Remarks:
the number logic that I laid bare is an algorithm strictly limited to the integrity of natural numbers; irrational and transcendental numbers do not occur, with the exception of Pi-Qute as factor to rationalize square roots of 2 -> 20V2/9 = 2.22222V2 = 3.14 2696805….. and Pi-Trite as factor to rationalize square roots of 3.
Pi- Trite= 3x11x11xV3 /2x10x10= 363V3 / 200 = 1.815V3 = 3.14 36 72 2…
So ratios of natural numbers are part of the core of this lean method.
The inevitable differences which occur in this method turn out never to exceed the thousandths and are predictable from the outset. Once a data is adapted to the rules of the system, it can go anywhere to sustain the system; this gives us the freedom to infer any numbers, usually after the second digit, which in most cases will be forming a repetition to create a natural number by means of 99.
( example: 5sqrt(2) = 7.07 106... changes to 7.07 0707... x99= 700 so 700/99 =~~ 5V2; all the odd numbers multiplied by V2, except 7, and adapted will, mutliplied by 99, deliver a striking natural number, like: 3V2= 420/99, 9V2= 1260/99. 11V2=1540/99; 13V2= 1820/99 ; etc.
Great and consequential exception is 7V2 = 9,89949, which forms the pivotal 9.9 and of course is a ‘root’ in the 99, that is the bridge between the 10/9 and 11/7 calculus and the key to the ambiguous rationality of V2, as either 99/70 = 1.4142857 or 140/99= 1.4141414…, where (99/70+140/99)/2 = 1.41421356 4 213564 213564. This is a rational approximation of 8 digits, what more do you ever want?
V2 = 1.41421356 2 …..
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In this chapter I want to explain the basis and background of the mathematics I have used throughout this website.
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This vignette symbolizes the the quintessential ratio: 10 : 9, with a circumference of circle 10 and square perimeter of 9, or, more explicit , an arc of 10 over a chord of 9. This is, I think, directly related to my most original finding,- although its ratio in diagonal and height of the great Pyramid was observed and mentioned by Lauer, but not elaborated upon, – this finding is that the square area can be interpreted as the surface area of a horn-torus.
This geometric relation, which because of resonance conditions has essential tolerance at the most profound levels, gives us the key
That is what I have done and it shows that the ratio is intrinsic in the geometry that unfolds from a Pi-value of the rational 22/7, 3.142857..142857.., the rationale of the Great Pyramid dimensions to express a half sphere, purportedly the earth itself of which they surmised a circumference only 200km short of the now established value.
Below we see the red circle 9, with the nineth square of same perimeter
The mathematical model I have ‘uncovered’ is a natural number logic with complete complementary consistency and, in that sense, unassailable. Its principles are very simple and you might enjoy these ‘clean and clear’ numbers , that , you never knew, were just around the corner in many designs you have appreciated for their inner harmonies. (“why did nobody tell us this?”).
The only fundamental difference with established mathematics is the crucial premise that only natural numbers are used and that the difference with established math never exceeds the ‘9-thousandths’, which turns out to be a matter of course of the model itself, it creates its own ‘laws’.
The integrity of the natural number and its ratios is paramount and preserved.
(new) Apart from being a tool of mathematical harmonics, think of music, the natural numbers are basic to the phenomena of resonance and wave interference. This already shows fields follow natural number order. This is the ultimate rationale for taking this odd “Kindergarten-model’ seriously, it may tell scientists something fundamental about the geometry of space, of ‘real’ space, not abstract space, which is the domain of the numbers.
[[[It is good to realize that the way we represent geometry with dots, straight lines and curves, does not make these tools intrinsicly real, we have objectified the distance between two points by drawing the shortest distance in between. In our conceptual frame work the line is rigidly straight, as it shows, but when there is no real line the shortest distance between two things, that are both in fundamental vibration, must vary; this variation may be small but it is there, it depends on the geometry of local space; so the ‘rigid preciseness’ of irrational numbers is superfluous in this environment, though their squares are welcome and their irrationality is ‘rationalized’.]]]
What for me became the deciding moment was when I found that the square is a 2-D representation of the surface area of a horn-torus, and that relation is expressed in the square having identical perimeter as circle. This transformation is one of the key dynamics the of geometry of space.
