It is time to discuss the cause of this geometry. Geometric Generalization’s viewpoint refuses the idea that the fundamental expansion character of space is a result of a potential, which is externally embedded into the system as if this expansion is caused by an explosion of energy. Since our abstract geometry does not contain matter and energy yet, there is no reason for us to assume that the expansion of space can be wound like a clock. Conversely, the expansion character is a natural result of the intrinsic property of this geometry. This geometry has an intrinsic logic, which creates the fundamental expansion character.

First, we must discuss absolute geometry so that we can describe its cause. Absolute geometry agrees with the first four postulates of Euclidean geometry, but it does not accept the (fifth) parallel postulate or any of its alternatives.

There are three commonly accepted variations of absolute geometry, which differ according to the nature of parallel lines:

From point A, which is not on the line L,

 - on elliptic geometry, there are no parallel lines through point A to line L, therefore elliptic geometry is closed.

 - on Euclidean geometry, there is only one unique parallel line through point A to line L, therefore Euclidean geometry is flat.

 - on hyperbolic geometry, there are infinite numbers of parallel lines through point A to line L, therefore hyperbolic geometry is open.

According to our daily observations, it can initially be assumed that the geometry of Nature should be Euclidean. However, it has never been possible to provide proof for Euclid’s fifth postulate, and it cannot be called self-evident like the other four postulates. Fifth postulate is just a special case of absolute geometry, and there is no reason or proof that suggests Euclidean geometry should be the choice for a basis of Nature. Additionally, perfect flatness can only exist within the condition of infinity, which is also an indefinable, abstract concept, and by no means an indisputable reality.

On the other hand, alternatives of Euclidean geometry (elliptic and hyperbolic) need additional reasons not to be flat and to be curved. For example, Einstein has given an example of a marble slab, which becomes a non-Euclidean continuum through a heating process. General relativity is also based on this idea, which suggests that space-time is curved by matter. However, it is important to notice that such a curvature (deviation from flatness) always needs an external cause (parameter) like heating or the existence of matter.

It can be presumed that the basic curvature of the geometry of physical reality is a consequence of the matter-energy content at the initial state of the universe. However, this approach inevitably concludes that matter-energy content is another primary constituent of physical reality, which has been inserted into physical reality from “elsewhere”. Geometric Generalization considers this approach as nonphysical, since it is not self-evident, and it is based on inputs or assumptions beyond definable laws of physical reality.

Curved geometries are always sub-geometry of a higher-dimensional flat geometry (even though curved geometries can be managed intrinsically by tensor calculus). For example, we may consider the surface of the Earth as a two-dimensional spherical geometry, where the shortest distance between two arbitrary points on the Earth’s surface is intrinsically curved. On the other hand, the Earth’s surface can also be defined on a higher order of three-dimensional flat geometry, where the shortest distance between these points crosses through the volume of the sphere without curvature.

Therefore, in the light of the above discussions, it is not possible that non-Euclidean geometries can exist independently.

Above discussions on the variations of absolute geometry lead us to a very interesting conclusion: There is no logical reason or a proof, which confirms that one of the commonly accepted variations of absolute geometry could stand alone, as the basis of physical reality.

In fact, there is another version of absolute geometry, which contains properties of variations of absolute geometry. Conversely, this geometry seems to be logically self-consistent (causal) to be real, unlike the other versions of absolute geometry.

This geometry has the basic properties of the two opposite alternatives of Euclidean geometry. These conflicting properties cannot be real independently by themselves (causeless); but they become the logical cause of each other’s existence. Space can be closed only if it expands, and it expands only if it is closed. These two basic conflicting properties can only be real together by causing each other.

According to Geometric Generalization, the fundamental logical principle of physical reality is a non-Euclidean geometry, and this geometry is the consequence of balancing the oppositeness in conflicting properties of non-Euclidean geometries. Hence, physical reality exists, because there is a geometry that can be logically self-consistent (rational and causal) to be real.

Interestingly, our geometry may appear to be flat (Euclidean) in large scale, since non-Euclidean opposite properties exist by causing and balancing each other correspondingly.

Please note that physical reality fundamentally involves both closeness and expansion properties even if it seems to be Euclidean in large scales after all. These properties are related with the curvature, and as we discussed above, “curved” geometries form a higher-dimensional flat geometry. Therefore, this geometry generates an additional dimensional parameter, which is perpendicular to spatial dimensions, to basic curved spatial three dimensions. This additional parameter is time dimension, or it is the radius of curvature of spatial dimensions.

Eventually, the most important point with this geometry is that space expands intrinsically because of its logical cause, not because it has built-in energy content. In other words, a constant and continuous expansion is the fundamental property of the geometry of physical reality.

The expansion explained in this chapter is the basis of the flux that is mentioned in the hypothesis. The flux is simply the constant and continuous expansion of closed spatial dimensions. As a result, our hypothesis can be expanded as “Whole physical reality is a complete and continuous flux of the expansion of space” However, at this point; it is still better to consider flux as an abstract geometric principle rather than a constituent of physical reality.

Figure 4.3 The flux – The fundamental expansion in space-time

Now, we may start to correlate our abstract geometry with the physical reality. After all, this geometry will be the picture of whole reality, and this geometry is not only an empty space where the real things can be, but also it describes what these real things are. However, at this point, we only have abstract and empty time and space dimensions. Let us keep on our adventure and discuss formation of physical concepts: “Formation of Mass and Energy”.

Readers, who want to have a deeper viewpoint on the mutual relationship between conflicting properties of non-Euclidean geometry, may examine Euclid’s second postulate: “Any straight line segment can be extended indefinitely in a straight line”. Simply, closeness causes the expansion as a must to prevent a direction to wrap around itself and to prevent a paradox. Additionally, closeness means there is a curvature in spatial dimensions; however, curvature is not a natural state in geometry, and it decreases with the expansion in spatial dimensions as a result.

Please note that this section does not discuss the mathematical consistency of Euclidean and non-Euclidean geometries. Instead, this section examines the causality of the space-time geometry, concluding that other versions of absolute geometry do not intrinsically involve the causality. According to this paper’s philosophical point of view, which we will re-examine in further chapters, it can be said that this paper accepts the concept of causality itself as a priori.