This chapter is a simplified summary, introducing contents of this paper. Your understanding is appreciated.

It seems that it is not possible to solve the mystery of Nature by dissecting the puzzle into even smaller pieces or by taking an even closer look at the picture.

Instead, if we are able to realize the logic in the whole puzzle, then we can try to derive well-known and well-formulated pieces of the puzzle from that whole picture. If we are able to compose a whole picture, which is logically self-consistent to be true and which contains and generates independent pieces of the puzzle, that whole picture should be the physics theory of everything.

First, we need a basic description of this wholeness that will lead us to a complete theory. In fact, there is a hypothesis, which develops the special and general theory of relativity and the theory of quantum mechanics. The philosophical basis of our hypothesis is as ancient as its opposite, atoms of Democritus, which treats Nature as a coincidence of scattered existing building blocks.

Geometric Generalization suggests a mathematical formulation of a major philosophical movement, which is probably first suggested by Heraclitus. This philosophy recognizes physical reality as a complete unity. According to this philosophy, physical reality is an ongoing process of constant movement and change, which is a consequence of balancing opposites.

In this chapter, we will present our hypothesis, which we will further develop in the following chapters to provide the briefest description of Nature. Simply, it is:

“Whole physical reality is a complete and continuous flux.”

To be more specific:

“All physical existence moves at the same speed, at the constant speed of light; however, final constituents of mass circulate at the constant speed of light in soft confinement volumes” (zitterbewegung).

According to this hypothesis, elementary particles with mass are circulating packages of energy, which are confined into local volumes within formations like knots or vortexes.

This chapter suggests that the basis of physical reality is a kind of non-Euclidean space-time geometry, and it examines the logical cause of this geometry. Eventually, the logical cause of our geometry that is discussed in this chapter is the principle logic of the basis of physical reality.

The Geometry of Physical Reality:

According to this paper, physical reality is based on closed space geometry, where spatial directions are curved to make great circles. This is similar to a line on the surface of a sphere, which becomes the circumference instead of extending straight into the infinity. However, this geometry fundamentally expands against the positive curvature.

The coordinate system that gives the best description of our geometry and maps the entire space-time resembles spherical coordinates (polar), where radial directions represent time dimension and circumferential directions are spatial.

Figure 1.1 The coordinate system

In our geometry, circumference (closed and curved spatial dimensions) expands with the increase in radius (time dimension). Any arbitrary point on spatial plane is static, but it flows towards time dimension, perpendicularly to spatial dimensions. This expansion is the fundamental act of physical reality. Readers may visualize this geometry by the example of an inflating spherical balloon by leaving one-dimension of the three spatial dimensions out.

Figure 1.2 Geometry of physical reality

However, flow of spatial coordinates towards time dimension is not a mechanic action that has energy content, and it is not possible to determine a rate of flow or speed of expansion. Simply, coordinates in radial (time) direction has sequential precedence (because of the decrease in curvature); while all coordinates are equivalent in circumferential (spatial) directions (at a time).

The Cause

We will discuss Euclid’s (fifth) parallel postulate in more detail, and conclude that elliptic, Euclidean, or hyperbolic geometries are not logically self-consistent (causeless) to be the basis of physical reality on their own.

As a result, we will suggest that the geometry of physical reality is a consequence of balancing the opposite properties of non-Euclidean geometries (closeness and openness). These opposite properties become the logical cause of each other’s existence, and they can only be real together by causing each other.

The most important point of this geometry is that space expands fundamentally (flux) because of its logical cause against the curvature, not because it has energy content like a wound clock.

Our space-time geometry is not only an empty space where existing things can be, but also it is the basis of the formation of the existing things. This chapter discusses mass and energy.

The Universal Strain on the Expansion – Wrinkling Epoch

Physically, our abstract expanding space represents the early inflationary epoch. However, the expanding space has collapsed onto itself, and the increase in the circumference’s (space) size has decelerated. This epoch is what we practically observe as Hubble’s cosmic expansion.

