## Philosophy of Pythagoras / Pythagorean Theorem

### 32 + 42=52 Pythagoras' Theorem is Caused by the Spherical shape of Matter as a Spherical Standing Wave in Space. Further, three dimensional space and spherical space are equivalent, as it takes three variables to describe a sphere. Pythagoras' Theorem is true because matter interacts spherically, as Albert Einstein realised.

(Albert Einstein, 1934) From the latest results of the theory of relativity it is probable that our three dimensional space is also approximately spherical, that is, that the laws of disposition of rigid bodies in it are not given by Euclidean geometry, but approximately by spherical geometry .... According to the general theory of relativity, the geometrical properties of space are not independent, but they are determined by matter. ... I wished to show that space time is not necessarily something to which one can ascribe to a separate existence, independently of the actual objects of physical reality. Physical objects are not in space, but these objects are spatially extended. In this way the concept 'empty space' loses its meaning. ... The fact that such a metric is called Euclidean is connected with the following. The postulation of such a metric in a three dimensional continuum is fully equivalent to the postulation of the axioms of Euclidean Geometry. The defining equation of the metric is then nothing but the Pythagorean theorem applied to the differentials of the co-ordinates.

## Introduction to Pythagoras' TheoremOn the Spherical Wave Structure of Matter (WSM)

Pythagoras' Theorem has been known for about 2,500 years. In that time it has been proven as logically true in many different ways. Significantly though, until recently humanity has not understood the metaphysics behind our perceptions of reality that would enable us to show why Pythagoras' theorem is true. With knowledge of the Spherical Standing Wave Structure of Matter (WSM) in Absolute Space we can now explain why Matter interacts Spherically, and thus why Pythagoras' Theorem is true. This also enables us to better understand why Albert Einstein used Pythagoras' Theorem as the foundation for his metric equations of Special Relativity. (Please see links to essays on the side of the page for summary and deductions of the Wave Structure of Matter.)

Geoff Haselhurst

## Pythagoras' Theorem and Einstein's Relativity

Physical objects are not in space, but these objects are spatially extended. (Albert Einstein)

Since I have embarked on the wide ocean, and given full sails to the wind, I say there is nothing in the whole universe that persists. Everything flows, and is formed as a fleeting image. Time itself, also, glides, in its continual motion, no differently than a river. For neither the river, nor the swift hour can stop: but as wave impels wave, and as the prior wave is chased by the coming wave, and chases the one before, so time flees equally, and, equally, follows, and is always new. For what was before is left behind: and what was not comes to be: and each moment is renewed. (Pythagorasâ€™s Teachings:The Eternal Flux)

Albert Einstein realised that matter could not exist as discrete particles (how do they interact with one another?). He tried (and failed) to describe matter as a continuous spherical force field in Space-Time.

The solution is to not only discard the particle conception of matter, but also the force field (which acts on particles, and as there are no particles there are no fields, they are merely mathematical approximations).

This is replaced with the Spherical Standing Wave Structure of Matter in Space which then explains the phenomena of particles and forces in Space and Time.

The correct foundation is to describe matter in terms of one thing - the Wave Motion of Space. Thus motion applies to space, not matter, i.e. the (wave) motion of Space causes matter, time and forces / fields (interconnection).
Note: We have a page of wave diagrams that will help you visualise the spherical standing wave structure of matter (WSM) in space. Basically, we only see the high wave amplitude wave-center and have been deluded into thinking matter was made of tiny little 'particles'. A very naive conception in hindsight - and quantum physics was telling us all along that waves were central to light and matter interactions!

The following article on Albert Einstein's relativity explains some of this, but with particular emphasis on his use of Pythagoras' Theorem as the foundation of his metric equations. The reason for Einstein's success is his realisation that matter is spherically spatially extended and thus matter interacts spherically with other matter (but as spherical standing wave interactions, which are in fact discrete and thus also explain the quantum properties of light and matter as found in Quantum theory.) The complete article can be found at;
Physics: Albert Einstein's Theory of Special & General Relativity
(The following section on Pythagoras' Theorem and its use in Einstein's Relativity is from this article.)

