"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
"The number one as the first and original number is strictly speaking not a number at all. One as unity and totality exists prior to the awareness of numbers which requires a capacity to distinguish between separate discrete entities. Thus, one symbolically corresponds to the uroboros state prior to creation and the separation of things. Two is the first real number."
Edward Edinger, Ego and Archetype
We consider numbers to be so familiar that we have nothing left to discover about them of any possible interest. But perhaps this is not so. 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 reflective and symmetrical as well as 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 behavvior. 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, transpalindromes squares, retrosquare composite, palindromic composite, retrocomposite prime, transpalindromic prime, retroprime square, palindromnic square, retrocomposite square, transpalindromic composite, palindromic prime.
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 his classification system, it is impossible to analyze the number field. Palindromes, or binomial reflection numbers are neither purely accidental nor without significance. Remember, OM (#108) 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 spiral 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.
#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 .
In Cosmic Fishing, Applewhite recounts Fuller's reaction to the great hexagonal court in the ancient ruins of Baalbek. He said, "The Phoenicians knew my principles."
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 and structure 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
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. And it is the task of Fuller's synergetic geometry to identify energy with number.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."
SYNCHROGRAPHICS
Synchrographics 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 arithmatic.
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 360360In 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 divided by 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:
We get a reversal of the initial number.3168
5445 : 55 x 99
8613The 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 octaes 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.
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 diagram
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 tranpalindromic 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.
They are graphically elegant due to simplicity of design and complexity of data. Visually attractive, they gather their power from content and interpretations beyond the immediate display of some numbers. The best graphics (most important and beautiful) are about the useful and important, about life and death, about the universe. They contain maximal information with minimal graphic elements. 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.
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 encompasse the full range of the 99 cycle. The cycle 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.
:Robert Marshall, Master Numeronomist
Under Synergetic Construction...
Created 7/10/99
Last Updated 10/06/99Web Design by Vickie Webb
