The term **space** is used somewhat differently in different fields of study. In physics **space** is defined via measurement and the standard space interval, called a standard meter or simply meter, is defined as the distance traveled by light in a vacuum per a specific period of time and in this determination the velocity of light $c$ is treated as constant.

In classical physics, **space** is a three-dimensional Euclidean space where any position can be described using three coordinates. In relativistic physics researchers operate with the notion space-time in which matter is able to influence space (the old idea of Riemann [1]: he stated that the question of the geometry of physical space does not make sense independently of physical phenomena, i.e., that space has no geometrical structure until we take into account the physical properties of matter in it, and that this structure can be determined only by measurement. In his opinion the physical matter determined the geometrical structure of space.)

In astronomy, **space** refers collectively to the relatively empty parts of the universe; any area outside of a celestial object can be considered as space.

In microscopic physics, or quantum physics, the notion of **space** is associated with an "arena of actions" in which all physical processes and phenomena take place. And this arena of actions we feel subjectively as a "receptacle for subjects". The measurement of **physical space** has long been important. The International System of Units (SI) is today the most common system of units used in the measuring of space, and is almost universally used within physics.

However, let us critically look at the determination of **physical space** as an “arena of actions”. In such a determination there exists, first, subjectivity and, second, objects themselves that play in processes that can not be examined at all (for instance, size, shape and the inner dynamics of the electron; what is a photon?; what are the particle’s de Broglie wavelength $\lambda$ and Compton wavelength $\lambda_{\rm Com}$?; how to understand the notion/phenomenon “wave-particle”?; what is spin?; what is the mechanism that forms Newton’s gravitational potential $Gm/r$ around an object with mass $m$?; what does the notion ‘mass’ mean exactly?, etc.).

Especially interesting are three next examples of the motion “on the arena of action, as a reservoir for objects”:

- When a vehicle suddenly jams on the brakes, an experienced physicist sitting in the vehicle will feel that something pushes him forward;
- Our experienced physicist comes to a playground and decided to go on the merry-go-round. The physicist raises the marry-go-round to a high speed and jumps onto it. However, holding hands tightly over the merry-go-round, he suddenly feels that something unseen grabs his legs pulling them out of the merry-go-round;
- The experienced physicist wished to have fun with a gyroscope and taking it in his hands, when the gyroscope’s rotor reached the speed of about 20 thousand revolutions per minute, he feels that this quietly functioning plaything for some reasons goes out of hand, which injures his muscles and tendons of his arms.

These three examples clearly give evidence of the existence of otherworldly forces at the scene of action among normal subjects.

However, this “arena of actions” can be completely formalised, such that those mystical forces (veiled under the force of inertia and the centrifugal force) will unravel explicitly, because fundamental physical notions and interactions are to be derived from pure mathematical constructions.

It is interesting to read Professor Vernadsky's work who back in 1920-1930s introduced the notion of *noosphera* [2] (from Greek *nous* — mind and *sphaira* — ball): a sphere of the arena of interaction between people and nature. In particular, he mentioned that Helmholtz probably was the first who noted that geometric space did not embrace all of empirically studied space—what Helmholtz called **physical space**; Helmholtz distinguished physical space from geometric space, as possessing its own properties, such as right-handedness and left-handedness; besides, Poincaré observed that geometry could not have been developed without solids. Further Vernadsky notes: "In discussing the state of space, I will be dealing with the state of empirical or physical space, which has only in part been assimilated by geometry. Grasping it geometrically is a task for the future". Vernadsky introduced such notion as *the state of space*, which in his opinion has to be closely connected with the concept of a physical field, which plays such an important role in contemporary theoretical physics.

All this means that **physical space** is a peculiar substrate that is subject to certain laws, which as has been seen below are purely mathematical. Such a view allows us to completely remove any subjectivity and all the figurants of fundamental physical processes will be hundred percent defined. The present article is dedicated to an elucidation of those somethings that form a primordial physical substrate and the determination of its mathematical properties.

## Starting point for a deeper understanding

A real physical space – is the term that stands behind such vague undetermined notions as Newton’s space (according to Newton, space was densely filled with solid balls), as a physical vacuum or ether, or ‘energofluid’. This last one is becoming more and more popular among high-energy physicists who have started to determine mass through the notion of energy, $m^2 = E^2/c^4 - p^2/c^2$, rejecting such fundamental things as the rest mass and the dependence of mass on velocity. This is due to the specificity of quantum chromodynamics in which quarks are massless particles, although protons, neutrons and so on are objects with concrete masses.

