A Treatise on Resonance

A treatise on resonance                                                           St.Dogs    Aug. 2003


Space in mathematics is quite different from space in physics, space in mathematics is abstract, space in physics is real, the everyday space around you, for instance

Whenever a sphere revolves in ordinary mathematics it revolves at any speed and holds its form

whenever though a sphere revolves in physics it transfigures and from its womb creates, given enough momentum, an independant shape, the torus

It is exactly because mathematics is primarily concerned with abstract entities, and not ‘reality’ as such, that seems to lie at the heart of the neglect of the torus that one finds in math.

No professional mathematician that I have asked knows the torus formula by heart, how much less probably the physicists

The mathematician will say that he can easily derive it, but that is not the point, they both don’t recognize it when it is in front of their eyes and this must be the reason that two major formula’s in the field of gravity are not recognized as toroidal geometries

In general one can say that every formula that features a “Pi ^2” signifies a toroidal geometry in space

The general formula for the ordinary torus surface is 4 Pi^2. k.r and for its volume 2 Pi^2. k.r^2,  in which k is the distance from the center to the center of the ring, so it is no big deal to remember, and especially not when we know there must be a  “pi^2’  involved

To make things easier though we can resort to an even simpler and more interesting shape and formula, that of the special case of the ‘horntorus’ (here called ‘corus’, torus with core)

This shape has all meridians of the tube go through one point in the middle, so ‘k’ becomes equal to ‘r’

The formulas now become resp. 4 Pi^2. r^2  (surface)  and  2 Pi^2. r^3 (volume)

Newtons formula for the acceleration of gravity is     a =  4 Pi^2 . R  /  T^2

Einstein wrote : “For this two dimensional globe there is a three dimensional analogon, the three dimensional spherical space, discovered by Riemann. The points in it are also all equal. It has a finite volume, characterized by her ‘radius’ R (2. Pi^2. R^3) “

The problem with Einsteins formula is that it was ‘constructed’ by Eddington and nobody in the scientific community has ever recognized it as a fundamental torus formula

Starting from the firm belief that here was no coincidence at work, but that the real thing about the geometry of space and gravity was connected with these two formulas, I have eversince been confirmed by my findings that here a grand avenue to a GUT (Grand Unified Theory) lies open

That this solution touches on the foundation of mathematics is a good thing, because it takes a different frame of mind than the ordinary to appreciate a calculus on circles without Pi, a theory of matter without the concept of mass and a theory of gravity without a force, and that is the kind of theory that unfolds here

The concept of gravity as a force is totally absent in this theory, (as it is not in Einsteins), every form of energy whether field or matter is considered here an excitation of space much in the sense of the 19th century English philosopher and mathematician William Cliiford , who wrote :

“I hold in fact : 1) that small portions of space are, in fact, of a nature analogous to little hills on a surface which is on the average flat ; namely the ordinary laws of geometry are not valid in them, 2) that this property of being curved or dis­torted is constantly passed on from one portion of space to another after the manner of a wave, 3) that this variation of the curvature of space is what really happens in that pheno­menon which we call the motion of matter, whether ponderable or ethereal, 4) that in the physical world nothing else takes place but this variation, subject (possibly) to the law of continuity.”

In mathematics ‘Clifford hills’ are a wellknown concept eversince, and in topology it is considered today that there are not only ‘hills’ but also ‘holes’ on the smallest scales and this brings us to the presentation of a fundamental concept used in this theory and this geometry, the space grain, or space pixel.

Without going into further detail here on the nature of the space grain, its essential geometrical shape is that of the horntorus and its field is similar to that of the earth magnetic field, layer upon layer of toroidal surface contracted and bounded by one point only

This by analysis extraordinary, (almost) hybrid, topological shape(oriented, non-oriented), that bounds a fully concave and convex shape in one point only, may be the key to the geometry of space

As we know the surface of a sphere is 4 times its greatest circle and so is expressible in two-dimensional space as the surface of a circle 4 times the section of the sphere

The surface of the ‘corus’ (horntorus) expressed in 2-D is an exact square from the length of its circumference

