**** A Treatise on Resonance
Preface
I wrote the core of this piece over 20 years ago, but it is a concise exposition of the relevance and deeper harmonies that surface when ‘natural numbers only’ are the tools to go by. The ‘integrity’ of the integer is the basic prerequisite of any natural resonance, from biology to astrophysics.
This piece is a concise overview and analysis of resonance in one.
A Treatise on Resonance St.Dogmaels Aug. 2003
Space in mathematics is quite different from space in physics, space in mathematics is abstract, space in physics is real, is ‘tangible’, 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’. This centrepoint is the borderpoint of the surface-area of the horn- (vortex-) torus and the borderpoint of space and the zero-dimension. (back to geometry space-pixall)
The formulas now become resp. 4 Pi^2. R^2 (surface area) 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) “
These two formulas are at the core of physics to this day and express by the Pi-squared a toroidal geometry underneath.
The problem with Einstein’s formula is that it was ‘constructed’ by Eddington and nobody in the scientific community has ever recognized it as also a fundamental torus formula! (To me Eddington’s construction did not feel like rigourous logic, more addition of chracteristics that suited well, but nowhere any allusion to space as a volume as the formula requires.) (I wished I had a link)
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 one to appreciate a calculus on circles without Pi, a theory of matter without concept of mass and a theory of gravity without 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 Einstein’s), 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- pixall. It is the ’embodiment of space’ the substance of a medium’. Its density ensures the perfect transmission of electromagnetic- and gravity waves
Without going into further detail here on the nature of the space-pixall, its essential geometrical shape is that of the vortex-torus (horn-torus) 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, that’s the geometry underneath, the vortex.
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 area of a sphere is 4 times its great circle and so is expressible in two-dimensional space as the surface area of a circle 4 times the great circle of the sphere. (This precision in a natural number has always baffled me, because one is flat, the other curved as a ball, and see how much it expresses the primary ratios, 4:1.
The surface of the ‘corus’ (vortex-/horn-torus) expressed in 2-D is an exact square from the length of its circumference: 4 . pi^2 . r^2= (2. pi . r)^2. It is indeed the square of the circle circumference 2.Pi.R.
Now there is a very special relationship between sphere and corus (horn-torus), and there respective surface areas
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 area and by analogy with aforementioned 3-D shapes. It means the transformation of sphere into corus and vice versa, by same surface area or by same 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 electro-magnetic field and its perpendicular component fields.
A torus basically is two perpendicular rings forming a dynamic surface area and volume, which is partly concave and partly convex.
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 caused by 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 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 and science in general. I often refer to ancient systems of thought or try to imagine them. For instance I try to imagine the capacity of logical thought in the late Stone Age on the basis of the buildings and architecture they left us. I claim that Maeshowe in Orkney was a scientific center and has a mathematical lay-out, the Measure of Maeshowe. I have shown the same mathematical principles apply at the pyramids of Giza)
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. This is an ingrained resonance in our system.
Aesthetics is sustained by mathematics. It is traceable to the patterns of the wave-fields 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 natural 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. (I call these ‘extra-primes’ because they form a crucial bond with unique ordering principles.)
a large contingent of numbers have these primes as natural number factors containing the number 2 = 50%, 3 = 33.33 %, 5 = 20 , together they people 50 + 16.66 + 6.34 = 73% of all natural numbers and 27% left for the rest without them, to which virtually all other primes belong.
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, true ‘integers’ only. This is the scientific ‘raison d’etre’ of this whole algorithm.
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, it is irrationality galore, it cannot be done. This is why what is done here in these pages has never been undertaken before as far as I know. No educated mathematician would break the rules I break. In as far as this queste for the natural number primacy is concerned, it is the requirements for whole numbers in ‘resonance’ that is the scientific justification and value of this mathematical model.
The transcendental nature of the number Pi, may give it a near divine aura, in fact it is blurring the transparency of geometry, as we will see and thereby hiding a celestial dome of mathematical relations that spans the whole universe.
