## On the Physical Foundations of Euler's Equation

*"There is a famous formula — perhaps the most compact and famous of all formulas — developed by Euler from a discovery of De Moivre: e ^{iπ} + 1 = 0. Elegant, concise, and full of meaning … It appeals equally to the mystic, the scientist, the philosopher, the mathematician." (Edward Kasner)*

*"Gentlemen, that is surely true, it is absolutely paradoxical; we cannot understand it, and we don't know what it means. But we have proved it, and therefore we know it must be the truth." (Benjamin Peirce)*

## Introduction

I believe that I have discovered the physical foundation of Euler's famous equation, e^{ix} = cos x + i sin x. However, I do not completely understand the consequences of these discoveries, so I write this hoping others with more knowledge than I can help to understand it. While I appreciate this is a radical claim, I have discovered three related things (below), each remarkable, that leads me to this conclusion.

It is well known that Euler's equation shows two equivalent ways to define points on a circle / rotation around a circle. The left side, the exponential function e^{ix}, represents the rotation of the arrow by the angle x (in radians). The right side, the trigonometric function cos x + i sin x, uses a complex number to represent a rectangle with sides cos x and sin x, the corner of the rectangle corresponds with the arrow / points on the circle.

However, it is a deep mathematical mystery as to why we use the number e to represent rotation on a unit circle (where the length of the rotating arrow is 1).

Given that e is this profound natural number, found in the Schrodinger wave equation, and thus clearly relates to some fundamental aspect of physical reality, it seems natural to consider this number in terms of three dimensional geometry and waves. This led to me to the following three discoveries.

## Three Discoveries

1. No one seems to have realized that this equation also represents the resultant of two orthogonal plane waves 90 degrees out of phase. i.e. Cos and Sin are 90 degrees out of phase, and the imaginary number i represents a rotation of the sin wave by 90 degrees (orthogonal). (This seems obvious to me, but perhaps this is because I had been studying Schrodinger's equation, which uses this natural exponential function to represent plane waves.)

2. No one has thought to transform this equation from a 2 dimensional plane (relating circles to squares / rectangles) to a 3 dimensional volume (relating spheres to cubes). When we do this, and ask what is the volume of a sphere relative to the volume of the cube enclosed within it, we find that the ratio, symbolized by (e) equals;

This is very very close to Euler's number e = 2.7182818284

(As a ratio (e)/e = 1.0008892447)

3. When we consider the simple case of a cube with length L =1,

then the radius of the sphere is sqrt3 / 2, and most profoundly, the properties of this sphere are unique - *this is the only sphere where the volume is 4/3 π r ^{3} = π r = (e).*

Further, the circumference C = 2(e), thus numerically, we can equate a 180 degree rotation (C/2) with (e) .

## Discussion

I find the following things all very remarkable ...

These are all profound universal relationships, all simple and obvious (once known), and yet no one else seems to have noticed these (based on searching Google).

The similarity between the geometric (e) and Euler's e is so close that I find it hard to believe it is a mere coincidence.

It is strange that Euler's e is calculated using a Taylor series, which related to calculating compound interest (Bernoulli), and later, to the alternative definition of e where e^{x} = its derivative, and yet the number is used in Schrodinger's equation to represent plane waves. It makes much more sense to use the geometric (e), which I suspect is converting from plane waves in two dimensions to three dimensions, from the geometry of cubes to the geometry of spheres.

I believe this relates to physical reality, where the electron and positron can be understood as two opposite phase spherical standing waves (from this foundation mathematical physicist Milo Wolff deduced the de Broglie wavelength and relativistic mass increase due to Doppler shifts of two spherical standing waves in relative motion to one another, thus, for the first time, uniting these fundamentals of quantum physics and special relativity).

It is odd that e, which converts a discrete relationship (interest added at the end of a unit of time) to a continuous relationship (continuously compounded interest) is calculated using an infinite sum of discrete numbers (Taylor series).

However, geometric (e), as the ratio of the volume of a sphere to the cube it encloses, is a continuous (and universal) relationship, it applies to all spheres and their enclosed cubes.

Pythagoras Theorem tells us that cos^{2}x + sin^{2}x = 1

When we factorize this we get;

(cos x + i sin x) (cos x - i sin x) = 1

and

(sin x + i cos x) (sin x - i cos x) = 1

This shows a deep relationship between Euler's equation and Pythagoras equation, and thus suggests a geometric / wave foundation for e.

This is also similar to how Dirac factorized Schrodinger's equation using 3 imaginary numbers, in so doing Dirac re-discovered complex quaternions (Hamilton, Clifford) which represent 3 orthogonal planes, and led to his prediction of electron spin and the existence of the positron.

Thus it is reasonable to conclude that the electron spin is related to the interaction of orthogonal plane waves which cause space to vibrate in circular / spiral patterns (spinors).

Further, since both factors can be either positive or negative, (negative times negative = positive) thus the positron, with its negative sign in the Dirac equation, is not negative energy (as Dirac thought), nor negative time (Feynman's view that the positron is an electron moving backwards in time!), but rather, the negative sign represents the opposite phase spherical standing wave to the electron standing wave. Thus the annihilation of a positron and electron is simply the destructive wave interference of two opposite phase standing waves.

Interestingly, the only reference I found for the geometric (e) on the internet, was for the ratio of the surface area of a sphere to the surface area of the inscribed tetrahedron (triangular pyramid), which also equals (e).

There is much more to write up, much that I do not fully understand. I hope that others can help untangle this fascinating mathematics and its relationship to physical reality.

Please feel free to contact me on Facebook.

Geoff Haselhurst

https://www.facebook.com/geoffrey.haselhurst

4th August, 2017

## Help Humanity

*"You must be the change you wish to see in the world."
(Mohandas Gandhi)*

*"When forced to summarize the general theory of relativity in one sentence:
Time and space and gravitation have no separate existence from matter. ... Physical objects are not in space, but these objects are spatially extended. In this way the concept 'empty space' loses its meaning. ... The particle can only appear as a limited region in space in which
the field strength or the energy density are particularly high. ...
The free, unhampered exchange of ideas and scientific conclusions is necessary for the sound development of science, as it is in all spheres
of cultural life. ... We must not conceal from ourselves that no improvement in the present depressing situation is possible without
a severe struggle; for the handful of those who are really determined to do something is minute in comparison with the mass of the lukewarm
and the misguided. ...
Humanity is going to need a substantially new way of thinking if it is to survive!" (Albert Einstein)*

Our world is in great trouble due to human behaviour founded on myths and customs that are causing the destruction of Nature and climate change. We can now deduce the most simple science theory of reality - the wave structure of matter in space. By understanding how we and everything around us are interconnected in Space we can then deduce solutions to the fundamental problems of human knowledge in physics, philosophy, metaphysics, theology, education, health, evolution and ecology, politics and society.

This is the profound new way of thinking that Einstein realised, that we exist as spatially extended structures of the universe - the discrete and separate body an illusion. This simply confirms the intuitions of the ancient philosophers and mystics.

Given the current censorship in physics / philosophy of science journals (based on the standard model of particle physics / big bang cosmology) the internet is the best hope for getting new knowledge known to the world. But that depends on you, the people who care about science and society, realise the importance of truth and reality.

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A new scientific truth does not triumph by convincing its opponents and making them see the light, but rather because its opponents eventually die, and a new generation grows up that is familiar with it. (**Max Planck**, 1920)

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