Now there is a very special relationship between sphere and corus, and there respective surfaces

And it is this relationship that seems to be fundamental to practically everything that happens in terms of energy and it is to do with squaring the circle by circumference as well as by surface and by analogy with aforementioned 3-D shapes it means the transformation of sphere into corus and vice versa, by like circumference or like volume

It is these two geometrical transformations that are seen as the originators of resonance and standing waves of resonance, or of longitudinal (spheroid) as distinct from transversal (toroid) waves, as, in short, the very basis of the electromagnetic field and its perpendicular components A torus basically is two perpendicular rings forming a dynamic surface


We have another broader look at the fundamental concepts we are dealing with here

In physics the sphere is known primarily, and certainly most impressively, from most celestial bodies, and bodies are mass, and mass here is identical with inertia, inertia then, excited space grains (as will be explained later)

So given the fact that the (near-)perfect sphere is rather rare in everyday nature ( pearls and berries are an exception and based on life), the aforementioned celestial bodies served as symbols of the perfect, the full moon, the setting sun, this is so from old and the sphere in math therefore is a form taken from heaven, like the circle before, it is also the straightness of the rays of the sun as they cast a shadow or fall through the clouds, that enlightens us on what is straight, and what a straight line

The circle and the straight line are the tools with which we measure reality, their relationship is an expression of the measure of heaven

Since heaven is harmony, the relation of the straight rays of light and the sphere of the sun they come from is harmonious (the term heaven is used rather taoistic and indeed meant to create a more spiritual environment for our investigations )

Harmony is most precious to our ears and when we transcribe the intervals that we experience as harmonious, that is musical, it turns out they are related with mathematical precision in whole numbers as 1, 2, 3, 4, 5, 6, etc  and that we hear something is wrong at , 7, 11, 13, 17 , disharmony

So here we find a direct and obvious link from our everyday experience of reality to our discerning a difference in quality by hearing harmony or disharmony, this is based on pure mathematics and has nothing to do with our habits or contingent likings, it is traceable to the patterns of the waves themselves, as we will see

The astronomer Kepler wrote around 1570 in his musical theory : “it is not sufficiently clear why the numbers, 1, 2, 3, 4, 5, 6 etc, produce musical intervals, while 7, 11, 13 and the like, don’t “.

it is this divide in numbers that forms the basis of this theory of resonance


What we hear as harmonics corresponds to a certain mathematical combination of whole numbers, what we hear as dissonants are prime number vibrations (from 7, 11 , onwards) that do not resonate with the other whole numbers that derive from the first three primes 2, 3 and 5

2, 3 or 5 are factors in all whole numbers except for the other prime numbers starting from 7, 11 onwards

Since resonance is always in some way related to whole number relationships, we need a calculus, a way of calculating, that restricts itself to whole numbers

That seems nice and easy, but there is a problem

Because we are dealing with waves our formulas contain the numbers Pi and Pi^2 and in ordinary mathematics the number Pi blocks all smooth transformation from circle to square or other polygons

The transcendental nature of the number Pi, may give it a near divine aura, in fact it is blurring the transparancy of geometry, as we will see and thereby hiding a celestial dome of mathematical relations that spans the whole universe


One of the oldest and certainly best known approximations of Pi is Archimedes’ 22 / 7, only a mere 0,001264….difference from canonical Pi, within nearly all practical tolerance of margin

the mathematical beauty here is, as far as I am concerned, that this perfectly expresses what Pi is about, a circle circumference 22, divided by its diameter 7

It seems a happy coïncidence, but in this theory coïncidence does not exist, because the relation 22/7, the ‘integral’ Pi, generates a whole spite of whole number relationships between circle and

square that remain hidden for canonical Pi, but only differ a fraction in value

Again Archimedes seems to have found that the proportion of the surfaces of square and inscribed circle is 14  :  11, and we see that the same numbers are involved, and this relationship implies that the square inscribed in the circle is half the size of the initial square, is  7

So we have two dimensions caught in whole numbers , 1-D = 22/7 , 2-D =  14 / 11 / 7 , is there a 3-D ?, and there is, the sphere in its cube relates as 11 :  21 , by surface ánd volume

