A Theory of Resonance


The Theory of Resonance I conceived in my first few months in South-West Wales in 2003, living in houses of friends on holiday and a hotel room, totally amazed by the things I found one after the other.
 I found all these things I was not looking for, but were pushed under my nose by the numbers and geometries. The proof of the scientific validity of this number-logic is in its application to the solar system and in particular the sharp calculation of the rings and gaps of Saturn, but also the masses and spacing of the planets and their orbits follow its principles; since recently also the dimensionless universal constants can be added to our arsenal of confirmations.
This is what makes me so sure that my mathematics is valid. It brought me to the conclusion that ‘gravity’ is a standing wave phenomenon, like all matter is. So the whole notion of ‘gravity’ combined with ‘mass’ goes overboard here. No forces, all geometry, and numbers. Scientists ’say’ that gravity is ‘geometry’ of space, but they ‘think’ of it as a ‘force’, they say ‘wave-packet’, but think in terms of ’solid’ enduring ‘entities’ with ‘Eigen-substance’ (self-substance), so called ‘elementary particles’, it’s all in the name already. They fool themselves.
Scientists found lots of new ‘fundamental’ particles and they are looking now for the ‘god-particle’, which would give mass and gravity to everything (Higgs-boson). What they look for is the ’space-pixall’, but its energy (mass) is indefinite and depends on the geometric configuration it is excited by.
Scientists would do well to see phenomena as different modes of space, devoid of self-substance (eigen-substance), but they can’t get around the fact that it is all the same stuff, (of the ‘same taste’) that space itself is the substance and the energy, creating myriads of modes, that space is the aether. (This is close to Buddhist analysis)

The substance-less notion of matter put forward here is probably a bridge too far for present-day physicists, because it equates ‘mass’ with ‘void’, ‘form’ with ‘emptiness’, but this notion goes back in the West to Schroedinger and Einstein, among the deeper thinkers on reality (see Great Minds). Einstein at the end of his life was sure that neither he nor any of his colleagues knew what light, the photon, the ultimate quantum, was.
 That situation has not changed over half a century on, because the quantum-confusion is still omnipresent and unabated.
 I think the solution to the ‘quantum-confusion’ is in the space-grain, the ‘space-pixall’, which I will go into in other pieces, possibly quite soon.

It is not the Large Hadron Collider which will give the answer to the physicists’ deepest questions, it is a change of mind; a different state of mind. (Now (2012) they claim they found the Higgs-boson, but all they have is meagre statistics, nothing ‘solid’, even Higgs himself did not look too happy about it all, but felt forced to go with the hype, they found a particle, but are not too happy with its ‘qualities’, it doesn’t do what it should theoretically, many are even very unhappy with it (Alan Turok), ‘who ordered this?’)
Whatever the case, the mathematics I will eventually present here is so convincing that I think it cannot be refuted, the data are much sharper than in my Stone Age studies and with much less tolerance.
 Since the concepts of mass and gravity get the axe here (they have never been understood because they are wrong), this would involve a revolution in science. Most people don’t like revolutions unless they are part of it, so my ideas can be certain of meeting resistance, which is already the case.
Publication of the calculations for Saturn, which is the ultimate proof, may take a while; data on the mathematics of the planet-masses could be published sooner though.

Here I can reveal that the quintessential pattern of our resonance geometry, the Rainbow Proportion, is sharply visible in the rings of Saturn, where it is embodied by the width of the light inner ring, this ring starts 5sqrt2 (7.071..) from the centre and ends at 9. In our model 9 : 7.070707… is virtually ~equal~ to 14 : 11 , the Rainbow Proportion, so it symbolizes the Perfect Ring proportion. ( ratio diameter : width, also Phi for area proportion, since 14/11-squared is ~virtually equal~ to 1.618… in our system, where differences of thousandths don’t count because of resonance tolerance).

Long ago when I first discovered the geometry, which I now call the Rainbow or Celestial Proportion, I said to myself: This is Big!
You watch and see how big this is, or work your way to the same insight.
This version, below, is of 12 years ago and there is no hint of Stone Age archaeology there yet, but the pyramids are already there and the geometries. Here only the Preface and Introduction plus the first part concerning primes and repunits is published now, but here also the repunit model of resonance in relation to the standing waves is given for the first time, the crucial link between torus and sphere, square and circle, is pointed out. This first part is titled ‘Origin of the Primes’, other parts are still in revision. The text has improved, but hardly changed.

The Preface and Introduction both deal with the general cosmology and ontology. They show the same kind of amazement I still can feel by doing the finds I do.


New sketches for
A Theory of Resonance
(primes, planets, pyramids)
Spring 2004
Copyright © 2004-2010 Yan Goudryan


Nothing I ever had in mind before I began on this quest for the secrets of the cosmos could compare to the outcome of what has become a kind of voyage of discovery in an ‘other‘ universe, different from the one I knew and started for.
It is far beyond my wildest imagination, that I would ever design a mathematical framework of distinct transparency, in the first place, because I had nothing with math to put it mildly and it defies any explanation why, as a complete outsider, I would be able to not only successfully apply the model to aesthetics, physics and astronomy, but could even show its logic is buried in the ruins of the pyramids and in the dust collected in planetary rings ; it is just ‘too much’.