However, this viewpoint completely rejects such disciplines as gravity and, in particular, Newton’s gravitational law. Abstract methods of modern quantum theories do not allow one to resolve the problem of the structure of physical vacuum. High-energy physics operates with notions experimentally obtained at the scale around the size of the atom $10^{-10}$ m, but then tries to extrapolate them up to $10^{-30}$ m where all kinds of fundamental interactions have to coincide. At the same time no less fundamental interactions – microscopic quantum mechanical and macroscopic gravitational – have already been lost somewhere when we approach the scale of the size of the atom. So it is clear that extrapolation does not help us to resolve the problem of the structure of physical space.

Moreover, the high-energy physicists focus on interactions between particles, but the study of quantum objects that generate these interactions, i.e. canonical particles and their interaction with space remain outside the attention of researchers. Below we explain the reasons why the interaction of quantum objects with space needs to be taken into account.

## Structure of mathematical space

So far in mathematics, a space is treated as a set with some particular properties and usually some additional structure. It is not a formally defined concept as such but a generic name for a number of similar concepts, most of which generalize some abstract properties of the physical concept of space. Distance measurement is abstracted as the concept of metric space and volume measurement leads to the concept of measured space.

Generalisation of the concept of space can be done [3,4,5,6] through set theory, topology and fractal geometry, which will allow us to look at the problem of the constitution of physical space from the most fundamental standpoint.

The fundamental metrics of our ordinary space-time is a convolution product in which the embedded part $\rm D4$ looks as follows:

(1)where $\rm dS$ is the element of space-time, $\rm d \Psi (w)$ is the function that accounts for the expansion of 3-D coordinates to 4-th dimension through the convolution $\ast$ with the volume of space.

Set theory, topology and fractal geometry allow us to consider the problem of structure of space as follows. According to set theory only an empty set $\oslash$ can represent nothing. Following von Neumann, Bounias considered an ordered set, $\{ \{ \oslash, \{ \oslash \} \}, \{ \oslash, \{ \oslash, \{ \oslash \} \}, \{ \oslash, \{ \oslash, \{ \oslash, \{ \oslash \} \}\}\}\}$, $\{ \{ \oslash, \{ \oslash \} \}, \{ \oslash, \{ \oslash, \{ \oslash \} \}, \{ \oslash, \{ \oslash, \{ \oslash, \{ \oslash \} \}\}\}\}$ and so on. By examining the set, one can count its members: $\{ \oslash \} =$ = zero, $\{ \{\oslash, \{ \oslash \}\} = 1$, $\{\oslash, \{ \oslash, \{ \oslash \}\} = 2$, $\{ \oslash, \{ \oslash, \{ \oslash, \{ \oslash \}\}\}\} = 3$,… This is the empty set as long as it consists of empty members and parts. On the other hand, it has the same number of members as the set of natural integers, $N = {0, 1, 2, ..., n}$. Although it is proper that reality is not reduced to enumeration, empty sets give rise to mathematical space, which in turn brings about physical space. So, something can emerge from emptiness.

The empty set is contained in itself, hence it is a non-well-founded set, or hyperset, or empty hyperset. Any parts of the empty hyperset are identical, either a large part $(\oslash)$ or the singleton $\{ \oslash \}$; the union of empty sets is also the same: $\oslash \cup (\oslash) \cup \{ \oslash \} \cup \{ \oslash, \{ \oslash \} \} \cup \{ \oslash, \{ \oslash \}, \{ \oslash, \{ \oslash, \{ \oslash \}\}\} \cup ... = \oslash$.

This is the major characteristic of a fractal structure, which means the self-similarity at all scales (in physical terms from the elementary sub-atomic level to cosmic sizes). One empty set $\oslash$ can be subdivided into two others; two empty sets generate something $(\oslash) \cup (\oslash)$ that is larger than the initial element. Consequently, the coefficient of similarity is $\rho \in [1/2, \ 1]$. In other words, $\rho$ realizes fragmentation when it falls within the interval $]1/2, 1[$ and the union of $\rho$ with interval $]0, 1/ [$ gives $]0, 1[$ . The coefficient of similarity $\rho$ allows us to estimate the fractal dimension of the empty hyperset; since this dimension contains the interval $]0, 1[$ as one of its components, it turns out that it is a “fuzzy” dimension.

