Intro
The ‘Natural Number Logic’ in geometry and arithmetic I present here is a vivid and aesthetic picture of the inter-relationships of ‘early’ prime numbers (2,3,5,7,11,13, 37, 101) at the roots of reality. What counts in this geometry is not the exact shortest distance as a measure, nor the exact match, as a verdict, what counts is the natural number/ratio that shows at hand. The convincing proof of this rule is that our new natural number system never differs more than a few thousandths or less, from the conventional values you calculate with your usual parameters. The system is also intriguingly and comprehensively complementary, which in itself shows mathematical truth and maturity and then ignoring what the natural numbers themselves seem to cry out, namely : ‘we are of the utmost importance! we are integrated and whole’, would be bordering on will-full negligence and lack of philosophical and scientific responsibility. Politics should not poison science.
This system is such a natural complementarity and is always with precision differences in the ‘thousandths’, only, it is scientific because of its tolerance. It functions because of the tolerance in the fusion of the resonance.
And see what precision and transparent insight you get back. Natural numbers and ratios galore.
Because the principle of resonance is based on natural number ratios, relationships and rhythms, it is evident that this natural number phenomenon at the basis of geometry must play a role in the wider geometry of space, that defines the parameters of the phenomenal world. We must find our way from the quantum dimension to the cosmic dimension in-one-go to unite our yet ‘inconsistent scientific pictures of reality’ today. What better light to guide us into ‘Resonance’ than our knowledge of ‘numbers’, of natural numbers, that is, the most authentic and integer of numbers, those that guide the ‘resonances’.
The ‘early’ primes
I have been and still am rather excited and overwhelmed by a combination of simple numbers I found that makes the irrationality of sqrt(2) ‘almost dissolve’ in geometric transparency (see below) and provides several interesting standard key numbers of this emerging new natural number calculus. Here 7, 8, 9, 10, 11, 13 (2x2x2; 3×3; 2×5) set the rules, no ‘irrationals’, no ‘transcendentals’, no ‘imaginaries’; all neat and proper integers, right from our early schooldays, weaving a network, a geometry, of fundamental wave forms. Some simple numbers, like 99, do a hell of a job in creating ‘natural number ratios’ from ‘irrationals’, like 5V2 = 7,071067.. changed into 7,070707... x 99 = 700/99. (7,0710… — 7,07070.. = 0,0003…difference.). just a faultless trick, that always works. The ‘boon’ of 99, right on the verge of rational/irrational is still fathomless. The intertwining of 99/70 and 140/99 is the strong pillared bridge from 11 eventually to 10 via 9 and 99. These are among the crucial ratios we get to deal with as we go along in our quest to uncover and may discover some undreamt of mathematical jewels just under the surface. The West will probably start to sell out the gold as an old-fashioned mug.
[[ in the following I introduce approximation of sqrt2 in rational ratios and it was the result of my own philosophy and math that I happened to find in the course of development. The Grok bot warned me not to present this equation as novelty because it is established and known to be so, so I do better to take away the suggestion that I did anything more than telling the world what had come out of my own calculations. The 2 crucial squares, differing in only 1. An ‘only 1’ that means a shift two an intrinsic pair.
In A lot is hidden in these numbers and in ‘their being together’. The one square relates to 80, the other by adding 1 to the twin as 81. what changes is the grit of the square we project
This 1 makes some tens, turn into the square of 3. The point here is that we have to squares of 80 and to one another one is added making 81 which is the square of 9. This is the decisive separation of 9 and 10, from the previous arrangement that were and are totally intertwined The simple point is the basic difference in grid of an odd number square is that the odd number grid has a single centre ‘one’, while the others have no solid centre, a 4-cluster, at best.
It makes us aware that the tendency is that it was about the grid.
and that there is an even and an odd grid, and that ‘s all, ( after all these years, sigh)
Which grid prevails is unpredictable and local in general.
ADDition 27/03/26:
Here follows the core of the geometry, here self-produced, but nothing ‘new’ in the ‘annals’. (of Grok)
19601 = 9801 + 9800, and these are 99^2 and twice 70^2 (4900) resp. divided by 13860 (2^2 x 5 x 3^2 x 7 x 11) = 1.414213564 213564 213564.… (period 213564) is conform V2 = 1.414 213562…. , these ‘early’– prime settings underpin the geometry of space, is the idea. For a mathematician there may be nothing special between the early primes.