This is why I find myself in a awkward position, I am an autodidact in nearly all the fields that are touched upon in this cosmology and still I dare propose far reaching solutions to outstanding scientific problems while the only tools I have as a philosopher and artist is my logic and my pencil.
The only reason I dare take the risk of ridicule is because this is a solid model with a consistent explanation, and even if science does not accept the model, rather than ignore it, it still stands as an important aesthetic tool, because it generates the ‘golden ratio’ of circles and squares, shows the mathematical spectre of musical tones and is highly practical and pictorial, besides that it has a beauty in its own right.
So if it’s not science, it’s art.

By way of pictures it is easy to show what by way of language is cumbersome to impossible. The pictograms show the geometrical harmonies that come with the number-logic, in which numbers are represented by colours and show that all the mentioned subjects fall under the same harmony.
The division in prime numbers that I have found, along with a theory of resonance which can bridge the gap between mind and matter (by dissolving matter), provides a philosophical and scientific approach to reality and the universe that may help to understand life better by opening our eyes for the ‘mystic view’.

Although the mathematics used is new, it is about the simplest one can imagine, otherwise I couldn’t have done it, and it is pure geometrical logic, a logic of natural numbers, in particular the logic of the primes; its fundamentals can be taught a toddler
Although the outcome may be very abstract, the elements of this cosmology are in general all very simple and designed to be understandable for any educated person
The problem that faces us in the confrontation with this pertinent number-logic is the problem of its ultimate significance.
While the number-logic goes against the grain of our present day logical and mathematical reasoning, it at the same time expresses an immovable intuitive body of logic of its own, and the question arises: what is its ultimate meaning, its true status ?

A theory of resonance
(logic and aesthetics of natural numbers)


Understanding resonance as the principle state of a fleeting reality is essentially understanding waves and fields as the basic constituents of the phenomenal world, doing away with all concepts of matter and mass, even gravity , and introducing the logic of prime numbers as fundamental to a theory of space, its geometry and waves, it’s energy and dynamics
The theory can be expanded by equating space and consciousness and by exploring recognition, memory and language as fundamental examples of resonance, so that here we have a mathematical model that bridges the gap between mind and matter by dropping matter out of the equation
This is the general perspective of the theory, it is a ‘theory of everything’ (toe), because it brings all natural phenomena from the solar system to esoteric mental phenomena back to resonance, to interacting fields
Resonance in its most rudimentary form is a natural number phenomenon, like a frequency, and this requirement of whole-number relationship is the starting point of this theory and its ultimate justification and argument, apart from all its compelling elegance and beauty.
Here we will explore the essence of reality by revealing the hidden logic and aesthetics of the natural numbers and the order of the primes and we will relate them to the continuity, geometry and vibration of the phenomenal world
Space is seen as a plenum, as the ultimate ‘ground’ of all energy and its geometry is seen as the blueprint of all transformation of energy
The ‘circular’ quality of certain pairs of natural numbers and their geometric peculiarities indicate that waves and wave patterns are produced and expressed by the numbers themselves in a most convincing way , like 12345678987654321 as square of 111111111 (R9)
Because of the extraordinary regularity of the used numbers and the transparency of their basic order, the model is, whatever else it is, a logic of aesthetics, and this the more so where it even entails its own ‘golden section’ for circles and squares, the perfect ratio (not 1 : 1,618.…, but rational-> 1 : 1,620.)
The elimination of Pi and other irrational numbers from all calculations, which is a major characteristic of this calculus, enables us to describe waves in whole numbers only, thereby bridging the chasm that canonical Pi creates between circles and squares, that is spheres and horn-tori, because of its irrational and transcendental nature.
It is this absence of Pi that enables a computation in which (topological) transformation of circles, squares, spheres, tori and cubes in one another are completely covered by whole numbers and an occasional square root key number to highlight the theory.
This logic shows that ‘squaring the circle’, in two ways, is at the heart of the dynamics of sphere and torus, of the origin of the waves and of the geometry of space itself, squaring the circle by equal surface in whole numbers is incompatible with squaring the circle by equal circumference, which is the gist of this model here.

It is the incompatibility of two number- and wave systems that creates a standing wave, which in its turn forms patterns of resonance, that under certain conditions ‘materialize’ space, so that matter is seen as a local excitement of space, as a particular state of space and its geometry. 
The standing wave is what the world is made of and it exists continuously because of the incompatibility of the number-systems, the interfering frequencies of an infinity of wave-fields in perennial vibration, cast in numbers and ratios that never go away, that are and always will be so, because it cannot be otherwise.
Here we will restrict our investigation to the logical, mathematical and physical consequences of the theory, by exploring the geometry that rises from the harmony of the natural numbers.
But before we explore the geometrical nature of the natural numbers, we can get accustomed to the regularity of the numbers involved by understanding the order of the primes and the ‘deep-structure-circles’, the rings of numbers, they form.