So we have whole number relationships for three dimensions, this seems to underline that in the logic of whole numbers there is also a certain type of circularity

If we have another look at this peculiar relation of a square 7 in a circle 11 , we are dealing with surfaces, but if we try to find a whole number relationship between either circumference by way of 22/7, it doesn’t work

If we resort to Pi and we set our circumference at 10, the inner square perimeter is 9,00316….., no integral, of course, but here the number 9 is very near, and here it happens, it turns out that treating the innersquare perimeter as integer 9, creates a whole new perspective on the relation of circle and square, and more importantly, on the relation of sphere and torus

It should be borne in mind that in this calculus every square or rectangle is primarily the 2-D representation of a toroid surface, and the factor that represents the torus in its essence is Pi^2

It is this Pi^2 that is the wholenumber relationship of the 10/9 calculus and its value is :


800 / 81   or               9,8765432098765432098………   or  9,87654321

the corresponding value for Pi here is :

sqrt 9,87654321=3,142696805…or 2,222222..sqrt(2)


The circularity that we find in the combination of 7 and 11 as the cardinal numbers of the 22/7-calculus thus turns out to have a similar pendant when we examine the relation between 10 and 9

Graphically this expresses itself most convincingly in a length of 11 bent over a length of 7 , and a length of 10 bent over one of 9, half circle and quarter circle in whole numbers, but incompatible

Here we encounter one of the principle arguments of this theory, namely that it is possible to make valid calculations with two distinct values for Pi, both derived from whole number relationships, and to show that patterns of resonance exist between these two whole number circularities although they are numerically incompatible, in number and geometry

This incompatibility is overcome by the very vagueness of the boundary of the eigenfrequency in resonance, so when the values of each system come very near each other and can resonate, they will, resonance is a kind of joining in, in a sense when the siren sounds the dogs howl in resonance, for some reason the silent full moon establishes the same effect in them

Resonance is not restricted to the exact eigenfrequency, as said, there is a small margin around each frequency, the overtones?, that transforms the difference in a standing wave pattern that basorbes the differences in frequency and thus allows for non conformity

Resonance is absorption of other vibration and it is also tuning, within set margins

It is easy to see that this model has great potential in formalising things like consciousness, language, memory, DNA-codes, you name it, in terms of resonance

Resonance means basically familiarity, being of the near same kind etc, so from resonance we can easily understand that kind attracts kind, that preferences in human behaviour and genetic similarities in organisms in general are easily explained in terms of resonance

Resonance bonds, resonance attracts, resonance absorbs, resonance emits, resonance orders

Numerically it seems everything is traceable to the difference of exactly one unit

This small difference is another cornerstone of a huge structure of number relationships that stands like a cathedral

This one unit is the difference between an even and an odd number square, grid of 4 or of 9

The odd number square allways has “one square” in the middle , this position sets the one out against all the others, although the one is not different from the others, it is different only by its position in the center, the even numbers have no central square, so the grids are geometrically incompatible

This changing of grid or lattice in squares, which is even more obvious in the changing shapes of polygons and their order, this change is probably one of the fundamental aspects of the geometry of space and the transport of energy

It is this change of grit that can be easily shown to accompany the curvature of geometry and consequently of space


To get a proper idea of a grid we will first examine the square, but the triangle, pentagon, hexagon and octagon will feature as well, indeed we revive the deep insights of Kepler who really was the first to intuit a clear geometry of space in polygons and polyhedra, as basic forms for all geometrical order

The five platonic solids, the tetrahedron, cube, octahedron, dodecahedron and icosahedron, still are what they were 2500 years ago, the only regular polygons that form a symmetric geometrical 3-D form

Today we know that for instance that some virusses have an icosahedron structure, with two, three and five fold axis of symmetry, so the whole of living nature up to the smallest organic creatures is pervaded with these geometrical harmonies and bondings

To make one more general comment on the polygons, it is essential that only triangle, square and hexagon can fully grid a Euclidean surface, that only the 12 pentagons of the dodecahedron form what comes near to a sphere

