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Iona Miller's SYNDEX 1: Introduction to General Numeronomy - Theory (Numeronomy) & Practice (Synchrographics)



           by Robert Marshall with Iona Miller, c1998

 "...I would  conjecture that the task of future mathematicians  will be to collect their characteristics and analyze, when possible, every number in its logical relationship to all others.  This research should be undertaken in collaborations with physicists, musicians, and psychologists who are conversant with the empirical facts about the structural characteristics of numbers in different mediums."

                                                        M. L. von Franz, Jungian  Analyst
                                                        NUMBER AND TIME

PRO:  "Unity is plural and at minimum sixfold..."

                                                        R. Buckminster Fuller

CON:  [There is a] "complete waste of time involved in the pursuit of a rational order to the distribution of prime numbers."

                                                        Professor  Kline
                                                        Harvard Mathematician

ABSTRACT:  Syndex Theory is about the intrinsic structural patterns that appear in the baseten continuum of number  A main feature of number theory has been the absence of any noticeable pattern of regularity among prime numbers.  Reversal is the key to number law.  Prime numbers play a fundamental role in the construction of geometrical form, and in turn of structures.  The Holotomic Sequence produces a graphic syntax that displays all the structural intricacies of the continuum of baseten numbers.  Syndex theory can graphically show 100% symmetry of prime number distribution within the context of its finite series of geometrical modules.  These mandalic modules are the minimal amount of consecutive factors of division.  Syndex is not number mysticism, but a non-reductive holistic way of viewing obvious realities about the continuum through  "circular unities."  Number is not an abstraction in synergetics.  Rather, each number has a geometrical identity plus a numerical identity, which is inter-transformative.

"We know that hidden symmetry groups exist in both physics and number theory but we understand very little of how they are related to one another."

                                      Yu I. Manin, Nov.,  Science 85

Numbers are what they are, not what we wish them to be.  They will not do what they cannot do, i.e. show symmetries where none exist.  Nor can they hide their inherent qualities forever from the astute devotee.

Well before Euclid's day, mathematicians had listed the first primes, looking for patterns or other clues to their distribution.  No particular patterns are immediately evident, except for the obvious one that all primes except 2 are odd numbers (since all larger even numbers have a factor of 2).

But a closer look suggests that the primes seem to be spreading out, or getting scarcer as the numbers grow larger.  For instance, there are eight primes between 102 and 120.  Further, note the gap of 13 consecutive composite numbers between 113 and 127.  There is no such long gap among the first 100 numbers.

(See table of  first primes and table of statistical distribution).

Due to inherent difficulties in mathematics and physics, these sciences have adopted the art of modelling as a working strategy.  Physical theories are provisional models of reality.  Mathematics itself is a model.  All physical theories are models.  A model should be able at the very least to describe certain phenomena fairly accurately.

A mathematical model is a complete and consistent set of mathematical equations, which may describe some physical, biological, social, or psychological entity...or conceptual entity.  We can substitute "structure" for "equations" when working with a non-numerical model.

Some of the purposes for which models are constructed are (1) to obtain answers about what will happen in the physical world;  (2)  to influence further experimentation or observation;  (3) to foster conceptual progress and understanding;  (4) to assist the axiomatization of the physical situation;  (5) to foster mathematics and the art of making mathematical models.

A model may be a convenient approximation, simplistic or sophisticated.  By creating models, or approximations, many researchers and philosophers think they have discovered the keys to the universe.  But even if those keys open only a small chamber on the periphery, they can be highly significant in their implications.

The whole object of mathematics is to create order from seeming chaos, to extract structure and invariance from disarray and turmoil.  In banishing chaos, we seek pattern, regularity, arrangement, understanding.  When it is there, we intuitively "feel it in our bones."  We can sense graphic or visual order, pattern, and symmetry in mathematics just as we do in nature.