(Picture squared circle, 14/11?)
Depending on the context the approximations may give a better view of the dynamics of any ‘real’ situation in the presence.
(picture)
You will know the formula for a circle circumference as 2.pi.r and you may have heard of the notorious ‘squaring the circle’, but you probably don’t know the square of the formula being 4.pi^2.R^2, which happens to be the square of 2.pi.r . And what might that look like you may wonder. Well, it is two circles in intimate relation, it is one circle tangent to an axis, circling its own centre, a horn-torus. The whole image is dynamic because the longtudinal bulge-circling is around the centre of the plane circle of the tangential circling, so to speak. These circles are fundamentally interlinked. They never touch, they symbolize the impenetrable space in these dynamics of the horntorus.
(picture horn-torus)
(Link met Daeumler , Remark on the dynamic horn torus model in (quantum) physics.))
new)
The small difference I found between 9 and 9.0031 (just over 3 thousandth) in the circumference of a square inscribed in a circle of 10, was the trigger of this whole endeaver to clear the way for natural-number-prority in the workings of nature. This can be shown in the ratio’s of the solar system and is spectacularly displayed by the rings and gaps of Saturn’s system razor-thin Euclidean ringsystem.
(pictures)
This model purports to explain the occurrence and function of resonance as one of the basic conditions for ordering space and ‘creation’ of matter. The model concentrates on the ratios of circles and squares, 2-D, as representatives of spheres and horn-tori in 3-D. It is these basic geometric forms which are supposed to comprise the backbone of the dynamics in the geometry of space. (There would be 5 dimensions, but no time dimension, because the two perpendicular circular (vorticular?) dimensions allow a measurement of time, that is, rate of change; this is a tentative explanation.
We see that the model has universal explanatory application. From the geometry and quanta of space, to the resonances of the solar system and lunar orbits around planets, from resonance to memory and awareness, to sympathy and agreement. What the model does is give an essential confirmation of Pythagoras’s conviction and insight that ‘all’ was a matter of number. Importantly rational numbers, comprising the ratio of two natural numbers, were seen as quintessential, which showed in the mathematical harmonics in music. The two pivotal rational numbers in this model are 11/7 and 10/9, and everything follows from their complementarity. It’s amazing.
The one is the half arc over the diameter, 11 over 7, the other is quarter arc over the chord as side of the square, 10 over 9 (2×5 / 3^2) : the arcs over triangle-, pentagon-, hexagon- and octagon-sides also show 11, 10 and 7 as factors in the ratios, resp. 11^2 / 10^2 , then 35 / 33 , then 22 / 21, and finally octagon 77/75, with a circle area ratio of 9:10 , so the complementarity is everywhere, in these early prime-number relationships. ( The pentagon is approximate to the 9/thousandths, but just within the rules.)
(picture, the arcs)
The points in space that relate to each other are not in a rigid environment, it vibrates and bends, so the mathematics of their relationships are equal, but not so are the fields they are operating in
This model ‘rationalizes’ the irrational square root of 2, which was Pythagoras’s nightmare, because the indefiniteness of the irrationality of square root 2 (V2) destroyed his belief in the integrity of number. Under Pythagoreans it remained tabu to mention the subject. If only they had known there are these two primal ratios that form a completely consistent integrated whole as is shown in this work.
Irrational numbers are only valid in Euclidean space, but that does not exist one can argue, it’s an abstraction. All space is curved, not flat, space without matter is een idea, not a reality.
Space is the ultimate medium that materializes in fields of specific character (resonance-frequencies).
(pictures)
The great advantage of this model is its pictorality. The pictures ‘show’ the truth of the mathematical formulas as do the pictures of Euclides express a^2 + b^2 = c^2, in the positions of the pictured squares of 3, 4 and 5, as 9 + 16 = 25.
(pictures, see proof pythagorean theorem, wiki)
You can see it before your eyes, the abstract proof is only in pictures, not numbers
Also the great pyramid is completely described by this model and is in a sense the most profound expression of it. (280/220= 1.618, golden mean).