The universal strain on the expansion can be visualized by an analogy of a growing tree. Growth rate of overall size of the tree will decrease, when branches start to grow in curled and jammed helical paths, instead of growing outwards in radial directions linearly. Hence, it is important that the increase in circumference’s size (growth of overall size of the tree) and the growth of branches are different phenomena, which are related to each other by the amount of curling and jamming in growth paths of branches.

Since, our fundamental expansion is a result of its logical cause only, the mechanism that compresses expansion is not a reason to decrease the expansion inherently. The expansion is constant and continuous in the wrinkling epoch. However, space expands on locally deformed (curved and twisted) paths by folding onto itself as if space-time locally wrinkles, ripples, buckles, bends, etc. In fact, these local strain formations (deformations) on the expanding space are the quanta of energy and matter.

Figure 1.3 The inflationary epoch to Hubble’s expansion

This figure (1.3) visualizes the core idea of this paper. The original radial directions of the inflationary epoch are no longer perpendicular to the circumference (spatial directions) in the wrinkling epoch. They either circulate locally and form quanta of matter or follow an oblique path in space-time. This expanding oblique path is also known as the path of light or the light-cone.

Mass and Energy

As we will discuss extensively in our paper that energy (quanta of matter and energy) is the local compression or confinement on the fundamental expansion. Simply, our concept of energy is an interpretation of Hooke’s law, which relates strain (on space-time geometry) to stress (on the expansion).

Briefly, tightness of the deformation on the expanding space (wavelength or Compton wavelength of the quantum) describes the energy content of that package. Similarly according to Hooke’s law, potential energy stored in a mechanical spring increases with the increase in deformation.

Although, this is a short summary, we should note a critical detail here: Strains and stress are not formed on a static space, and it cannot be so. Strain packages emerge on expanding geometry, and they have energy because there is a confinement or compression on the expansion.

Most simply, quanta of energy (photons) are like surface wrinkles whose stress is directed towards spatial directions. On the other hand, quanta of matter are strain formations like knots (hadrons) and vortexes (leptons), whose stress is directed towards perpendicular time dimension.

Figure 1.4 Photo of a knot on balloon

Formation of matter and energy can be visualized by an example, where a balloon has knots tied on its surface. Simply, knots that are tied on the surface of the balloon represent matter, and surface wrinkles that emerge around knots represent electromagnetic field. Please note that there are many important differences in our balloon example, such as the balloon surface is made from an elastic substance (medium), and there exists a pressure difference between the interior and the exterior of the balloon. On the other hand, our medium is a plain space, and it is not able to contain such properties of stress, wall tension etc.

Eventually, the formation of the knots causes the universal strain on the expansion, since the knots behave like mechanical springs, which lock and confine the expansion into local volumes. Therefore, there exists a mutual relationship between the total stress in the knots (amount of matter content in the universe) and the ratio of the universal strain on the expansion (e.g. Hubble’s constant). Similarly, each knot tied on our balloon’s surface decreases the circumference of the balloon, and increases its inner pressure.

In this chapter, we will examine the concepts of distance and time in the geometry of physical reality and we will derive relativistic transformations as a natural result of our hypothesis.

Unit definitions in our modern unit systems have been chosen arbitrarily considering the earth and human scale. Naturally, this arbitrariness may seem to be the only way at first. For example, in Euclidean geometry, distance can only be measured by comparing it to another arbitrarily chosen standard distance (metric), because Euclidean space does not have any intrinsic properties, which can be taken as a standard metric.  

However, in reality, such an arbitrary choice of the unit standards of the physical concepts like distance, time, mass, energy, etc. ignores the principles of Nature. First, all of these basic concepts are directly related and dependent on each other. Second, our space-time geometry and its knots and vortexes have intrinsic structural properties, which strongly stand to be accepted as a standard.