Albert Einstein's Metric equation is simply Pythagoras' Theorem applied to the three spatial co-ordinates, and equating them to the displacement of a ray of light. Einstein writes;

Special relativity is still based directly on an empirical law, that of the constancy of the velocity of light.
dx2 + dy2 + dz2 =(cdt)2 where cdt is the distance traveled by light c in time dt.
The fact that such a metric is called Euclidean is connected with the following. The postulation of such a metric in a three dimensional continuum is fully equivalent to the postulation of the axioms of Euclidean Geometry. The defining equation of the metric is then nothing but the Pythagorean theorem applied to the differentials of the co-ordinates. (Albert Einstein, 1934)
In the special theory of relativity those co-ordinate changes (by transformation) are permitted for which also in the new co-ordinate system the quantity (cdt)2 (fundamental invariant dS2) equals the sum of the squares of the co-ordinate differentials. Such transformations are called Lorentz transformations. (Albert Einstein, 1934)

The reason why Special Relativity works mathematically is twofold:

i) Special relativity assumes that the velocity of light is constant, and thus if there is no change in the velocity of the In-Wave then there can be no acceleration of the Wave-Center. This explains why special relativity is limited to relative motion between matter that is non-accelerated. (Inertial reference frames)

ii) In Albert Einstein's Metric Equations the displacement of the light beam is determined by cdt, thus it makes no difference, mathematically speaking, if the velocity of light is assumed constant, and thus time is changed to keep the metrical equation true (as Albert Einstein did) or conversely, to assume a constant Time, and that the velocity of In-Waves (Light) is changed. As it turns out, it is this latter case which is true, and this differing velocity of the In-Waves (from one side of the Wave-Center relative to the other) is the cause of the apparent motion of Wave-Centers.

Significantly, Albert Einstein confirms this view, that the velocity of light is not always constant, when he writes;

(Special relativity is founded) on the basis of the law of the constancy of the velocity of light. But the general theory of relativity cannot retain this law. On the contrary, we arrived at the result that according to this latter theory the velocity of light must always depend on the co-ordinates when a gravitational field is present. (Albert Einstein, 1954)

Most importantly though, despite these changes in the velocity of the In-Waves, their velocity is always measured to be the same.
This curious phenomena occurs because for any relative difference in the velocity of the In-Wave from one side of the Wave-Center to the other, there is a corresponding change in wavelength (which determines length), such that the same In-Wave always meets at its Wave-Center at the same time. As velocity is length/time then the velocity of the In-Wave (velocity of light c) is always measured to be the same, and the difference in wave velocity from one side to the other causes the apparent motion of the Wave-Center through Space. It seems that many people mistakenly assume that the velocity of light is constant, it is not, but is always measured to be the same (irrespective of motion) - this fact has caused much confusion.

Because Albert Einstein misunderstood time (as his geometry of relativity had no dynamic Wave Motion, which is the true cause of time) this explains one reason why he so badly judged Quantum Theory (as it turns out Quantum Theory was correct in its use of standing waves to explain the discrete energy states of light and matter).

... the methods introduced by quantum mechanics are not likely to give a useful basis for the whole of physics. In the Schrodinger equation, absolute time, and also the potential energy, play a decisive role, while these two concepts have been recognized by the theory of relativity as inadmissible in principle. (Albert Einstein, 1954)

Now it is this relationship about the change in wavelength and ellipsoidal dimension with Motion that is at the heart of Relativity so it is important to hear what Lorentz has to say on the subject;

...the simplest course is certainly to consider the electrons themselves as wholly immutable, as perfectly rigid spheres, with a constant uniformly distributed surface charge. .. But, unfortunately, it is at variance with our theorem. ... It is for this reason that I have examined what becomes of the theory, if the electrons themselves are considered as liable to the same changes of dimensions as the bodies in which they are contained. ... the explanation of Michelson's experimental result, ... admit, for moving bodies, only a contraction, determined by the coefficient in the direction of the line of motion. The electrons themselves become flattened ellipsoids.
This would enable us to predict that no experiment made with a terrestrial source of light will ever show us an influence of the Earth's motion.
It is clear that, since the observer is unconscious of these changes, ( the contraction of a measuring rod in the direction of motion), relying on his rod, he will not find the true shape of bodies. He will take for a sphere what really is an ellipsoid,

Attention must now be drawn to a remarkable reciprocity that has been pointed out by Albert Einstein. ... Let us now imagine that each observer and (one is moving with constant velocity relative to the other) is able to see the system to which the other belongs, ... It will be clear by what has been said that the impressions received by the two observers and would be alike in all respects. It would be impossible to tell which of them moves or stands still with respect to the ether. ... This is a point which Albert Einstein has laid particular stress on, in a theory in which he starts from what he calls the principle of relativity.

