paraphrase the elements of numerical progression in a graphic sysstem that involves the primary elements of plane or two dimensional geometry. The two axes provide a base for symmetry as a reference to sysstematically involve the cyclic and wave functions of the numeric continuum.
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