It is shown in this model, that by revealing the wave-function of specific basic numbers and ratios, 11 : 7 and 10 : 9, and lately 14 : 11, particularly, a whole set of ratios comes to the fore that are ‘moulding the resonances’ of reality.
The solar system reveals many striking resonance relationships, especially in (planet moon-) orbitals and, even more enigmatic, as I have found out: in mathematical progressions in the masses !! of the planets, whereby these seem to grow and decline in mass between each other by some resonant fashion, that guides these rhythms into the ‘deep dense future’.
(I calculated and wrote about it regarding the solar system, I hope to present that more properly and prominently presently, it stems hopeful)
From ‘quantum jump to planetary gravity‘ in one comprehensive logical and mathematical ‘sketch’, that is what we are after.…. for over a century now.
Approximations for sqrt2 (1.414 21356 213……)
19601 / 13860 = 1.414 21356 4 213564 213564….
19601 x sqrt(2) = 27720.0000360… 27720 = 2 x 13860
13860 x sqrt(2) = 19600,9999999= 19601
Here is the strongest proof of how close we can get to the conventional with our unconventional natural number-logic. Now see this number as 2772, this is twice 1386 which is a conspicuous product of early primes, 2, 3, 7, 11, but it is also de denominator to our rational approximation of sqrt(2) (19601/13860). So the sum of 2 (3) different squares (99^2+2×70^2 =9801+4900+4900= 19601 ) results in a number that is a real cleavage in scales of precision. It shows that the 19601/13860 is a powerful approximation of V2, it is 1.414 21356 4 213564 21….., and thus correct to eight digits and repeating with periods of 6 digits (213564). It also carries new factors in the period ; this period carried 2^2, 3 , 13 and 37^2. Another corroboration of the pivotal prime 37, that owes its status as the direct link to a 3 when around to form their repunit 37 x 3 =111 (R3) –> 37 is a structural prime like 101 and all the other earlier primes . (101= 10^2 +1; 11 x 101= 1111(R4), that underpin the logic and the system
The factors of this period are 2^2 x 3 x 13 x 37^2. The square of 37 is remarkable here , because with the 3 there is at least one repunit bond present in the 37 x 3 = 111 (=R3). The repunit is the direct link to ratio 10 : 9 (10/9= 1.111111….) and the repunits hold the order of the primes as I show. A certain power of 10 minus 1 gives by repdigit 9 (…9999…), then divided by 9 , gives the repunit (….1111….) that holds the prime you are after. It is my contention that all prime numbers, are part of a repunit( even 3 is:) : 3 x 37 = 111 (R3)) the third Repunit, mind you. There is hierarchy in the numbers, because of their use.
These slight differences are considered ‘immaterial’ in the real world, it is what fits a resonance, not some ‘exact’ number without solid reference, it is permanent fine-tuning of similar vibrations at this level. Resonance has a ‘tolerance’, is not necessarily exact in the conventional sense. Nature’s Sensitivity’ starts here, with the resonances, the perceived signals of the environment, the most basic reactions and interactions, the inter-connection of it all, the communication sec, and its effect on the ‘appearance of reality‘.
The natural number geometry that I have found is not only mathematical but would also have physical implications as shown in data-adaptations. Such adaptation can occur in the standing waves of Saturn’s rings as well as some striking approximations of the universal constants. This goes to the heart of the really big unknowns of physics, the ‘framework of universal constants’.
Maybe my work opens a door, notwithstanding the massive resistance and ridicule this model will probably encounter with professionals . When the connection I propose between this basic geometry and real world processes can be established by quick and solid data, what I expect, from quantum to cosmos, then a valuable tool is created to analyse and influence, maybe even re-balance, natural processes. This would mean a revolution in Resonance and in Science for that matter. No doubt with its own dangers. Resonance can be all-devastating.
I think by clever enginering you can still control the wildest waters, but you have to practice with models and these are expensive. Why the huge tide differences on the french coast are not exploited for energy production. Tons and tons of water are moving, through sluices, massive energy. too difficult, maybe, already because oft he same high tide.
Squaring the circle
A very striking way of ‘squaring the circle by compass and ruler‘ is here actually realized in two dimensions, by circumference (1-D) and by surface-area (2-D). Squaring the circle by area is the notorious ‘impossible’ one, but squaring a circle to an equal square-perimeter is mathematically more interesting, because revealing intricacy and rule. (So circle circumference 40 = perimeter 40 square -> is 4 times a side 10, square 10 , 2-D )
The discovery that the length of a circle over the side of its inscribed square is practically 10:9 has changed my life when I was 50.