Moreover, variations in these structural properties are the reason for the relativistic mechanism in Nature. Energy content of the matter quantum (mass) and time as quantity of clock-ticks (but not time as a dimension) emerge as a local property of the knots and the vortexes, and they change as a function of the tightness of the internal confinement (circulating - zitterbewegung) mechanism.

In this chapter, we will discover how our definitions intrinsically generate the constancy of speed of light, and derive (gamma of) relativistic equations. This chapter explains the mechanism (principle) of relativity at the smallest scale, and it physically formulates the concept of simultaneity along with Einstein’s imaginary elements called “measuring rods” and “clocks”.

Three versions of the derivations gamma of Lorentz Transformation Equations and re-construction of N.D. Mermin’s “light clock” experiment are presented.

This chapter details the formation of fundamental forces and gravity, and it is the longest chapter of our paper. However, we will keep its introduction very brief by glancing at the stress relations in our balloon example only. (Here, we will ignore all the differences of the balloon example, but please note that our space-time is not an elastic substance with static knots on it.)

Simply, surface wrinkles that emerge around the knots or the vortexes illustrate the electromagnetic field.

Figure 1.5 Photo of a knot on a balloon and wrinkles around it

Please note the direction of the crests that are formed around the knot here (figure 1.5). These wrinkles do not resemble water waves that propagate outwards from the point source, forming a series of concentric circles. Here, the balloon’s surface is wrinkled as if there is circumferential stress around the knot. Crests of these wrinkles extend and flow in radial directions to the knot or to the source.

When the relaxation of wrinkles around a knot or a vortex (with charge) is intercepted by another nearby knot or vortex (with charge), this interception affects the stress content in these knots and vortexes (electric potential). Fine structure constant represents the formation density of these surface wrinkles, which is related to the ratio of the universal strain on the expansion.

The locking stress mechanism in balloon knots symbolizes the nuclear (strong and weak) forces. Simply, the tendency to expand in the knot compresses itself, and keeps the knot locked.

Our balloon example is imperfect here because this knot on the balloon’s surface is tied and stabilized with a small string at the base of the knot. These balloon knots tend to unknot immediately, and they can remain stable only if the pressure/wall tension in the knotted volume properly balances the overall balloon pressure. Practically, this relation has a very important consequence: It is the reason why certain types of the knots or the vortexes (elementary particles) always have the same mass (ignoring the relativistic cases).

Finally, the mutual relationship between the (local) stresses in knots and (global) strain (geometric deformation) in overall state of the balloon represents the gravitation. In our balloon example, the knot deforms the geometry of the balloon and varies the wall tension around it.

Figure 1.6 Photo demonstrating how a knot on a balloon deforms its shape

This paper’s approach is similar to general relativity, however this paper does not assume that space-time can have a capacity to resist being curved or deformed, but it assumes that the expansion resists compression. Additionally, this paper clearly distinguishes the effect of the existence of a single knot on space with the secondary but the amplified effect of the lumping and gathering of knots and vortexes (matter).

In this chapter, we will also examine the perfect balance in Nature. In our balloon example, the total stress in knots and the shrinkage in balloon’s circumference (increase of pressure inside the balloon) are connected to each other. Similarly, we will discuss that the matter-energy content in the universe was not determined initially, and it was not scattered because of a mysteriously embedded energy potential (the big bang). Instead, matter content in the universe was formed during the wrinkling epoch against the universal strain on the expansion.

As a result, Hubble’s constant emerges as a function of the amount of the total matter content in the universe, and it only varies with the variation of the amount of the total matter content in the universe (which has mostly happened at the end of the inflationary epoch). We will also examine the two opposite mechanisms that affect the rate of Hubble’s expansion within physical reality, and we will discuss how Geometric Generalization simply explains the “dark matter” and “dark energy” mysteries without suggesting unobservable things.

We will conclude that energy is always conserved, since the total energy and mass content in the universe always equals zero by balancing the largest against the sum of the smallest.

