Nune Chilingaryan
email: nune.chilingaryan@lyon.archi.fr

The passéist, modernist and futurist features of some social housing ensembles built during the transitional period of 1960-1970: the case of the Maurelette in Marseille"

Keywords: big residential complexes, resilience of modernist urban concept, historical tendencies, city structure, postmodernism planning, urban design, architectural projects, reconstruction

The history of architecture and urbanism is classically represented as a sequence of major doctrines. However, the well-known architectural "-isms" are bound together with productions of the so-called "transitional" periods, the legacy of which deserves special scientific interest. A significant number of these kinds of "in-between product" in the field of social housing was built during the 1960s and 1970s, between the periods of modernism and postmodernism.

At the end of the 1980s in many European countries, particularly in France, massive renovation processes were started, which continue to this day. Due to political, social and aesthetic changes, a great deal of post-war residential heritage has been radically reconstructed or demolished. This process touches not only ordinary residential groups (so-called grands ensembles), but also harms some of the more interesting ensembles. Many of them are undervalued and have not been rehabilitated since their creation, with some often doomed to disappear.

The current paper is an attempt to analyse the historical, urban and morphological aspects of the Maurelette residential complex, built in the northern suburbs of Marseille during 1963-1965. The design particularities of the Maurelette complex demonstrate the ambition to create a "non-ordinary" ensemble using ordinary and inexpensive construction means and materials. The original interpretation of the traditional square, street and rampart could be considered as early applications of postmodernist ideas of free "expression" of historical urban forms. This case study intends to raise awareness about the heritage constructed between the major architectural periods, which deserve to be included in the contemporary urban structure without negating or altering their authentic concept. Its existence will contribute to the continued urban environment, thereby making the modern city more resilient.

Introduction

The history of architecture and urbanism is classically represented as a sequence of movements and major doctrines. However, the actual evolution of architecture and urbanism does not follow this kind of linear character. The well-known architectural "-isms" are bound together with productions of so-called "transitional" periods, the legacy of which deserves special scientific interest. Thanks to its unique mix of previous and present tendencies, as well as techniques and compositional approaches, this architectural layer is shaping a foundation for future movements. A significant number of these kinds of interesting "in-between product" in the field of social housing were built during the 1960s and 1970s. These ensembles were constructed between the periods of modernism and postmodernism, when the postulates of modernism (particularly the ones incarnated in social housing groups) were severely criticized for having poor urban and architectural design. The basic principles of the modernist movement are accused of being incomprehensible to the middle and lower classes for whom it was intended. It even goes towards the idealization of the defects of the historical city. "Sun, silence, fresh air? This is exactly the opposite that we should wish for. A city consists of little sun, and maybe bad odours and noise" [START_REF] Benevolo | Histoire de l'Architecture Moderne[END_REF] , proclaims the famous French architect, Emile Aillaud, criticizing the hygienist principles of CIAM.

By the mid-1950s, young architects were beginning to challenge the principles defined by the Athens Charter, particularly the rejection of traditional urban planning methods, the denial of the historical memory of the cities and the implementation of impersonal forms into the existing urban context. In 1953, an international group of architects was formed, known as Team X (Jacob Bakema, Georges Candilis, Giancarlo De Carlo, Aldo van Eyck, Alison and Peter Smithson, Shadrach Woods). They criticized the radical modernist approach and proposed to reintroduce a human dimension and continuity into urban habitat architecture. One of them, Aldo van Eyck, thought that the real enemy of architecture was modernist urbanism, as well as the complexes it had created [START_REF] Haddad | A Critical History of Contemporary Architecture: 1960-2010[END_REF] .

A few years later, these ideas would lead to a form of cell-structured housing, without the usual towers and bars. This would eventually be widely interpreted in the French PAN (the acronym for the French translation of New Architecture Programme) annual competition entries and finally realized in the new French new cities during 1970-1980. But, in early 1960, the towers and bars, which were colourful, sometimes linked and sometimes dismembered and broken, would still be predominantly found in housing concepts. The Maurelette complex is one of those "compromised" architectural products, which demonstrates an ambition to create a "non-ordinary" ensemble using ordinary and inexpensive construction means and materials.