Origins of the primes

Since all numbers are primes or products of primes, it must be the primes that hold the key to the structure and interaction of numbers. (Prime numbers are only divisible by themselves and 1)
If we hold that numbers are produced by preceding numbers and that all numbers are essentially the symbol of a ’sum’ of ´ones´, the ‘one’ they are all divisible by, like 2 is 1 + 1 and 2 : 1= 2, likewise 3 is 1 + 1 + 1 etc. , but if we also hold that numbers, if not primes, are a unique product of primes, then it is easier to appreciate that plus or minus 1 makes all the difference in a non-primal number, even if it is a big number, because the ‘character’ of the number changes dramatically, not only changes the number from odd to even or vice versa, if not to prime, also the factors change completely.
Factorisation defines every number’s unique character and basic factors are always primes, although here powers of 10 will be seen as an exception, and can be expressed in factors 10^n ( ten to the power n; n is any natural number)
The difference in ‘factors’ between 1000 and 1001 is telling : 10 x 10 x 10 and 7 x 11 x 13
That 10 is not a prime but the product of 2 and 5 makes it a special number, because 2 and 5 are the only primes that have no period in the reciprocal (inverse, so 7 and 1/7).
The period is a repetitive returning pattern of numbers in the inverse of the prime (1/7 = 0.142857 142857 …..), but with 2 it is ½ = 0.5 and with 5 it is 1/5 = 0.2 and no repetition, no period, unique, because they are each others reciprocal, so only in their case inverse ends immediately and results in the other as reciprocal number, only in their case the product of prime and reciprocal period number is not 10^n – 1 (99….99, nine-only numbers, rep-digit), but 10 , no more, no less. This is what makes 10 exceptional as a number and the basis of the decimal system.
Because of this outstanding feature these two primes, 2 and 5, are called here ‘extra-primes’ as opposed to ‘proper-primes’, that is, all other primes, except for 3, which is also an ‘extra-prime’, but here also called the ‘root-prime’ of all ‘proper primes’, it is also the root of 9 , the pivotal number of this calculus.
This difference in character of the first three primes and the rest is so fundamental, as we shall see, that most general rules that are formed here hold only for the ‘proper primes’ and that 2 , 3 and 5 are not seen as of the same category. Only 2, 3 and 5 partake in musical harmonies, the other primes don’t, as was already observed 2,500 years ago.

Although 3 , 9 and 27 shape all the ‘proper primes’, and 3, as a root-prime, produces itself as ‘proper prime’, it is the only number with itself as reciprocal, through division on 10 (3 x 3. 3 3 3 3 3… = 9. 9 9 9 9 9 9….. = 10)
3 is the one and only prime number that is identical with its inverse period-number and that shares most rules with the ‘proper’ primes, the order of which it creates and shapes.
The inverse of 3, its reciprocal, is in decimal expression 0,3333333……, but the period length is only 1 figure (0.3 3 3 3 3 3) and the produced period-number is 3, the ‘ proper root-prime’. So 3 is extra-, root- and proper-prime in one.
The period-number multiplied by the prime produces 3 x 3 = 9, the first nine-only number, and this 9 is 1 short of 10 and so the division repeats itself and so the repetition, the period, the number-wave, is created, and infinite.
By squaring the root-prime 3 we get 9, then dividing the ‘square of 10’, a 100, by the square of 3 , the 9, produces another inverse , with a ‘hidden period’ because the period of the reciprocal is not obvious as it is in most cases, because it looks in decimals like 1/9 is 0,1 1 1 1 1… , so for 10/9 the period is 1. 1 1 1 1 1… , but for 100/9 it is 11, 11 11 11 11…
The period-length here is found to be two figures, two ‘ones’ and the produced second ‘proper- prime’ is 11, the true starting number of the ‘proper primes’.
Eleven is the first number beyond the initial 10 digits, the first power of 10 + 1 and the first repetition of digit, the first repeated unit and called a ‘repunit’, a number consisting of only 1’s.
In symbol “R(n) “ or also 11…….11 , so 11 is the prototype repunit, and moreover the first repunit that is also a prime number, which are relatively rare ( R19 and R 23 are also, but then the next are R317 and R1031 and many more are known now, but there must be infinitely many)
The cube of the root-prime 3 is 27 and dividing the cube of 10 , a 1000, by the cube of 3, produces the reciprocal period of 37,037037037…., the period-length is 3 figures, the period-number is 37 and here the third ‘proper prime’ is created , that together with the root-prime 3 form the next repunit after 11 (R2), which is 3 x 37 = 111 (R3). The length of the period 037, is 3 digits and is always the ordinal of the repunit, here R3.
The first three primes 2 , 3 and 5 are outside the body of general rules for primes, as said, but the peculiar relationship of the three first primes produces and shapes the order of all other primes.
All proper primes are the result of operations of 3 respective powers of 3, the root-prime (3, 9, 27), on 3 respective powers of 10, the product of the extra-primes 2 and 5 (10, 100, 1000).
These three operations in three dimensions by three powers of 3 on three powers of 10 set the stage for the order of the primes, where 9 is omnipresent as representative of 3 and is factor to all periods of primes taken as whole numbers, except the period of 3.