What is shown here is that space is considered the vibrating substance of reality and that its functioning is ordered along natural number relationships and specific ratios of pivotal numbers
One of the oldest and certainly best known approximations of Pi is Archimedes’ 22 / 7, only a mere 0,001264….difference from ‘canonical Pi’ 3.14159265.., within nearly all practical tolerance margins. Archimedes knew it was not exact and calculated another ratio smaller than Pi, 223/71 = 3.140845… and another rational number.
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. The Half arc of 11 over the diameter of 7 is a perfect insight in what is the matter, it turns out to be one of the key ratios of the algorithm.
The other is 10 : 9 and that then is a Quarter arc of 10 over a side of the inscribed square as 9.
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 surface areas 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 14, so is 7. This ratio 14:11 is only reached by using 22/7.
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 by volume, which is exceptional
So we have whole number relationships for three dimensions between straight lines (line, square, cube) and curved lines ( arc, circle, sphere)
This seems to underline that in the logic of natural numbers there is also a certain type of circularity.
If we have another look at this peculiar relation of a square area 7 in a circle area 11 , we are dealing with areas, 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 natural number 9 is very close, and here it happens, it turns out that treating the inner-square 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, the two circles involved in the shape.
It is this Pi^2 that is the whole-number relationship of the 10/9 calculus and its value is :
800 / 81 or 9,87654320 987654320 98……… or 9,87654321.
the corresponding value for Pi here is :
sqrt 9,87654321=3,142696805…or 2,222..sqrt(2) or 20V2 / 9
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 eigen-frequency 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 as when the siren sounds the dogs howl in resonance, for some reason. Anyhow it shows how extremely subtle the vibrations are we are dealing with and how fundamental in causing a reaction. The dog cannot help howling, it’s its natural resonance with the environment. It’s a reflex, like barking at a noise.
Resonance is not restricted to the exact eigen-frequency, as said, there is a small margin around each frequency, the overtones.
Resonance is absorption of other vibration and it is also tuning-in, 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 and (some standard) ratios.
Resonance expresses 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 (inhereted) resonances.
With people of generally the same background, even culture, finding common ground is easier because of resonance, so the preference often has a practical origin, easier communication in all manner of ways. The stranger is ‘unknown’, so caution is called for. Since resonance is ‘unvoluntarily’ the seeds of racism lie in this preference for the own kind. It’s the safest, supposedly).
Much of the so-called ‘racism’ is rooted in the perceived possible ‘threat’ by the unknown ‘strange’ persons. The negativity stems originally from ‘insecurity’ and ‘fear’, not from intrinsic evil-mindedness.
Resonance bonds, resonance attracts, resonance absorbs, resonance emits, resonance orders, but also: resonance destroys. It can be dangerous in many ways. Mechanical by increasing amplitude, mental by stirring cruel memories.
Numerically it seems everything is traceable to the difference of exactly one unit. The difference of only 1 produces a number with a totally different set of factors in the reciprocal. This is of importance for all differing resonance frequencies in one object.
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, a grid of 4 or of 9, of 2^2 and 3^2, elemental products.
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 centre, the even numbers have no central square, only a point marked by a cross. so the grids are geometrically incompatible. This dichotomy in geometry is between odd and even and that is the difference of 1
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, the quantum jump.
It is this change of grid that can be easily shown to accompany the curvature of geometry and consequently of space
The space-pixall is by nature neutral, it is a carrier of waves, and it cannot have eigen-mass, it is empty, but geometrical, it is the substance of the basic medium of reality, space.
To get a proper idea of a grid we will first examine here the square, but the triangle, hexagon and octagon will feature as well, indeed we revive the spirit and 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 cosmological 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, 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 (flat) surface area, that only the 12 pentagons of the dodecahedron form what comes somewhere near to a sphere. If you replace its 6 pentagons by hexagons you get a near perfect sphere as you see it in a ‘football’ (truncated icosahedron).
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 crossing of lines, in 4-square formation, while the 9-square formation sets the grid for odd-number ordering, with the central ‘one’ as the true difference, that stands out.
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 centres 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 can be 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 centre-circle, the wave pattern of the the 4-grid is determined by the radius of the centre-circle.