An a mbitious progra mme: to create a form of modernity that respects the genius loci

The North Highway, constructed in Marseille in 1950, makes the city centre accessible from the northern peripheral area and creates favourable conditions for the installation of a new large residential district. Very soon, this area will reflect the typical suburban image of French cities of the post-war period. The La Tour district, which has chosen for the construction of the Maurelette complex, is surrounded by a heterogeneous urban environment, on the site of the former St. Joseph village with the two-to three-storey houses. Meanwhile, the new neighbourhoods to the west of the highway represent some variations of "band" construction, with disseminations of still-surviving castles and parks. In the 18th and 19th centuries, the district was one of many bastides (the country residences of the rich bourgeoisie) in Marseille. This sector kept its rural character until the middle of the 20th century (Fig. 1). On the district plan, we can see two castles (Castle Tower in the northern part, which surrounded by walls, and the House of Senior, in the centre of the district), terraces and a sycamore tree lane extending from the Gay-Lussac Street (formerly St. Joseph Way) to the House of Senior. The reason for and the exact date of the demolition of the Castle Tower are unknown. It appears on the Napoleonic cadastre plans and other plans until 1957 (Fig. 2).

The design of the ensemble for 745 apartments was entrusted to architects Eugène, Pierre and Jacques Chirié. The workmaster was the La Tour real estate company. Special importance was attached to this operation, as it was considered as an exemplary site. The realization was followed by the Ministry of Construction, General Planning and Equipment, the Productivity Commission, the Ministry of Health and Population, and the Municipality of Marseille. The programme was set up by the Chirié architectural agency after sociological, architectural and technical studies had been conducted for over a year.

The objectives, which were expressed by the authors in the "Maurelette Programme" and the "Basic Principles for the Choice of the Ground Plan" were:

-to create a pleasant environment to live in, allowing families to escape the feeling of banishment that is too often associated with grands ensembles -to conserve cherished property elements: the beautiful House of Senior and trees that rise in the field or grove aisles -to produce a simple and inexpensive structure -to conserve the outline and fresh appearance of the Maurelette, when seen from the North Highway in Marseille, as a quiet and green area [START_REF]Les Matérieux du Projet de la Maurelette[END_REF] The creation of a well-integrated urban structure, clearly standing out on a heterogeneous and discontinued site (Fig. 3), resulted in a programme conceived by the Chirie architectural agency in collaboration with landscaper Jacques Sgard and colourist Bernard Lassus.

Fig.3. The Maurelette on the site structure (in red)

Ground plan for a new "collective bastide" (fortified residential property)

Despite the fact that the designers used a very limited number of well-known and inexpensive typological elements (78 bars with a height varying from two-to-four storeys, as well as one 15-storey and four 11-storey towers, all of which were 11.2 meters long and articulated in an orthogonal system), there is no symmetry and repetitiveness in the structure of the Maurelette; neither is there a brutal application of building systems according to the héliothermique axis. [START_REF] Rey | North-south Orientation of Residential Buildings[END_REF] Also, there is no demonstration of sovereignty in relation to the existing urban context (a typical approach used by the pioneers of the modernist movement). The morphology of the terrain is taken in consideration in a very original way. The new structure is not completely separated from the existing urban context; however, it is represented as independent and complex, in that is "folded in on itself" (Fig. 4).

Fig.4. The Chirie ground plan for the Maurelette project

The broken line of buildings placed along the eastern and northern boundaries of the district, which acts as the ramparts along with the main entrance from old St. Joseph Street, and accentuated by the towers, evokes the ambiance of an autonomic and well-defended area. Two building groups (eastern Maurelette I and northern Maurelette II) are organized to suggest a movement towards the centre of the composition. Both groups represent spatial "arrows" flying towards the House of Culture (the renovated House of Senior) and the new Social Centre. Thus, the role of the House of Senior and the old sycamore tree lane goes beyond the simple "conservation of cherished elements of the district", as mentioned in the programme. Here, they are the main elements predetermining the composition of the eastern part (Maurelette I), which, in turn, suggest that the northern part's structure (Maurelette II) is a spatial and functional counterweight to the Social Centre.