Our knots and vortexes are not hard objects (e.g. atoms like marbles) whose existence can be observed directly. Instead, they are the integral constituents of space-time continuum.

The local strain formations like knots or vortexes are volumes, where the expansion in space-time geometry is confined and compressed. These (confinement) volumes are soft volumes; in fact, the uncertainty range in Heisenberg’s equation describes the size of that knot or vortex.

With the universal strain on the expansion, local strains are formed in the expanding space geometry. These strains create local deformations in space-time geometry like vertical displacements of buckles, wrinkles etc. Geometric Generalization accepts the whole strain package itself as the basic constituent of physical reality, and assumes that the wave function describes the magnitude of (vertical) deformation at a location at a time in a strain formation, instead of probability of finding a “point particle” at a location at a time.

Therefore, energy or mass does not emerge as a property of a point particle, but the property of the entire strain on space-time. However, energy in the strain formation is released at a point similar to a mechanical spring, which breaks down at a single point under overload.

However, our definition of physical existence is based on the existence of a vertical deformation (like wrinkle, curvature, buckle etc.) on the expanding space geometry, since it is the deformation, which practically exists at a location.

Please note that formulations of our strains are rather different from waves. In strains, vertical displacements are related to the deformations (contractions) in longitudinal directions, which are ignored in wave equations.

Finally, this chapter contains our paper’s most difficult section to comprehend. We will examine why even a complete strain (energy) package itself does not have a certain location in space-time. Simply, we will conclude that in the wrinkling epoch, the location that a strain package can exist expands equivalently and isotropically with the expansion. In other words, it is the location itself, which is not local in the wrinkling epoch of space-time.

In the last section of this chapter, we will also take an overview of the philosophical consequences of our deterministic approach to Quantum Mechanics, and we will examine the possibility of free will.

Finally, in this chapter, we will conclude Geometric Generalization.

First, we will summarize this paper’s philosophical deductions, and we will examine the ultimate question of philosophy: Why is there something instead of nothing? Eventually, we will conclude that there is no absolute existence of material things at all, but existence is formed as a phase of nothingness by deviating from the nothingness-flatness state towards a balanced oppositeness.

Afterwards, we will sum up the basic principles of Nature, list those points that should be included in the formulation of quantum mechanics, and we will expand our hypothesis into its full physical meaning, which stands for the briefest formulation of the mechanism of Nature.

The third section of this chapter is a striking one; it examines the exact meaning of relativity, and then, it suggests the connection between gravitation and quantum mechanics.

We will also present the list of properties of the universe at large scale in this chapter.

According to our paper, matter is a chain of geometric deformations (knots and vortexes) on the continuous space-time geometry (similar to knittings), instead of a heap of point “particles” with uncertain locations.

In fact, quantum of matter and energy (elementary “particles”) can be classified mathematically according to a few basic principles (curvature and torsion). In this chapter, we will examine these principles and discuss the basic classification of elementary “particles”.

Afterwards, we will present the physical meanings of the concepts like charge, spin, exclusion, etc., and we will explain why and how they form.

Please note that it is possible to produce many kinds of elementary “particles” by reapplying basic geometric deformations like the repeated folding of a piece of paper, like origami.

This chapter presents one of the most practical deductions of our paper.

According to our unified point of view, all basic physical concepts like distance, time, mass, energy, etc. are directly related to each other. However, our contemporary unit systems (e.g. SI) are human and earth size centric. They were not designed considering the basic principles of Nature. Therefore, many constants appear in equations of physics. Equations full of constants make it very hard to comprehend laws of Nature, and basic concepts of Nature seem to be unrelated and independent.

In this section, we will clearly identify constants, some of which emerge because of the inconsistencies in unit systems as opposed to others that represent the current state of balance in the universe.

Finally, we will suggest our Universal (natural) Unit System according to our unified point of view.

This was the quick overview. Further chapters give a detailed explanation of this theory with essential discussions on many topics.

Next chapter is “Philosophy and Methodology”…