4-D mathematical spaces have parts in common with 3-D spaces, which yields 3-D closed structures. There are then parts in common with 2-D, 1-D and zero dimension (points). General topology indicates the origin of time, which should be treated as an assembly of sections $S_i$ of open sets (Poincaré sections).

Due to fuzzy dimensions generated by fractality, the general part of a pair of open sets $W_q$ and $W_l$ with different dimensions $q$ and $l$ also accumulate points of open space. For instance, it is impossible to put a pot onto a sheet without changing the shape of the 2-D sheet into a 3-D packet. Only a 2-D slice of the pot can be a part of a sheet. Therefore, infinitely many slices, i.e. a new subset of sections with dimensionality from 0 to 3, ensure the raw universe in its timeless form.

Primary topology is a topology of open sets (in particular, the empty set $\oslash$ is an open set, but its topological ball is not open). That is why primary topology cannot be a physically measured space. However, the availability of closed intersections (timeless Poincaré sections) of abstract mathematical spaces creates properties typical for a physical space. What happens to these sections $S_i$ if all belong to an embedding 4-space? A series of sections $S_{i}$, $S_{i+1}$, $S_{i+2}$, etc. resembles the successive images of a movie, and only nothing does not move. Therefore, the difference of distribution of objects within two corresponding sections will mean a detectable increment of time. Hence *time* will emerge from order relations holding onto these sections. And hence space-time acquires a topological discrete structure.

## Measure, distance, metric and objects [3,4,5,6]

The concept of measure usually involves such particular features as existence of mappings and the indexation of collections of subsets on natural integers. Classically, a measure is a comparison of the measured object with some unit taken as a standard. The “unit used as a standard” is the part played by a gauge $J$. A measure involves respective mappings on spaces, which must be provided with the rules $\cup$, $\cap$ and $\complement$.

Any space can be subdivided in two major classes: objects and distances. In spaces of the ${\rm R}^n$ type, tessellation by balls is involved, which again requires a distance to be available for measurement of diameters of intervals. Intervals can be replaced by topological balls, and therefore evaluation of their diameter still needs an appropriate general definition of a distance.

In physics, a ruler is called a metric. As a rule, mathematical spaces including topological spaces have been treated as not endowed with a metric, and properties of metric spaces have not been the same as those of non-metric spaces. However, Michel Bounias [3,4] showed that all topological spaces are metric. In fact, union and intersection allow the introduction of the symmetric difference between two sets $A_i$ and $A_j$

(2)i.e. we have the complementary of the intersection of these sets in their union. Symmetric difference satisfies the following properties: $\Delta (A_i, A_j)=0$ if $A_i = A_j$, $\Delta (A_i, A_j)= \Delta (A_j, A_i)$ and $\Delta (A_i, A_k)$ is contained in union of $\Delta (A_i, A_j)$ and $\Delta (A_j, A_k)$. This means it is a true distance and can also be extended to the distance of three, four and so on, sets in one.

The complementary $\rm \cup _{i\neq j}(A_i \cap A_j)$ is closed in a closed space; it is also closed even when it includes open components with non-equal dimensions. In this system the characteristic $\rm \mathfrak{m} \langle \{A_i\} \rangle = \cup _{ i\neq j}(A_i \cap A_j)$ was called “instant” by Bounias, because it is responsible for the state of objects in a timeless Poincaré section. Since distances $\Delta$ are complements to the objects, the system looks like a manifold of open and closed parts. The mapping of these manifolds from one to the other section, which preserve the topology, corresponds to a frame of reference in which topology will describe significant changes in the configuration of some components. If morphism takes place, we can compare the state of the section with the state of mapping of this section and these changes can be interpreted as a phenomenon similar to motion.

Since the definition of a topology implies the definition of the mentioned set distance, every topological space is endowed with this set metric. The norm of the set metric is $|| A|| = \Delta (\oslash, A)$. Therefore, all topological spaces are metric spaces, $\Delta$-metric spaces, and they are measurable.

Now we can explore the intersection of sets. If we have sets of non-equal dimensions, then their intersection will be closed – the intersection of closed space is also closed, ${\mathop \cup _{i\neq j}}(A_i, A_j)$, which means the availability of physical objects. Since distances $\Delta$ are complements to the objects, the whole system becomes a manifold of open and closed subparts. Such a procedure subdivides a universe to two parts: distances and objects.