In fact, the ancient private country residence (bastide) was replaced by a modern collective bastide in the same image of autonomy. This fact was considered by most inhabitants as an advantage. In the early 1980s, the inhabitants, wishing to "defend" the autonomy of their "Maurelette universe", organized a petition against the decision to make the Social Centre accessible for the inhabitants of the surrounding neighbourhoods. [START_REF] Testimony | Director of the Social Centre of Maurelette[END_REF] Moreover, this urban approach generated different estimations of different parts. This is not typical for most ordinary, socalled "statistical" residential complexes of the post-war period. The tenants of Maurelette II consider living in a quieter area, away from the noise and bustle of the mall. In fact, Maurelette represents a kind of "concentrated remix" (in both senses of the word: space and time) of a historical residential district with more or less active areas, whose disposition is usually the result of a secular spontaneous development.

We see the same approachthat is, both traditional and modernistin the road organization. The inner side of the area is treated as a continuous pedestrian walkway that avoids crossing the traffic routes. The car service is organized from the Gay-Lussac and Chatelier Streets, with the six-metre wide alleyways, which are distant from the buildings and have no crossings in-between (Fig. 5).

Fig.5. The road organization in Maurelette, with morphological analysis by N. Chilingaryan

The obvious ambition to shield the Maurelette ensemble from any motor noise has led to the creation of a free zone, which is 50 metres wide, along the Northern Highway. Similarly, most of the parking is organized in underground garages. Outdoor parking lots are arranged near the walkways outside the buildings groups. Thus, the application of a classic modernist principlenamely, the separation of pedestrian and automobile tracksin the case of Maurelette could also be considered as an additional measure to protect the privacy of the "secure collective property", which is non-penetrable by others.

Space organization: apparent spontaneity, hidden hierarchy?

Some documents compare the spatial organization in Maurelette with the La Viste residential complex, which was built in the same period and, like Maurelette, was realized in the framework of the construction programme for 4,000 new residences in Marseille. Some researchers claim that, in Maurelette, "the influence of Team X is clearly evident, especially the principles applied by Candilis, Josic and Woods at La Viste". [START_REF] Durousseau | Patrimoine du XXe siècle en PACA, Ensembles & Résidences à Marseille 1955-1975[END_REF] However, a detailed study of these two complexes reveals the very different approaches of their designers. In fact, this similarity is limited to a bird's eye view of the plans, due to the use of bars, towers and angled buildings, which are juxtaposed to comparable proximities. This similarity could be explained by the application of an orthogonal plan system, which facilitates relatively inexpensive construction. The two complexes are well inscribed in the surrounding urban context, albeit in very different ways: La Viste has an open morphology, which is penetrable from the south and the north. The plan represents the building branches, the traces of which strictly follow the territory limits to form the incomplete structures. These branches could be continued if necessary (in line with the basic principle of Team X about indeterminate, continuous urban forms). This transparent, accessible and continuous composition of La Viste (conceived by the militants who opposed the indifferent and exposed structures of the CIAM pioneers) paradoxically makes it similar to the phalanstery of Le Corbusier (founder of CIAM), which was presented in a project competition for the reconstruction of the unsanitary sector, known as No. 6, in Paris in 1936 (Figs. 6 and7). Unlike La Viste, the creators of Maurelette used all means "to finish"; in other words, to give an identity and personality to the spaces. The names found around the squares (names of alleys and patios refer to the ancient domain and major events from the history of Provence), as well as the landscape design, which not only maintains the existing beautiful trees, but doubles the surface of planted areas, reflect an ambition to create a real Marseille site, which is attached to its context and not an abstract modernist residence model that is more or less humanized. The structure of Maurelette does not contain any sign of evolution or continuity. If the basic element of La Viste's composition is the continuous bar that is punctuated by towers (in other words, the volume module), the structure of the Maurelette remains a space module, meaning it is sometimes enclosed, sometimes open, but always clearly defined. The analysis of this spatial element (called a "square" by its authors) demonstrates the very specific approach applied by designers and deserves a special attention.

