CHAPTER I: NORMAL SYMMETRY

Willing and the Physical World
Fractal Geometry and Group Theory
Energy Conservation and Classical Mechanics

And of all things We created two pairs, that you may be reminded! Holy Quran, [51:49] - as it is symbolically emphasized by the numbers 51-49, we shall see that, despite the perfect symmetry, the two pairs are not identical, but they complement each other just like male and female, and that s due to the indivisible unit of the Single Monad that cannot be on both sides at the same time!

Since the beginning of physics, symmetry considerations have provided us with an extremely powerful and useful tool in our effort to understand nature. Gradually they have become the backbone of our theoretical formulation of physical laws. Tsung-Dao Lee, Particle Physics and an Introduction to Field Theory (1981), 177

How could it be that the same technique applies to the Internet, the weather and the stock market? Why, without particularly trying, am I touching so many different aspects of many different things? Benoit Mandelbrot, Interview at The Edge (20 December 2004)

In most of the exciting scientific discoveries throughout the previous century, symmetry was the principal feature that framed the mathematical models and motivated novel experiments. Most of the modern theories in physics and mathematics, are stimulated by symmetry, which subsequently predicted further symmetries that eventually led to more advanced models. The cosmos, however, does not seem to be perfectly symmetrical; otherwise, we would not be seeing all this consistent spectrum of varieties and differences, even between individuals of the same kind. Therefore, symmetry has to be broken at various levels in order to produce these evident variations in nature, which are the result of small recurring fluctuations around some critical point, which then determine how the system develops along the course of time. Any small deviation from the original symmetry could result in large transformations at later stages, as it is usually exemplified by the butterfly effect.

Therefore, symmetry breaking is the critical process behind the formation of patterns, through a series of apparently chaotic transactions that are very sensitive to the initial conditions, which are then determined dynamically as the system interacts with the surrounding environment. For this reason, these chaotic systems are described by fractal geometry, because the mathematical models must involve a relation with iterative computing, where the output is repeatedly applied to determine the new input. This is the main reason why fractal dimensions are main characteristics of all patterns in nature, in both their outward physical features and other intangible behaviors [[19]].

In addition to the abundance of all kinds of fractal structures in nature, there are also many astronomical observations which show that the whole Universe exhibits a fractal aspect over a fairly wide range of scales, and that large-scale structures are much better described by a scale-dependent fractal dimension. For this reason there has been many attempts to describe space-time itself in terms of fractals, and this is where the Duality of Time Theory has succeeded.

Due to the dynamic formation of dimensions , in the inner levels of time, the Duality of Time Theory leads to granular and self-contained space-time with fractal and genuinely-complex time-time geometry, which are the key features needed to accommodate both quantum and relativistic phenomena. Many previous studies have already shown how the principles of Quantum Mechanics can be derived from the fractal structure of space-time, but they either do not justify the use of fractals, or they are forced to make new unjustified assertions that may lead to fractal space-time.

In the Duality of Time Theory, since the dimensions of space and matter are being re-created in the inner (complete) levels of time, the final dimension becomes multi-fractal and equals to the dynamic ratio of inner to outer times, which is nothing but the normalized speed of light in the corresponding medium, and it approaches  in absolute perfect vacuum. For the same reason also, space-time becomes genuinely complex , since both its real and imaginary components have the same nature of time, which itself becomes as simple as the recurrence , or counting the number of geometrical nodes as they are re-created in one chronological sequence. Without postulating the inner levels of time, both the complex and fractal dimensions would not have any genuine meaning, unless both the numerator and denominator of the fraction, and both the real and imaginary parts of the complex number, are all of the same nature.

In this manner, normal time is an imaginary and fractional dimension of the complete dimensions of space, which are the real levels of time. Because they are complete integers, the dimensions of space are mutually perpendicular, or spherically orthogonal, on each other, which is what makes the perfectly symmetrical (isotropic and homogeneous) Euclidean geometry, that can be expressed with normal complex numbers , in which the modulus is given by . In contrast, because it is fractional, incomplete or non-integer, dimension, (normal, or the outer level of) time is hyperbolically orthogonal on the dimensions of space, and thus expressed by the hyperbolic split-complex numbers , in which the modulus is given by . This complex hyperbolic geometry is the fundamental reason behind relativity and Lorentz transformations, and it provides the required tools to express the curvature and topology of space-time, away from Riemannian manifolds, in which the geometry becomes ill-defined at the points of singularities.

Infinities occur because the observer is situated inside the spatial dimensions, which then appear to be continuous and infinite, while the whole space forms only one discrete state in the encompassing outer arrow of time. Since it is a fractional dimension of space, when the outer arrow of time is complemented by the orthogonal arrow, they produce together a new homogeneous spatial dimension, which is perfectly symmetrical. Inversely, when a complete dimension of space is split into two orthogonal time directions, symmetry is broken and physical configurations can now form as fractal structures that have dimensions ranging from  to .

As we shall see in Chapter III, General Relativity is the first approximation for inside observers, necessarily situated within the  spatial dimensions, but since the Universe is evolving we need to describe it by hyperbolic numbers , from the time perspective. So General Relativity is valid every single instance of time, because the resulting instantaneous space is continuous, but when the outward time flows these instances will form a series of discrete states that should be described by Quantum Field Theory. If we combine these two perspectives properly, we can eliminate the singularities from General Relativity as well as infinities from Quantum Field Theory. In the Duality of Time, this is achieved very easily because, being fractal dimensions, each of the two arrows of time is expressed as infinite divergent series, but because they are complementary, they always sum up to unity, since each one is the reciprocal of the other. As we have seen in the Introduction, this critical concept has been referred to in most ancient cosmologies, as the splitting between heavens and earth, but in Islamic cosmology this is a creative process that is happening perpetually at every instance of time.

Actually, the word for heart in Arabic is qalb , whose root means reciprocal or inverting, is directly related to symmetry and mirroring, because the process of perception occurs in the heart as a result of the meeting between the two complementary physical and psychical worlds, as we shall discuss further in Chapter II. So the heart is like a mirror between these two worlds and each side appears in it inverted into the other, just as we see our inverted image in the mirror. So the breaking of symmetry is not an isolated event that happened at some point in the early development of the Universe, but it is continuously occurring, and it is the fundamental reason for the emergence of (the outer) time. We shall also see in Chapter II that, when we take this fact into account, super-symmetry will become already included in the Standard Model, as the same normal symmetry between particles and anti-particles, which will also explain the evident asymmetry between matter and anti-matter.