Among the integers, the prime numbers play a role that is analogous to the elements of chemistry.  They are nature's building blocks, but unlike the elements, the list never ends.  Euclid proved that there are an infinite number of primes.  Classically, the main feature of the list of primes is the absence of any noticable pattern or regularity.

The attraction of this unsolved mystery of the prime number problem is that we are able to see what is going on long before we can prove it.  As in Popper's theory of scientific knowledge, we formulate a "bold conjecture," not likely to be true by accident.  Then this conjecture is subjected to the test of numerical calculation.  If the test fails to refute the conjecture, it is strengthened--proved so to speak, in the sense of natural science.  Prime numbers are the key to the required rate of growth in the Holotomes.

In physics or chemistry, experimental agreement with theory to 8th place accuracy is considered strong confirmation of the theory.  The agreement is hardly accidental, therefore, the principle by which the calculation was made must be right.

In Platonism, mathematical objects are considered objectively real.  In a certain sense, this is the realist or Platonist philosophy.  The regularity which has been predicted and confirmed is not illusory--there is something there which is lawful and regular.

But in another way, Syndex also conforms to the constructivist dogma where genuine mathematics can only be obtained by a finite construction.


Confidence in the orderliness or rationality of the number system underlies all research.  This realism is more than convenient; it is indispensable.  Proofs are a way of understanding why and how conjectures are true.

A "wave-particle" complementarity theory of the number field reveals a key (here termed THE AURIC KEY) to the orderly, rational distribution of prime numbers.  According to Fuller, the occurance of a basic wave has the significance that despite their discrete nature numbers as integers have wave-like properties.  It is a metaphysical wave with reciprocity potential.  In fact, there is an abundance of waves occuring on all levels of numbers.

Fuller demonstrates that "...nature is always operating in her own modular system of four progressively additive, then four progressively subtractive event octaves with a ninth null event, altogether constituting an octave nine system..."

REVERSAL is the key to number law.  SYNDEX THEORY is about the intrinsic structural patterns that occur in the BASETEN CONTINUUM of number. RETROCITY, as a dynamic of the number continuum, has been ignored by orthodox science.

This study of the BASETEN number continuum through an innovative system of graphics is capable of illustrating the interrelationships existing between the chain of numbers and the elemental aspects of geometry --  specifically CIRCULAR UNITY.  These graphics so excited the late Dr. Fuller that he asked to include them in his planned publication of SYNERGETICS III.

The theoretical aspect of SYNDEX is termed NUMERONOMY; the practice or practical side is contained in the graphic depictions whose creation and employment is known as SYNCHROGRAPHICS.

One of the graphic keys to NUMERONOMY is the TRIAXIAL RETROGRADE INTERFACE.  It represents the graphic foundation of the synchronicity between geometry and number and sets the stage for the series of circular unities designated as the HOLOTOMIC SEQUENCE.  These geonumerical modules also represent the minimal numerical sums which accomodate the maximum amount of consecutive divisors.

Each of these modules, presented in the form of a spiralic SYNCHROGRAPH, is a half positive, half negative retrograde octave wave cycle and reveals an ideal symmetrical circular unity.

The HOLOTOMIC SEQUENCE begins with the three ambidirectional axes of the TRIAXIAL RETROGRADE INTERFACE.  Then by the involvement of prime numbers in their natural order of occurrence, the following series of true circular unities (HOLOTOMES) emerge:


The HOLOTOMIC SEQUENCE revolutionizes the science of description by producing a GRAPHIC SYNTAX that involves all of the structural intricacies of the continuum of baseten number.

The scientifically systematic discipline of  NUMERONOMY via SYNCHROGRAPHICS establishes a whole new phase in  GENERAL SYSTEMS LOGIC.  SYNDEX is an ancient yet new wave in the realm of Physiomathematical Epistemography.  It deals with a unified series of transfinite idea interactions in a closed loop, (IDEAL SYNTAX), sans paradox.