## Tessel-lattice and the generation of matter [4,5,6,7]

Providing the empty set $(\oslash)$ with mathematical operations $\in$ and $\subset$, as combination rules, and also the ability of complementary $( { \mathop \complement})$ we obtain a magma (i.e. fusion) of empty sets: Magma is a union of elements $(\oslash)$, which act as the initiator polygon, and complementary $(\complement)$, which acts as the rule of construction; i.e., the magma is the generator of the final structure. This allowed Bounias to formulate the following theorem:

*The magma* $\oslash^{\oslash} = \{ \oslash, \complement \}$ *constructed with the empty hyperset and the axiom of availability is a fractal lattice.*

Writing $( \oslash^{\oslash})$ denotes the magma, and reflects the set of all self-mappings of $\oslash$. The space, constructed with the empty set cells of the magma $\oslash^{\oslash}$, is a Boolean lattice, and this lattice $S(\oslash)$ is provided with a topology of discrete space. A lattice of tessellation balls has been called a tessel-lattice [4], and hence the magma of empty hyperset becomes a fractal tessel-lattice.

Introduction of the lattice of empty sets ensures the existence of a physical-like space. In fact, the consequence of spaces $(W_m)$, $W_n)$, … formed as parts of the empty set $\oslash$ shows that the intersections have non-equal dimensions, which gives rise to spaces containing all their accumulating points forming closed sets. If morphisms are observed then this enables the interpretation as a motion-like phenomenon, when one compares the state of a section with the state of a mapped section. A space-time-like sequence of Poincaré sections is a non-linear convolution of morphisms. Our space-time then becomes one of the mathematically optimal morphisms and time is an emergent parameter indexed on non-linear topological structures guaranteed by discrete sets. This means that the foundation of the concept of time is the existence of orderly relations in the sets of functions available in intersect sections.

Time is thus not a primary parameter and the physical universe has no beginning: time is just related to ordered existence, not to existence itself. The topological space does not require any fundamental difference between reversible and steady-state phenomena, nor between reversible and irreversible process. Rather relations simply apply to non-linearly distributed topologies and from rough to finest topologies.

So real physical space can be presented in the form of a mathematical lattice: the tessel-lattice is regularly ordered such that the packing has no gaps between two or more empty topological balls. This is guaranteed by a set ${\not}{\rm {\!\!} (c)}{\!\!\!\!\!}$, which does not have members and parts. Such tessel-lattice accounts for the existence of relativistic space and the quantum void (vacuum), as: 1) the conception of distance and the conception of time are defined and 2) such space includes a quantum void, because the mosaic space introduces a discrete topology with quantum scales and, moreover, it does not have “solid objects” that would appear as real matter. The tessel-lattice with these characters has properties of a degenerate physical space. The sequence of mappings from one structural state to the other of an elementary cell of the tessel-lattice generates an oscillation of the cell’s volume along the arrow of physical time. However, there is also an option of transformation of a cell under the influence of some iteration similarity that overcomes conservation of homeomorphism. For $N$ similar figures with the ratio of similarity $1/\rho$ the Bouligand exponent $\rm (e)$ is given by expression

(3)and the cell of an image changes its dimension from $D$ to $D^{\prime} = {\rm ln}N/{\rm ln} \rho = {\rm e}$ where $\rm e > 1$. A change of the dimension means an acquisition of properties of “solid” objects, i.e. the creation of matter.

Figure 1: The continuity of homeomorphic mappings of structures is broken once a deformation involves an iterated transformation with internal self-similarity, which involves a change in the dimension of the mapped structure. Here the first 2 or 3 steps of the iteration are sketched, with basically the new figure jumping from (D) to approximately (D + 1:45). The mediator of transformations is provided in all cases by empty set units.

The universe can be treated as a tessel-lattice composed of a huge number of cells or topological balls. The measure includes such notions as length, surface and volume. Because of that a loop distance $l$ of the universe (i.e. the perimeter that would be measured by means of a ruler in principle) can be related to parameters of $N$ balls.

Indeed, let $\mu$ be a measure of balls (their length, surface or volume with the corresponding dimensions $\delta$ = 1-D, 2-D or 3-D). In the middle part of the universe with the dimension $D$ we have $N$ times $\mu^{\delta}$, which equals approximately $l^{\rm D}$, so that we estimate the dimention of this part of the universe:

(4)Thus, from expression (4) we can see that at least a part of the universe having different dimension $D$ can be distinguished from the other universe, which can be perceived as the presence of dark matter there.