Being the symbol of the 9-number system that is integral to the primes and their order, the repunits, the number 9 is complement in the ratio 10 : 9, the ratio that dominates this calculus and the order of the primes (10/9 = 1.11111111………..)
Nine in a sense has two periods, as 9 on 10 it has the period 1, as 3^2 (9) on 10^2 (100) it has the period 11, this dual nature of the period of 9 provides it with a versatility that is capable of combining straight and squared categories in whole numbers for circles and squares alike
The number series 9-10-11 is crucial ( as are its squares 81-100-121), and the same numbers in their bilateral combinations dominate the geometric structure of the primes, and that of squares and circles as well, the relation of 9 to 10 and 9 to 11 , as well as 10 to 11 is fundamental in this logic, and in the logic of the primes
The rudimentary and very first relations are
10 : 3 = 3 + 1 (remainder)
10^2 : 3^2 = 11 + 1
10^3 : 3^3 = 37 + 1
10^9 : 3^4 = 12345679 + 1
What we see here is the formation, the emergence, of the basic proper primes 11 and 37 , they, as factors of R2 and R3 are connected in deep-structure to most primes, like the numbers 2 , 3 and 5 are connected as factors to most natural numbers (some 60%, 2 already 50%)
But there is more here, there is a structural evidence for a fourth geometrical dimension, because the powers of 3 can easily be related to the powers of 10, only in the case of the fourth dimension , the powers of ten here are the cube of the third dimension values of powers of ten (10^3)^3 = 10^9 , but we also see a whole number correspondence between 9^2 (81) and a power of ten,
81 x 12345679 = 999 999 999 + 1 = 10^9
A nine-only number (999…999, rep-digit) divided by 9 always produces a one-only number (111…111) , a rep-unit (or repunit = repetitive-unit), so these nine 9’s correspond to the repunit 111 111 111 (R9), and 9 is here its ‘ordinal’ number.
Many repunits are composites of smaller repunites and new primes as we shall see
The divisors of the first three powers of 10 are the first three powers of 3, the root-prime
10 is divided by root-prime 3 and produces ‘proper-3’ and a remainder of 1 : 10/3 = 3 , rem. 1
100 is divided by 9 and produces the proper prime and repunit 11 : 100/9 = 11 , rem. 1
1000 divided by 27 produces the (extraordinary) proper prime 37 : 1000/27 = 37 , rem. 1
We know that a remainder of 1 in division is the sure sign of a period in the reciprocal, it is the 1 where the division starts all over again ad infinitum
It is easy to see why prime divisor and period must always have a product of 10^n – 1, a number of only-nines, like :
9 x 11 = 99;    27 x 37 = 999;      101 x 99 = 9999 ;
but also 7 x 142857 = 999 999 (R6).
this is because divisor and result reach a product of 99……….99 and not of 10^n, 100….000, and so create the remainder of 1 for the next division which creates the period, the cycle, the wave.
The length of the period of the prime’s reciprocal can never exceed in digits the number of the prime minus 1 (this is a mathematical necessity, it shows how a number is a collection of ‘ones’) and the period must, to be a period of a prime, secure a remainder of 1 in a division on a power of 10 , and must, multiplied by the prime, always produce a number of 9’s only, 999………999 = (10^n) –1
Since the prime cannot be divisible by 9 as a matter of definition and the product of prime and period, the nine-only number, is always divisible by 9, it must be the period of the proper prime that is always divisible by 9, because no new factors get added.
So the rule is that the period of the proper prime (’s reciprocal) is always divisible by 9 (3×3) and never by 2 and 5 (the factors of the powers of 10) and that, once the period-number is divided by 9, it becomes a ‘deep-structure number'(dsn).
[[  1/7=0,14285714…     142857/9 = 15873 (dsn)  this x7 = 111 111 (R6)   15873= 11 x 13 x 37 x 3     (37×3=111) (11x13x7=1001)]].

This deep-structure number is a product of primes which, when multiplied by the prime produces inevitably a repunit.
The primes are tied together in this base-repunit and will not occur other than in this combination in other repunits.
Combined they are a set of factors characterizing their base-repunit and defining their relations.
Every proper prime is factor to a repunit, inevitably, Q.E.D.
Nine and eleven

The period of 11 is 1/11 = 0,09090909……… and this period-number 9 is shared by all repunits, however big, they have only one 9 at the end of the period of the repunit’s reciprocal, it is this one 9 that all periods of primes share and the 9 that incorporates the 9-fold-system in the decimal system
This feature stresses again the link between repunit and 9, that is best symbolized in the rational number 100 / 9 = 11, 11 11 11 11 11 11 11…………..
Exceptional here is that in this primal beginning it is not other primes that produce new primes, as is the soon emerging rule, but a second and third (dimension) power of 3, and this is because there are no other primes to do it yet and this is the best illustration of why 3 and 9 dominate the deepstructure of the primes : because they produce it.
All even repunits are always linked to 9 , the odd ones when multiple of 3 are linked to 27 that produce the first proper primes 11 and 37 in this way and lay the base for the order and its rules