(pictures, see proof pythagorean theorem, wiki)
(picture, skeleton great pyramid, diamomd)
The quintessence and intrinsic beauty of this model is expressed in my treasured emblem of this mathematics as the orange square of circumference 9, inscribed in a blue circle of circumference 10. (it is here that established math gives a circumference of 9.0031.. for the square). So in terms of ratios here the side of the square as cord relates to the circular arc as 9 : 10, instead of 9.0031.. : 10. When we have accepted the priority of the natural 9 over the irrational 9.0031… (since V2)
(picture emblem)
When we accept the new principle, we are forced to accept a new value for Pi (3.1415 926535 897…) (or 3.1416) in this relation, which turns out to be a universal factor for transforming any value with V2 into a natural number as circumference of a circle. This factor happens to be 2 x 10/9 x V2, or written 2.2222…V2, to be 3.14269680…, the difference with Pi is 0.00110… , just over 1/1000. This number happens to be the square root of 9.87654321, or 800/81 or 8.(10/9)^2 , which we will encounter later on. But mind, we now have entered the domain of the ratio 10:9!
In that respect I make the claim that Saturn’ s oblateness is geometrically a spindle torus with equatorial diameter of 10 and polar axis of 9 (120792:108325 = 10:9). This ratio exactly coincides with the geometry of the ring patterns of Saturn’s incredibly rigid equatorial field with its gaps and its wayward moons feeding on the rubble of the rings.
But before we go there I must introduce you to another ratio which is crucial to our model, a ratio that was first calculated quite accurately by Archimedes who taught (c. 250BCE) that the value of Pi (ratio circumf/diam) must be between 22/7 and 223/71. The latter value, 3.1408450…., is closest to canonic Pi (3.141592…), but we will concentrate on the 22/7 as 3.142857 1428…, because here we find the other primes we need , the 11 and 7 (2 x 11/ 7), next to 10 and 9 (2×5/3^2). These first 5 primes are, with 13, the foundation of this number logic. 13 also is part of the pythagorean triangle, 5^2 + (3×4)^2 = 13^2 (again only the early primes involved here.)
Archimedes used polygons in his calculations for Pi, an, in principle, pictorial method. So also here the visible geometry is the decisive insight to ‘prove’ something valid. Polygons are central to the logic of this work and even some rare hypothesis on the nature of the polygons that they may be the 2-D representations of 3-D dynamics, like what might be the geometric basis of (cell) division, a , what I call ‘Siamese twin horn-torus’.
The octagon has a square root 2 relation to the area of its inscribed square. In this state the re are 2 sqrts in one whole, eventually forming two simple square based horn-tori, which I call ‘core’-i (torus with core-point bordering the whole 3-D field, the corus, cori).
(picture)
The sphere and the torus are supposed to be in this cosmology and ontology (metaphysics), the two representatives of the basic dynamics in the wave-geometry of space, they though are treated here as circles and squares to simplify the basic logic that sustains the dynamic. The natural numbers and there underlying geometric relations is what interests us here. A planetary orbit is a ring, a tube trough which rolls a hurtling sphere, in resonance with the specific location created by the interplanetary resonance-field
We can show that the numbers 9800 and 9801 are crucial in understanding division in nature mathematically, the 1 difference creates from 9801-1 = 9800, a double square area of 4900, the square of 70. The number 70 is 10 x 7 and one of those basic numbers we deal with only.
9801 happens to be the square of 99, another one of our pivotal numbers, which together make up our expression of sqrt2 in rational numbers, these are 99/70 and 140/99, resp. : 1.4142857… and 1.4141414141…… It depends on which denominator you you need in the calculation/situation.
Every wholesome value is predestined to have its function, whatever it is.
Mind you, a very beautiful rational approximation of sqrt.2 is 9801 + 9800 = 19601 over 2^2 x 3^2 x 5 x 7 x 11 = 13860 = 1.414 21356 4 213564 213564, that is 8 digits right and rational, that is enough to cover any curvature of space, practically. And it involves again only the first 5 primes and nothing else.
The real great thing was recently that I could comprime 24 years of work in 1 comprehensive geometrical picture with crucial numbers that sum up all basic relations in this geometry of space, easily on the side of a credit card
It is about vibrating in a certain standing wave, which makes it seem to be standing still but in reality it’s waves in resonance.