If we know the universe’s components, that is if we can describe sizes and shapes of topological balls, we will be able to re-establish an invisible structure of a large size.

The present theory of space predicts the formation of sub-universes, or clusters that embrace $10^x$ cells of the tessel-lattice where the exponent changes from $x = 1$ to $x=60$. Thus, $x=1$ (Planck’s scale) corresponds to the size of an elementary cell of the tessel-lattice; a radius that covers $10^{10}$ cells describes the size of a quark; a radius that includes $10^{17}$ cells depicts the atom size; a radius with $10^{21}$ cells is characterized as the molecular size; a radius that includes $10^{28}$ cells represents the humanoid size; a radius that consists of $10^{40}$ cells features the solar system scale; a radius covering $10^{56}$ cells corresponds to a cluster of a cosmic structure. Of course the universe suggests different arrangements of the organisation of matter at each of those scales.

Figure 2: A topological ball is represented as a triangle, figuring 3 dimensions in a metaphorical form. A degenerate ball keeps the same dimension in contrast with a particled ball endowed with a fractal substructure. A complete decomposition into one single ball (*k* = 1) conserves the volume without keeping the fractal dimension. The von Koch-like fractal has been simplified to 3 iterations for clarity.

The organisation of matter at the microscopic (atomic) level has to recreate a sub microscopic spatial ordering. Hence the crystal lattice is also a reflection of the sub microscopic ordering of real physical space that can be associated with the tessel-lattice of tightly packed balls – elementary bricks of the primary substrate of the universe.

In the tessel-lattice balls are found in a degenerate state and their characteristics are such mathematical parameters as length, surface, volume and fractality. Evidently, the removal of degeneracy must result in local phase transitions in the tessel-lattice, which creates “solid” physical matter. So matter (mass, charge and canonical particle) is immediately generated by space and has to be described by the same characteristics as the balls from which matter is formed. The behaviour of a canonical particle obeys submicroscopic mechanics that is determined on the Planck’s scale in the real space and is wholly deterministic by its nature. At the same time, deterministic submicroscopic mechanics is in complete agreement with the results predicted by conventional probabilistic quantum mechanics, which is developed on the atomic scale in an abstract phase space. Moreover, submicroscopic mechanics allows the derivation of Newton's law of universal gravitation and the nuclear forces starting from first sub microscopic principles of the tessellation structure of **physical space.**

**Bibliography**

[1] See Riemann's talk in: B. Riemann: *Collected Papers*, R. Narasimhan, (ed.), Springer Verlag, 1990.

[2] V. Vernadsky, On the States of Physical Space, *21 ^{th} Century Science & Technology*, Winter 2007-2008 Issue, pp. 10-22. http://www.21stcenturysciencetech.com/Articles%202008/States_of_Space.pdf

[3] M. Bounias and V. Krasnoholovets, Scanning the structure of ill-known spaces: Part 1. Founding principles about mathematical constitution of space, *Kybernetes: The International Journal of Systems and Cybernetics* **32**, No. 7/8, pp. 945—975 (2003); (also http://arXiv.org/abs/physics/0211096).

[4] M. Bounias and V. Krasnoholovets, Scanning the structure of ill-known spaces: Part 2. Principles of construction of physical space, *Kybernetes: The International Journal of Systems and Cybernetics* **32**, No. 7/8, pp. 976—1004 (2003); (also http://arXiv.org/abs/physics/0212004.)

[5] M. Bounias and V. Krasnoholovets, Scanning the structure of ill-known spaces: Part 3. Distribution of topological structures at elementary and cosmic scales, *Kybernetes: The International Journal of Systems and Cybernetics* **32**, No. 7/8, pp. 1005—1020 (2003); (also http://arXiv.org/abs/physics/0301049).

[6] M. Bounias and V. Krasnoholovets, The universe from nothing: A mathematical lattice of empty sets, *International Journal of Anticipatory Computing Systems* **16**, pp. 3-24 (2004) (also http://arXiv.org/abs/physics/0309102).

[7] V. Krasnoholovets, The tessellattice of mother-space as a source and generator of matter and physical laws, in: *Einstein and Poincaré: The physical vacuum*, Ed.: V. V. Dvoeglazov (Apeiron, Montreal, 2006), pp. 143-153 (also http://arxiv.org/abs/1010.3894).