10 / 3 = 3 rem.1 3 x 3 = 9 / 9 = 1 R1
100 / 9 = 11 rem.1 9 x 11 = 99 / 9 = 11 R2
1000 / 27 = 37 rem.1 27 x 37 = 999 / 9 = 111 R3

11 is factor to 11 (R2) and is one of the rare repunit-primes, because repunits are usually composite numbers and here 111 (R3) has two factors, factor 37,( the third centred hexagonal number after 7 and 19) and root-prime 3
R2 and R3 are factor to every second (even) and third repunit so the majority of repunits has 11 (half of them) as factor and 3 and/or 37 or 333667, occur in all repunits of ordinal multiple 3, that is a third, so the  majority of repunits has at least one of these early proper primes as factors.
Now that we have this system of repunits and primes we can see that R4, which is 1111, must be divisible by R2 and this produces the next prime 1111 / 11 = 101.
This prime 101 is special as first prime in the hundreds, like 11 is in the tens, and it forms a new and decisive step in the formation and further presence of the order of powers of (10^n) +1. (101 = 10^2  +1)
11 is the first, 101 is the second, but 1001 is the third and then suddenly we can even derive part of the primes from another stock as well , because 1001 is factorised 7 x 11 x 13, so here 7 and 13 are new and here we find primes together that only occur again when 111 x 1001 = 111111 (R6) with all the early factors, 3, 7, 11, 13, 37.
In general these are the primes that are factor in even repunits, because a major new rule is now that any repunit Rn multiplied by a power of 10^n +1 , will always produce a repunit twice the initial one: R2n, so :
R3 x 10^3 +1 = R6 and in formula (Rn x 10^n +1 = R 2n)

One by one, with every step, we find new rules to apply and new numbers to incorporate in the body of the primes, a body that has a rather different character from that of the production of composite natural numbers, because the amount of primes becomes less and less and the amount of composite numbers grows at their expense and to find the primes is rather more cumbersome, than just adding 1 to find the next number, but we are greatly helped by the iron-cast internal structure of the repunits which makes the ‘majority’ of interesting primes within easy reach of calculation, depending on the capacity of a computer

Prime ordinals

A repunit like R5 and all other repunits with ‘prime ordinal’ numbers like R7, R11, R13, R19, are more difficult to factorize and they add only new and apparently unrelated primes to the already calculated ones, like R19 and R23, which are among the rare repunits which are primes themselves.
R5 produces the primes 41 and 271 as factors which are 40 + 1 and 270 +1 , and doubtless here there is again order because 41 follows 37 as prime and 40 +1 is strongly connected with 36 + 1 = 37 and we can see 10 and 9 in there, as we see in 270, so there is lots to be found in different layers.
Primes often are part of a new substructure themselves, like 37 relates to 333667, seen as 33…….67 and the same can be traced in the prime repunits R317 and R1031 , which are reminiscent of 37 and 101, among the earliest proper primes (3….7, 10……1)
[ even as co-factor in 37 x 333667 = 12345679 !! (10/9)^2 x 10^7) ]
It is the R6 that really combines and produces the early primes like 7 and 13 (7×13=91) as new primes and those are then related to R2 (11) and R3, so we have
11 x 111 (37×3) x 91 (7 x13) = 111 111 (R6) or
111 x 1001 = 111 111

Like 37 also 7 is a centred hexagonal number and 13 is a centred dodecahedral (12-)number (ball-packing), this means that there is a 1 unit , a centre, surrounded by multiples of 6 or 12, like 6+1=7, 12 +1=13, 18+1=19, 36 + 1=37 etc. and since all primes are of the form 6n +/- 1 and probably half of them are thus of the form 6n + 1 , a substantial amount of centred hexagonal and dodecahedral numbers are primes, and relate to packing (densely) (Kepler’s conjecture). At quantum level the hexagon is of profound importance, this has no doubt to do with the number 6 in relation to all primes, except 2 and 3 it is made of.
The pentagonal dodecahedron is the symbol of a twelve-sided centred compression of a sphere. The rhomboid dodecahedron (crystal) emerges from perpendicular field contraction and fills all space between its units, something the pentagonal dodecahedron cannot.
All of these early proper primes have geometric qualities and they shape the order by their specific qualities and here we see that the hierarchy of the primes is not numerical but geometrical, these primes define geometric ordering principles, and it is this that defines their prominence and frequent occurrence in analysis and their weight in the ‘hierarchy.’ (37, 101, 9901, 9091 etc)
The ‘early primes’ are all short period primes, but dominate the complete logic of the primes
This we will expand on presently, here we continue the analysis of the deep-structure of the primes