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 pressure of the sun that affects the Deep-field geometry, nevertheless the mathematical progression is not affected. The progressions move some 60 million away from the Sun, but the standing wave of 45 million is not affected either. Mercury is the pivot of the resonances expressed in concentric circles (ellipses), including Jupiter. This Deepfield disturbance by the concentrated energy presence of the Sun, which is strong enough to dominate the combined deep-field geometry up to and including that of Jupiter is not appreciated by science because of the failing concept of space as vacuous, as empty. This is one of the most clear evidence the nature of Space is not understood and a dangerous fallacy, which shows most dramatically in the elusive ‘dark matter’ concept. This is why the heating of the atmosphere by the recent microwave tsunami is not registered at all as such by science. The elephant in the room is not noticed. “Don’t touch our mobiles”, we’d rather perish than give up our phones.
The regularity of this deepfield analysis is much more revealing and thus much better than the failing Titius-Bode Law.
In words the resonances are visible in the following sequence :
First we see Mercury pushed out by about 60m.kms, then we see Venus as the first regular distance at 45 million from Mercury pushed by more or less the same 60 million to 105 ( data: 108), and this remains the unit, Earth at 90 pushed with 60 to 150 (Spot on 149.5), Mars from 180 tot 240 (d. 228), the asteroids from 360 to 420 and finally Jupiter from 720 to 780 (data: 778), so there is a basic pattern of a doubling of the radius, disturbed by an initial push out of about 60 million by the sun. This push in the deepfield includes even the orbit of Jupiter, the planet that has most gravitational/systemic influence on the Sun (‘s innards).
The interesting thing here is that the orbit of Saturn lies about where it should in the centre-less grid, obviously 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 (d. 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 mill., so the sequence is the follwing, top row are scientific data, second row the mathematical groundwork, the resonances, the standing wave of 45m.kms.
Merc V E M Ast. J S U N
58 108 149 228 778 1427 2870 4497
60 105 150 240 420 780 1440 2880 4500
0 45 45 90 180 360 720(660) 1440 2880 4500
0 1 1 2 4 8 16 32 64 100
The regularity here seems beyond reasonable doubt and it is a variant of the doubling of the failed Titius-Bode Law, revealing more deep structure and possibilities of resonance, more elegance and transparancy, I would say.
This first example of a cosmic implication of a basic mathematical 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 the resonance of inertial fields
In this theory of continuity, it makes sense to say that gravity is “the cooling of the cosmos” (article) and that, necessarily, radiation is the “heating of the cosmos”, is even the energizing of space, that space is energy, consists of energy, produces and absorbs energy. Space as the ultimate ‘ground’ of all transformation, of the One.
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
Inertial fields, when coming to relative rest contract 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-number field of influence extends and includes the orbit of Jupiter, while Saturn is so far away that the Sun has no relevant ‘substance’ anymore in its geometrical grid and has become a pointfactor, so Saturn , Uranus and Neptune, are part of the even number geometry (grid). It shows how orderly the Solar system is at its extremes.
Pluto is considered to be outside the primary planetary system, not a real planet, similar is recently discovered Quaoar.
(I wrote this after Quaor was found, but before Pluto was kicked out of the primary system! I concluded it on the basis of the mathematics, the space-geometry, before the facts, the scientists come with arbitrary external appearances and conditions and after the facts.
It is held here that it is the interaction of these two geometries of odd and even number-grids, that effectuates the geometry of the solar system, up to the exact masses of the planets and their apparent sizes. (really incredible)
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 probably all cosmological ringsystems. The relevance of tidal friction in the solar system should get into the picture with respect to the inner frictions in the molten core of Earth in relation to the temperature of the atmosphere (and the length of day in msecs). There is a definite longstanding link between ‘climate and core’, which is again embedded in the ebs and flows of the solar system as a whole with respect to the influence of the inner sun
I am still hoping against hope that some temporary interplanetary friction, and related effects this would have on the Sun, drives a temporary climatic temperature rise on Earth, I mean, that, because of exceptional friction, the temperature of the molten core of the earth is rising and affects the atmospheric temperature and would fade with the dissolution of the planetary conjunction, there would be a slowing of the acceleration we see now.
But this is a very lonely hope.
Still I hope that mankind comes to its senses and together to mitigate the worst of the destruction that is in the cards now.
These are the lines along which is reasoned in this treatise on ‘resonance (and climate change’, by now)
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