The AURIC KEY excavates many vital properties of numbers.  Its very existence and ultimate form hinges on numbers which occupy space.  By arranging numbers in synchrographics, a dynamic number field or matrix is formed.  Regarding numbers in terms of matrices or fields is important, and acknowledging that numbers have spatial qualities is vital to fully appreciating their signficance.

Prime numbers play a fundamental role in the construction of geometric form, and in turn of structures.  These structures are manifested from the microscopic to the macroscopic, and comprise our physical reality.  As far as the realization of the implication of numbers is concerned, a knowledge of prime numbers is crucial.

SYNCHROGRAPHICS emerged from the notion that geometry and numbers are interwoven disciplines emerging from a mysterious or unified source.  By isolating basic relations between number and geometry something may be learned about that source.

HOLOTOME  E (2520) may be considered the cornerstone to the various MANDALOGS that comprise the AURIC KEY.  It reveals the pattern of compouind synchronicities that occur in the behavior of the base digits in all their permutations which end at 2520.  It graphically shows 100% symmetry of prime number distribution within the context of each Holotome.

The wavecycle begins near the center and spirals outwards, a process which reveals even more inherent symmetry and produces tantalizing rays.  There are seven spiral bands, each divided in 360 subsections of circular unity.

(Insert picture, Holotome E synchrograph)

Cycles in numbers can represent dynamic feedback processes, just as discrete numbers can represent "things,"  entities, or quanta.   In the living "journey" of ongoing manifestation the rigid structural elements of number may be considered male (or yang); the complementary, dynamic cycle process (recursion) can be considered female (or yin).

This cyclic process can be demonstrated in the distinction between palindromic and transpalindromic numbers.  It can be graphically depicted with minimal elements in a short series of self-reflecting mandalas, revealing the frequencies and harmonies inherent in the natural number sequence, the marvelous in the mundane.

The psychological mechanism that we refer to as the BASETEN concept of quantitative notation, commonly refered to as ordinary number is the ultimate foundation of the science of mathematics which in turn is the solitary foundation of industrial civilization.


Analogs with consciousness reveal the process of psychological transcendence unfolding from the O (circle-zero) of undifferentiated "objective" wholeness which depicts the eternal principle of universe and the inner being, or Self.  Wholes can be modeled as circles, cycles, spheres, or hyperspheres.  The natural number  sequence can lead our consciousness from point, to line, to plane, to multi- and hyper-dimensional representations.

The SYNDEX perspective regarding the geometry of number, (and visa versa, or the interrelationship of plurality and form, including structure), is predicated upon a series of geometrical modules that involve a discrete plurality determined by the natural order of prime number occurrence.  These modules, (here designated as HOLOTOMES), are the minimal amount of consecutive factors of division.  They display the progressive and regressive movement of the pilot-wave or pulse inherent in the number sequence.

The various properties of the so-called baseten system of numerical notation have been the objects of serious study, (under differing titles), for many centuries: numerology, number theory, etc.  The singular most common element of all such systems is that one cannot contemplate numeracy to any extent without an automatic involvement with geometry.  The reason for this is clear:  A triangle is an expression of the number three and a square is an expression of the number four, i.e. number and geometry are two sides of the same coin.

Beginning with this simple observation, the discipline of Synchrographics was contrived to analyze the geometrical properties of number and conversely the numerical properties of geometry.

The solution to the question of how to find the most evocative points of synchronicity between the disciplines of quantitative notation and the essential expressions of geometry occurred through an observation of such simplistic proportions that it is a mystery of profound depth that even such masters as Euler and Dedekind (or anyone else, for that matter ) did not employ such thinking long ago.

That a synchronetic relationship exists between the so-called baseten system of numerical notation and the fundamental principles of geometry was clearly suggested by Pythagoras in his series of rectangles with whole number commensuration between specified rectangles and their diagonals.  The 3:4:5  right angle triangle is the initial example of triplets.