It is the repunit that forms the deep-structure of the prime, in the repunit the prime finds its connection to other primes, the repunit can be considered a ring of connected numbers that will always appear in combination, and it is essential to see this in perspective with the waves and their patterns.
This ring is symbolised by the repunit 111111..……111111  which reveals its ring-like quality easier in the graphic notation..….11111111111111111…….  ‘Ones’ can be added in front, at the end, in the middle, the number doesn’t budge and expresses the same, an infinite number of ones can be added and so an infinite number of primes, because every new repunit produces at least one new prime, but can go up to 7 , but there is a definite decline here, because the numbers grow by powers of ten, whereas on average 4 primes are added in that amount, this is the proof that relatively less and less primes occur (which is no surprise; but it is a clear mechanism expressed this way).

The repunits are ever expanding rings of numbers, that, with every new 1 added, add more than a power of ten new numbers under their number and, as there is no end to the wave, there is no end to the numbers to describe it, so this is the infinity of the wave, of the period of the prime.

Example of deep-structure

To give an example of analysis of the deep-structure of primes we analyse the prime 7
We start with calculating its reciprocal (inverse)
Its inverse is 1 / 7 = 0,142857 142857 142857 14…… . The length of the period is 6 digits and a new rule is that the length of the period of a prime’s reciprocal is always a number that is identical with the ordinal number of the ‘repunit of first occurrence’ (rofo), the base-repunit of the prime, in this case R6.
The period number of 7’s deep-structure is 142857 and it is this number we always can and do divide by 9 to find the ‘deep-structure-number’ (dsn), in this case 15873 (7 x 15873 = 111111 = R6)
The next step is that we analyse on divisibility by 3 by taking the sum of the digits and see if they are a multiple of 3 , the sum is 24, so 15873 / 3 = 5291 and because of the extra divisibility by 3 (after being divided by 9 already) we know the prime 37 must be at hand, because 3 (here actually 27) will never appear ‘alone’, will always be accompanied by 37 (R3 = 111) ,or 333667 (R9 as in 1001001), and indeed 5291/37 = 143 , which is 11 x 13
The subsequent step is that we try to divide by 11, because R2 is factor to all even ordinal repunits, thus half of all repunits.
Only after these steps have been taken do we know what kind of repunit we have, is it a composite repunit or one with prime as ordinal, in this case dividing by 11 gives 143 / 11 = 13 and that is a new prime, so R 6 is the base-repunit (rofo) of 7 and 13
We now have localised and brought into contact all the first 6 primes, 2 and 5 as the bearers of the decimal system that ‘creates’ the primes by forcing repetition in the inverse, 3 as the universal (repetitive)operator (divisor) (root-prime), together called the ‘extra-primes’, and then 7, 11 and 13, all in very close connection (R2-R6), called ‘early proper primes’ with small periods and being factor to early short repunits, R2-R9. (R8= 11,101, 137, 73)
We have found that the period length of the prime is always the same as the ordinal of its base repunit, so if any prime is at hand, by calculating its inverse and getting its period length we also know to which base-repunit the prime belongs, and the ordinal number of the repunit then tells us if it is even or odd, if it is a composite ordinal or a prime ordinal and in this way of any prime its deep-structure is ready at hand, and can be the start of analysis.
Another feature of the repunits is that all its new factors, that is, those factors (primes) not related to earlier repunits, are equal to or multiples of the ordinal number of the repunit plus 1

Example of analysis

R11 has only factors : 21649 and 513239 
so (21649 – 1 )/11 = 1968 and (513239 – 1)/11 = 46658
Here we can discern another layer in the deep-structure because 1968 is factorised , 2, 3, 41 and 46658 is 2, 41, 569 , so both carry 41 as factor in a deeper layer still, but this is of no direct concern to us here, it only shows that the numbers have deep layers, that do not show readily, because only now it is obvious that R11 is linked to 41 which in its turn is linked to R5 as factor, so there is more there, again, but what interests us here in this rule is that we can always find the factors of a repunit however difficult it may seem
Suppose we have a large prime-ordinal-numbered repunit , then if we create all multiples of the ordinal number + 1 between them all the factors of the repunit will be present, so the repunit defines its own primes, and deep-structure, by its ordinal multiples + 1
The deep-structure of the prime is the base-repunit which defines the length of its period and the other primes it is directly linked to in the hierarchy of the periods (and the repunits consequently), so the primes are each others irreplaceable co-factors in the repunit, in the reciprocal actually, so we see that the repunits mirror the order of the primes.
Although it may seem different these primes define each other exhaustively in the products they form together, their place is as solid as that of 7 between 6 and 8, because 7 as a prime is inevitably linked to 13, 11, 37 in the R6, the length of its period, 6. (Note these prime numbers as pivotal in the fabric of space)
Returning to the important repunit R6, important because of the ´weighty´ early primes it combines, it reminds us of a hexagonal order and indeed the primes are hexagonal-connected, 7 is, like 37, a centred hexagonal number, while 13 is a centred dodecahedral (12-)number, the first scale of a ball-packing, a spherical order.
So we see here in this early ring of crucial primes 7 (6+1), 13 (12+1) and 37 (36 + 1),all together in R6, the base of the hexagonal grid that is further highlighted by all primes (except 2 and 3 (2×3=6)), being of the form 6n +/- 1.
All early proper primes have definite geometrical qualities, because here can be added 11 that is the key to squaring the circle and in this system 11 forms a tandem with 7 to cover all circles and squares, spheres , tori and cubes in whole numbers.
[[ squaring circle by area with 2800/891 is :->  140^2 = 19600 // 2800= 7 x 891= 6237 =Vf = 78.974=~79, we see here a circle with diameter 2 x 79 = 158, whereas the diagonal of the square is 140V2= 198, so the difference on each side is 20, so a hypothetical outer circle would have radius 99 with an area of 2800/891  xx 99^2 = 30800 ]]
This all underscores the claim that the primes are directly related to geometry, the geometry that rises from the repunits and that casts the primes in a definite geometrical frame with great regularity and transparency.
The repunits in this geometry symbolize the infinite circle that ever expands and that holds all the primes as its factors, like Riemann’s zero’s rippling away at sea level.
That the primes and their deep-structure are more than only regular in repunits is shown by the convincing products that show ever more layers in the structure of the primes, like the primes that form the powers of