But first the square

All prime numbers except 2 are odd numbers and it is the square of  two , 4 , that sets the grid for all even number squares, with a dimensionless point as center, a crosing of lines, while the 9-square formation sets the grid for odd-number ordering, with the central ‘one’ as the true difference

So here we see a difference in ordering of concentric squares, the one with the obvious centerpiece, the other, without substantial centre ( 9, 4 ) (pict. of grids)

The centers of the circles will always be 1 unit radius apart, concentricity will never merge, unless you can devide your 1 into 4, your 9 into 36, your 25 into a 100

All these numbers themselves will again and again pop up as the absolute pillars of the geometrical and proportional framework of this theory

What we can see is that the 4-grid is a quartering of the 9-grid

While the wave pattern (concentricity) of the ‘one’ and of all squared odd-numbers is defined by the diameter of the centercircle, the wavepattern of the the 4-grid is determined by the radius of the centercircle

this difference in grid is dramatically shown in the first 6 orbits (asteroids included) of the solar system, which are pushed out by the tremendous radiation of the sun, but as mathematical progression they centre on Mercury in concentric circles

Venus as the first distance at 45 million from Mercury pushed by more or less 60 million to 105 (108), and this remains the unit, Earth at 90 pushed with 60  to 150, Mars from 180 tot 240 (228), the asteroids from 360 to 420 and finally Jupiter from 720 to 780 (778), so there is a pattern of doubling of the radius, disturbed by an initial push out of about 60 million by the sun

The interesting thing here is that the orbit of Saturn lies about where it should in the centerless grid, without the pressure of the sun, at 1440 , it is at 1427, less than 1% off the mark and Uranus makes good by being near perfect at 2880 (2870), Neptune completes the ringsystem as nineth orbit and as eighth planet at the enigmatic, but rather precise 4500 million km mark, which is exactly 100 times our initial unit of  45, so the sequence is the follwing
M          V        E       M      Ast.   J         S         U         N

56       108    149     228             778  1427   2870  4497

60       105    150   240    420   780  1440  2880    4500

0           45    90     180   360   720  1440  2880    4500

0              1      2      4      8    16     32    64      100

sqrts     1    sqrt2    2    2sqrt2  4     4sqrt2    8       10


M      V       E        M      ast.        J         S        U          N


56   108    149    228              778        1427  2870    4497


60   105    150    240    420   780       1440  2880    4500


0     45       90     180     360   720       1440  2880    4500
1          2        4          8       16          32       64        100

sqrt  1    sqrt2      2    2sqrt2       4        4sqrt2     8       10



The regularity here seems beyond reasonable doubt and it is a variety of the doubling of the Titius-Bode Law, but with more deep structure and possibilities of resonance, more elegant.

This first example of a cosmic implication of a basic framework is only a playful example of a truly grand picture of the solar system as a system of standing waves in resonance and of gravity as essentially a condition of the electromagnetic field that pervades all space above the temperature of the cosmic background radiation and is the effect of attraction that results from resonance of inertial fields

In this theory of continuity, it makes sense to say that gravity is “the cooling of the cosmos”, that radiation is the energizing of space, that space is energy, consists of energy, produces and absorbs energy

Now that we have ventured into the solar system, we might as well have a closer look at several aspects that we have just touched upon


In physics inertia when coming to relative rest contracts to a sphere, this means that their (inertial) field of direct influence will geometrically be ordered like the odd number grid, only by interference of the even number grid, (the overall dominant order) this fases out

Analysis of the solar system has just shown that this odd field of influence extends to the orbit of Jupiter, while Saturn is so far away that the Sun has no relevant substance anymore in its grid and has become a pointfactor, so Saturn , Uranus and Neptune, are part of the even number geometry (Pluto is considered to be outside the primary planetary system, not a real planet, similar is recent Quaoar)

It is held here that it is the interaction of these two geometries of even en uneven numbergrid, that effectuates the geometry of the solar system, up to the exact masses of the planets and their apparent sizes

The solar system is the interference pattern of two distinct resonance systems, locked in a standing wave pattern, as are the rings of Saturn, and all cosmological ringsystems

These are the lines along which will be reasoned in this treatise



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