That no clear mathematical progression was found that established a systematic serial law was the catch in all attempts to establish a synchronetic order to the distribution of such triangles of whole number commensuration.

The key to the solution of this problem of Pythagorean triplets is contained in another mathematical enigma, that is, the distribution of prime numbers.

The  most elegant and beautiful solution to this pair of classical mathematical mysteries has appeared through an innovative discipline labeled synchrographics  which was rediscovered and redeveloped by Marshall over the past thirty years.

Instead of dealing with a series of triangles or diagonally divided rectangles, SYNCHROGRAPHICS deals with a series of spirals that are generated by numerically commensurate sums that are generated with the prime numbers beginning with half of the first Pythagorean rectangle.

These  sums, the HOLOTOMES, are the actual sequence of minimal numerical factors.  The sequence begins with twelve as Holotome A.  Its nave, or midpoint, being six, is classically referred to as the first perfect number.

The first synchrograph then, which depicts the initial Holotome is twelvefold (as is the ancient Zodiac) of six positive and six negative symbolic events, and is the first sum divisible by four base digits: 2, 3, 4  and  6.

Doubling HOLOTOME A, we get B or number 24 which is the first number divisible by five base digits: 2, 3, 4, 6, and 8...

From there on we multiply each resultant sum by the next prime number in its natural order of occurrence 24 x 3 = 72 x 5 = 360 x 7 = 2520 x 11 = 27720, ad infinitum.  Thereby we release the HOLOTOMIC SEQUENCE from obscurity.

I will list here the four pairs of two-digit retroprimes or transpalindromic primes, or reversible primes, to see if I can add any signficant enhancement to their importance to the behavior or structure of prime numbers:

13      31;    17     71;    37    73;    79    97 and a fifth pair of near prime composites that act as a latching pair that  COMPLETES THE SYMMETRY GROUP WHICH DOVETAILS WITH THE OCTAVE THAT IS FOUR POSITIVE AND FOUR NEGATIVE EVENT IDENTITIES THAT CREATES A CIRCULAR UNITY IN THE  9/11 retrograde wavecycle:


A VERY CLEAR AND NATURALLY SYMMETRICAL THREE COLUMN RENDERING OF THIS NUMERICAL LATCHING ORDER  EXISTS.   The chances of this being an accidential coincidence is as likely as finding an 8000 lb. elephant inside the nose of a mouse.  Interesting in  view of how the multiples of nine turn retrograde at 49.5.

The same pattern continues with the 3, 4, 5, 6, etc. digit transpalindromic primes which indicates clearly that the progression of prime numbers is totally an orderly and coherent progression with no sign of disorganization, whatever.

Does the Number Seven Produce an Irrational Sequence?

Industrial civilization has been, in effect, dealing with a short deck.  360 (Holotome C) is an example of circular unity which accomodates all of the base digits except number seven.  This renders physiomathematical epistomography eternally baffling, at least until SYNDEX is understood.

To the traditional number theorist, SEVEN (7) appears to generate an irrational sequence.

Several years ago, Bob Marshall came to realize that 1428571 repeating is NOT AN IRRATIONAL SEQUENCE, but that any number in this sequence added to the third following number produced nine (9):

Note that the only digits not represented in the seven sequence are 3, 6, and 9, (also no zero).


By the simple expedient of multiplying 360 by prime number seven we get 2520, the very first and very smallest number divisible by ALL EIGHT BASE DIGITS.  Remember, singularity or ONE IS NOT A NUMBER.  [insert Hebrew character for Aleph] is Aleph, THE FOOL.

When One (singularity) is divided by seven, the resulting decimal .142857 is formed, which is a six-digit cyclation.  3, 6, and 9 are the only base digits missing from this cycle, but they can be accounted for by the observation that it is a six digit cycle.  Another way of viewing how 3, 6 and 9 are accounted for is that any digit in this sequence when added to the third following digit produces 9:
                                    + 142857142857142857142857

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