10^n + 1
11 = 11
101 = 101

7 x 11 x 13 = 1001
73 x 137 = 10001
11 x 9091 = 100001
101 x 9901 = 1000001
11 x 909091 = 10000001
17 x 5882353 = 100000001
7 x 11 x 13 x 19 x 52579 = 1000000001

19 x 52579 = 999001
again mostly the same numbers recur
bigger repunits are of R4 1111 = 11 x 101
structure R10 1111111111 = 11111 x 100001
or R12 111111111111 = 1111 x 100010001
R14 11111111111111 = 1111111 x 10000001

So the ‘hierarchy’ and framework of the primes is very much related to the powers of 10 plus or minus 1
(10^n) +/- 1
Different combinations of this same pool of initial primes, gives rise to numerous symmetric number patterns that can be of great help with computation
Because of the extreme regularities in the deep-structure of the primes all answers to computation already exist in this model, only the search to connect the intermediate answers is the work of the computer, and all numbers are based on natural numbers, so this model works extremely fast, already now, without substantial input, its factorisation beats Maple easily
It connects square and circle without the barrier of Pi, and this is how sphere surface and corus surface can both be expressed in whole numbers and whole number relationships
This makes the decimal system and its notation into not just another convenient contingent choice, this system has intrinsic qualities that are peculiar in relation to the primes, including the logic of its notation and symbol, it is because of the central role of 9, 10 and 11.
The decimal system stands out for transparent deep-structure, its capability of creating waves of numbers and numbers of waves because of hierarchical positioning in the notation.
Squaring (and mutually multiplying differing) repunits gives characteristic wave-like number series

R2 x R2 = 121
R3 x R3 = 12321
R4 x R4 = 1234321

R7 x R7 = 1234567654321

R9 x R9 = 12345678987654321

One can interpret this last sequence as a complete wave, in which rarity and density come to their maximum in the lowest and highest numbers, and it is R9 itself that holds the key to the wave structure of the repunits.
Only in R9 times R10 123456789987654321, there is still symmetry, although the chain of alternation seems broken by two 9’s , seemingly still connected, but this connection stops with the following square repunit
R10 x R10 = 1234567900987654321
We find the 8 missing in the first sequence and this will remain the characteristic of that number sequence, because123456789 + 1 =123456790
Moreover we now see also a definite characteristic ending of the squared repunit from R9 onward, the perfect number sequence : 987654321
So the squared repunit has the general form :
12345679 0 123……….432 0 987654321
The first half of this number 12345679 0123…. is the same as that of the inverse of 81 (that is 9 squared)
1/81 = 0, 012345679 012345679 012345679 012345…………

The second half of this number, the ending : 0987654321 , is not only a remarkable number in its own right, it turns out to be the key to the geometry of the torus and it is the number that is the digital of the rational number 800 / 81, again the inverse of 81
9,87654320 987654320 987………. ( here 987654320 + 1 = 987654321 )
It is this rational number that is a substitute of Pi^2 (pi-squared) and taking the square root of this number you get a number that is about the same as Pi (3.14159…), that is : 3,142696805…….. = 0,00110…. more than Pi , it is also written 2,2222…..V2,
Here maybe for the first time a square root number becomes a valid substitute for Pi , and it is called : Qute (pron: “cute”) , by me, symbol Q, because it is a ‘cute’ number, as will be shown, and because the Q, as symbol, so perfectly symbolizes what happens in its geometry, as will be explained.
Two rings of prime numbers create, when squared or multiplied, a torus, a square, a rectangle
This is where the new mathematics takes off.