I cannot speak here of the many highly interesting applications which Albert Einstein has made of this principle. His results concerning electromagnetic and optical phenomena agree in the main with those which we have obtained in the preceding pages, the chief difference being that Albert Einstein simply postulates what we have deduced, ... from the fundamental equations of the electromagnetic field. By doing so, he may certainly take credit for making us see in the negative result of experiments like those of Michelson, Rayleigh and Brace, not a fortuitous compensation of opposing effects, but the manifestation of a general and fundamental principle.
Yet, I think, something may also be claimed in the favour of the form in which I have presented the theory. I cannot but regard the ether, which can be the seat of an electromagnetic field with its energy and its vibrations, as endowed with a certain degree of substantiality, however different it may be from all ordinary matter. (Lorentz, 1906)

Thus Lorentz was correct;

In order to explain this absence of any effect of the Earth's translation, I have ventured the hypothesis, that the dimensions of a solid body undergo slight change when it moves through the ether. (Lorentz, 1906)

Most profoundly, Lorentz first deduced the foundations of Albert Einstein's Relativity from the assumption of a rigid Space (ether), and that the cause of the electromagnetic field effect that he was using was in fact vibrations in this Space / Ether.
Though Albert Einstein related relative motions of matter only to other matter and not back to an absolute Space like Lorentz did, (which is mathematically simpler I suppose) the important point is that the Logic of Relativity is founded on, and completely consistent with, an Absolute Space. (Contrary to current opinions)

From Lorentz's purely mathematical foundation (of the change in matter's ellipsoidal shape with motion) Albert Einstein then developed his Theory of Relativity, which assumed that matter existed as a spherical spatially extended field which changes ellipsoidal shape with motion and thus also with acceleration (which leads to the ellipsoidal geometry which underpins General Relativity and gravitation).
Albert Einstein took one further step than Lorentz though, and assumed (like Leibniz and Mach) that all motion of matter was relative only to other matter, he writes;

It has, of course, been known since the days of the ancient Greeks that in order to describe the movement of a body, a second body is needed to which the movement of the first is referred. (Albert Einstein, 1919)

By doing this Albert Einstein effectively renounced the concept of a fundamental Space separate from matter (as a field), as he explains below;

Since the field exists even in a vacuum, should one conceive of the field as state of a 'carrier', or should it rather be endowed with an independent existence not reducible to anything else? In other words, is there an 'aether' which carries the field; the aether being considered in the undulatory state, for example, when it carries light waves? The question has a natural answer: Because one cannot dispense with the field concept, it is preferable not to introduce in addition a carrier with hypothetical properties. (Albert Einstein, 1950)
Physical objects are not in space, but these objects are spatially extended. In this way the concept 'empty space' loses its meaning.
The field thus becomes an irreducible element of physical description, irreducible in the same sense as the concept of matter (particles) in the theory of Newton. (Albert Einstein, 1954)

By using Albert Einstein's own words it is now possible to show that his ideas need only a slight modification - from his foundation that matter is a spherical spatially extended 'field', to a foundation based upon Space rather than matter, and that matter is caused by Spherical Standing Waves in Space.
Albert Einstein is correct in asserting that matter is spherically spatially extended, and thus to reject the concept of the particle;

According to general relativity, the concept of space detached from any physical content (matter, objects) does not exist. The physical reality of space is represented by a field whose components are continuous functions of four independent variables - the co-ordinates of space and time. Since the theory of general relativity implies the representation of physical reality by a continuous field, the concept of particles or material points cannot play a fundamental part, nor can the concept of motion. The particle can only appear as a limited region in space in which the field strength or the energy density are particularly high. (Albert Einstein, 1950)

Albert Einstein is nearly correct when he says that the particle can only appear as a limited region in Space in which the field strength / energy density is particularly high, for this is simply the high Wave-Amplitude & Density of the Wave-Center of the Spherical Standing Wave. This obviously explains why Matter can never exceed the Velocity of Light. As the 'particle' is in fact the Wave-Center of a SSW, it is impossible for this Wave-Center to ever move faster than the velocity of the incoming waves, which is the velocity of light.