For over 25 years now I have pursued the consequences of the above postulate, in direct connection with the ratio of 11:7 for a half circle spanning its diameter. These are the absolute core and inner incompatible resonances of this system. It rises from the logic of these first 6 primes and blossoms into a variety of ‘fields’.
The ratio 10 : 9 is the key to the peculiar resonances that develop from the entanglement of 2 circles in a toroid setting.
The square of the number ‘Qute’ (3.14 26 9680) = 9.87654321 (exact) ; this is the rational number 800/81.
This is another striking instant of the fundamental dominance of the ratio 10:9, here even ‘squared’, with 8 as cubic ingredient.
This deeply layered number from which the factors 7 and 11 have disappeared , is the hallmark of the ‘torus’ geometry, with two circles involved, one with the Pi of 22/7, but also one Pi with an 11 denominator this is :
(800/81) / (22/7) == 2800/891 (28 x 100 / 11 x 81 = 3.14 25 36 47 58 69 etc. inherent succession perfection)
The Pi’s of 22/7 and 2800/891 disappear in the rational Qute–squared of 9.87654321 = 800/81 (Q^2), the 10:9 and (10/9)^2 ratios here are evident again, even squared.
square and circle ; torus and sphere
Here we pass on to the geometry of the square and its encompassing circle.
The mathematical friction, the incompatibility may be just what powers the system; it can never come to rest. This is my ‘metaphysics of physical mathematics.’
This whole presentation is very pictorial which places it square in our idea of what geometry should be like, visualize spatial relations, but actually what I propose here is as a parallel ‘universe’. This upends all we learned about the rigorousness of mathematics. Here beauty rules supreme and ‘exactness’ is sacrificed for the sake of the integrity of the natural numbers and their relations (= ratios = resonances). (Is anything more exact than a natural number?)
It is the total consistency and complementarity which are the most impressive characteristics of this approach with ‘whole and rational numbers only’ and then the ease of practical implantation, as is shown in the international standards on A4 paper (3x99=297; 3×70=210mm, 594/420=sqrt.2), sizes based on a square root 2 ratio as the cutting parameter (Lichtenberg,1870). (A0=1189 x 841; 1189/99=12).
99 is our pivotal number here. I had my system before I got acquainted with Lichtenberg, he was a great confirmation.
This universal standard is precisely the materialization of the theoretical prescription from the number logic I describe here, with the relatively large, but essential numbers ( 140, 198) to keep the natural numbers untouched and in view, like we also find in megalithic Giza and Orkney.
So the core logic of these specific numbers and combinations is known since mathematician and aphorist Lichtenberg suggested it, but the mathematical and geometrical implications were never pursued maybe, because what I present here would then be common knowledge, like 22/7 for Pi was to engineers and architects before the calculator came into use.> Since Archimedes actually.
But this logic of natural numbers may go back to the late Stone Age on the British Isles, in Orkney as well, where it flourished in architecture and cosmology and maybe ‘ the Creation of Time’ as the cultural highpoint of its cosmology, mathematics and architecture.
It is of some importance to realize before we set out to explore these ultimate regions of reality that both ‘quantum’ and ‘cosmic’ are states of our being, body and mind, we create concepts to express our control of inner and outer worlds and their unity. But we fail.
How can we compare a solid human being brain and all, dead or alive, with a star. The star, which comes with the majesticity of the whole cosmos, is absolutely ‘infantile’ compared to the complexity of the human brain and reality. So what are we whimpering at our insignificance compared to a star? Because it is big?. gets old? What is old when nothing is alive? What is big when beyond imagination. And so what? It is after all of an alien order we cannot be measured by.
It is the Quality that counts in this comparison, if anything. The definite reason was to intimidate with the celestial, but our quality is the Mind we’re exponents of. All of life is of a different order than the cosmic mechanics it sprouts from, continuously….. and higher at that.
Life is where creative intelligence springs from the immeasurable amount of possibilities and resonances in existence Our Environment that finds a fertile ground for outside resonances to bond together and then multiply. Resonance and changing environment are the basis of life. The whole concept of evolution is empty without resonance as the driving force of adaptation, repetition and recognition.
In this maths we discover the complementarity in resonances of basic natural orders.
Resonance is the glue that binds reality together, it is ‘the’ ‘interaction’, as such and probably the most basic one for any evolution.