Number waves

We have seen how the square of a repunit reveals a new deep-structure 123456790…. and ….987654321 and both are centred on 81, the square of 9 , because
100/81 = 1,23456790 123…… and 800/81 = 9,87654320 987654320 987……
Given that the repunit as single number represents a circle circumference, then the squared circle, the squared repunit, is the image of a torus
The formula for the torus is 4.pi^2.k.r , where k is the distance from the centre of the torus to the centre of its ring, but in this special case k = r, so we get the formula
4 . pi^2 . r^2 , the “corus”, which is the square of 2. pi . r , the circle circumference.
This special case of the torus, as squared circle, where all radials go through one point in the middle, is here called “corus” , contraction of ‘core’ (one point) and ‘torus’ , but is in mathematics called : ‘horn torus’, and its surface is equal to a square, the square of its circle circumference 2. pi. r. = 
4 . pi^2 . r^2
It is this square’s perimeter that is equal to the great circle of the ‘corus’, but also to the great circle of the sphere that encompasses the ‘corus’  (mathematically).

the relation between corus and encompassing sphere, which is the same as ‘squaring the circle by circumference’, turns out to be a central issue in the geometry of the whole numbers and in the here exposed geometry and dynamics of space
The repetition of the period in the squares of the primes beyond R9 is the sure sign of an existing period and is therefore another example of a wave, because the change in the numbers spreads from the centre of the number symmetrically to both sides while the outer ends of the squares of the primes remain the same, the circle widens but remains closed, the centre pulses the numbers to both sides which shows graphically:
R 9 ^2 = 12345678987654321

R10 ^2 = 1234567900987654321
R11 ^2 = 123456790120987654321
R12 ^2 = 12345679012320987654321
R13 ^2 = 1234567901234320987654321
R14 ^2 = 123456790123454320987654321
R15 ^2 = 12345679012345654320987654321
R16 ^2 = 1234567901234567654320987654321
R17 ^2 = 123456790123456787654320987654321
R18 ^2 = 12345679012345678987654320987654321

R19 ^2 = 1234567901234567900987654320987654321
R20 ^2 = 123456790123456790120987654320987654321
R21 ^2 = 12345679012345679012320987654320987654321

If we do not square but multiply different order repunits, that is different number circles, then we see a different pattern because the ‘number wave’, is characterised by the smallest repunit of the two
R3 x R 6 = 12333321
R3 x R 16 = 123333333333333333333321
R8 x R 10 = 12345678887654321
R11 x R 12 = 123456790110987654321
R11 x R 13 = 1234567901110987654321
R11 x R 21 = 1234567901111111111111111110987654321

Because these last series is all to do with unequal circles we are dealing with a proper torus here, that seems to express another type of wave and its surface is a rectangle in 2 dimensional space
Another feature that is fundamental is that the repunits follow a 9-number order, because the patterns of the numbers are the same for R9, R18, R27, R36 etc, so we see that the 9-number pattern that was already visible in the origin of the primes as factors of repunits, now as a second wave propagates through the numbers, not through one number individually, but through the whole body of numbers
There is no better way of visualizing the wave than the nine-number system pulling through the decimal order by lagging behind one point on every ten, it is supposed that the intricacies of the two wave systems creates resonance and standing waves and that the ratio of 10 : 9 is the quintessence of the numerical and geometrical order that is shown here

{{ To make the ring of repunits clear it is good to start with the best possible example which is a square of side 100 which is encompassed by a circle of 111.11…, this according to our geometry of 10/9=1.111… as square and circle perimeters. (We don’t count the decimals, because less than a thousandth: 111,111….~111)
We see here immediately how the repunit as circle is connected to the power of 10, in this case 100 and 1000, because the formula here is 100 : 111 =  10^2 : (10^3 – 1)/9  so (10)^n : {(10)^n+1)}/9.}}

In the repunits is hidden the geometrical order of the natural numbers.
It secures that resonance is discrete and sustains the standing wave that everything is made of.
We have become familiar now with the fact that it is possible in mathematics to find all kinds of numbers that not only express harmony in a hidden, often geometrical way, but even show it in and by themselves
With this knowledge, that the elegance of numbers expresses something of ‘real’ visual beauty and harmony,- an objectivity that is even more striking in our intuitive perception of musical versus false tones-, we may better appreciate why the geometry of the natural numbers that is presently revealed was born of aesthetics and found, and still finds, one of its major arguments in what it shows.
Whatever this harmonious model may mean in practical use time will tell, but the fact that it is possible to construct the existence of such an ‘eternal’ model in the first place and then even apply it successfully to the solar system, as I will show, should in itself be a fact of considerable importance to science.
That the geometry used here so strikingly resembles the geometry of the old Egyptians, the formidable builders of the Giza site and pyramids, who show to have been aware of the crucial ‘early’ primes, 11, 37, 101, 137, that only adds to the overall suggestion that something of great value may have been unearthed here again.

This is the end of Part 1 of A Theory of Resonance.
Copyright 2004-2010 Yan Goudryan
Published on internet September 2010


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