Unfortunately Albert Einstein incorrectly assumed that a mathematical description of effects, the spherical, spatially extended continuous force field, was the best way of representing reality. In fact these force field effects are caused by the changing velocity of the In-Waves which determine the future position of the Wave-Center (and thus the apparent force and accelerated motion of the particle). In essence the field theory is a continuous mathematical approximation of effects which are caused by many discrete (quantum) standing wave interactions. Hence the Wave Structure of Matter (WSM) explains the 'cause' of both the 'field' and the 'particle effects'.

Though most of Albert Einstein's discussion of Space is in terms of matter interactions described by fields, it is important to realise that Albert Einstein actually knew that Space must somehow exist and have properties that caused these force fields, he writes;

Recapitulating, we may say that according to the general theory of relativity space is endowed with physical qualities; in this sense, therefore, there exists an ether. According to the general theory of relativity space without ether is unthinkable; for in such space there not only would be no propagation of light, but also no possibility of existence for standards of space and time (measuring-rods and clocks), nor therefore any space-time intervals in the physical sense. (Albert Einstein, Leiden Lecture, 1920)

In ending this summary of Special Relativity, it is important to acknowledge the great power of this mathematical theory, as Albert Einstein explains (for it leads directly to Albert Einstein's famous E=mc2). But now we realise that this equivalence of Matter and Energy is simply because they are both manifestations of the same thing, Wave-Motion of Space.

The heuristic method of the special theory of relativity is characterized by the following principle: only those equations are admissible as an expression of natural laws which do not change their form when the co-ordinates are changed by means of the Lorentz transformation (covariance of equations with respect to the Lorentz transformations). This method led to the discovery of the necessary connection between momentum and energy, between electric and magnetic field strength, electrostatic and electrodynamic forces, inert mass and energy; thus the number of independent concepts and fundamental equations was reduced. (Albert Einstein, 1934)

### Links to Deduction of Pythagoras' Theorem / Information / Biography of Pythagoras

http://www.cut-the-knot.org/pythagoras/index.shtml - Pythagorean Theorem and its many proofs
Pythagoras, biography. Eric's Treasure Trove features more than 10 proofs; A proof of the Pythagorean Theorem by Liu Hui (third century AD) An interesting page.

http://www.sunsite.ubc.ca/LivingMathematics/V001N01/UBCExamples/Pythagoras/pythagoras.html - UBC Mathematics Department. An Interactive Proof of Pythagoras' theorem. This Java applet was written by Jim Morey .Home page of the grand prize winner in Sun Microsystem's Java programming contest in 1995.

http://www.math.ubc.ca/~morey/java/pyth/ - Pythagoras' Haven. The following window shows a geometrical proof of Pythagoras' Theorem.

http://www.arcytech.org/java/pythagoras/ - The Pythagorean Theorem. This lesson will allow you to figure out the Pythagorean Theorem all by yourself.

http://scidiv.bcc.ctc.edu/Math/Pythagoras.html - Pythagorean Theorem. Pythagoras was a teacher, a philosopher, a mystic and, to his followers the Pythagorean Theorem exhibits a fundamental truth.

http://dir.yahoo.com/Science/Mathematics/Mathematicians/Pythagoras/Pythagorean_Theorem/

http://www.geocities.com/CapeCanaveral/Launchpad/3740/ - Pythagoras' Theorem - by Seth Y-Maxwell. Pythagoras was a great Mathematician who was the first to create the music scale of today. He also created theorems.

http://www.davis-inc.com/pythagor/proof2.html - An excellent animation showing proof of Pythagoras' Theorem.

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