The use of a square in its classical sense as a structural element of a traditional city requires the presence of two other essential elements (streets and courtyards), which are absent from Maurelette. The concept of a square, when selected by the architects as a foundation unit, also loses its original meaning because of its overuse beyond the traditional city: 10 squares formed by 80 buildings located on the territory with a total surface area of only 11 hectares. Thus, the squares in Maurelette have an ambiguous nature: being firmly attached to the context of the site, they are absolutely decontextualized, in the original sense of the term, in the same way as traditional urban components. This allows us to consider the Maurelette as a kind of precursor of the postmodernist approach by reintroducing elements and forms of the past as symbols freed from their historically formed "support".

The typological analysis of the Maurelette squares allows for them to be classified into three typological groups linked by hierarchical logic. Square-types are divided into the following three classes:

1. The first type is a square with low buildings on opposite corners, with one or more towers (with 11-and 15-storeys). We will refer to it as a "main square" because the two spaces of this type (namely, the Commerce Square and the Baussane Square) play a decisive role in the morphology due to their dimensions and positions, as well as the presence of common services (Fig. 8).

Fig.8. The "main squares", with typological analysis by N. Chilingaryan

2. The spaces surrounded on three sides by low buildings and accentuated by a tower. We will refer to this type as a "current square"; five of the 10 squares in Maurelette belong to this type. These are Charles Bichi Square, Portique Square, Old Sycamore Square, Cadran Square and the Square of 14 December" (Fig. 9). 3. The third type is the "piazzetta" (smallest square), which is deprived of towers and bounded by angled bars. We find three squares of this type: Bastide Square in Maurelette I and two in Maurelette II (Autures Square and Ben Quihado Square) (Fig. 10). These are the most modest, but indispensable, public spaces in the hierarchy. Being located right next to the central dominant volumes (House of Culture and Social Centre), they provide a kind of "pause space", meaning a consecutive passage to the spaces with more active public vocation (Fig. 11). Thus, this polyvalent space module, which is really neither a square, court nor street in the classic sense, practically assumes the roles of these three main components of a traditional urban habitat. The creators were able to give every gradation of the ambiance of a traditional city district to the site: from private to public, from agitation to relaxation, from greenery to mineral surface. All these were simply made by the intelligible alternation of the same, delicately changed spatial element.

The Maurelette is free of "unexpected surprises", such as windows-leaves or big sculptures of snakes in the middle of a court, similar to Emile Aillaud's Pablo Picasso complex in Nanterre (Fig. 12). Rather, it is inspired by everyday poetry, where the nuances take priority over the singular event. The Maurelette complex, which was created in the interim period between the "time of rationality" and the "end of interdictions", represents a remarkable architectural symbiosis. This ensemble, in the larger sense of Kisho Kurokawa's definition, is itself an "intermediate space", where experience of the past matures and crystallizes in order to give life to the future. The Maurelette residential complex in Marseille could be considered as an early application of the postmodernist ideas of decontextualization and free "expression" of historical urban forms. It appears as a "cultural bridge" connecting the past, the present and the future. Fortunately, until the present day, this ensemble is almost intact, while, in 2006, it received the "Heritage of the 20th Century" label as Maurelette Park. However, its "survival" could be considered as a rather extraordinary phenomenon than a typical one.

Since the end of the 1980s in many European countries, particularly in France, massive renovation processes have taken place, which continue to this day. Due to political, social and aesthetic changes, a great part of post-war residential heritage has been radically reconstructed or demolished. This process touches not only ordinary residential groups (so-called grands ensembles), realized right after the Second World War with very limited means and techniques, it also harms some of the interesting ensembles. Many have been undervalued, not rehabilitated, since their creation, while some are often doomed to disappear.

The Maurelette case study intends to raise awareness of the heritage constructed between major architectural periods (for example, modernism and postmodernism), as it represents an interesting architectural symbiosis with hidden values. Such heritage deserves to be included in the contemporary urban structure, without negating or altering its authentic concept. Moreover, its existence will contribute to the continuous, uninterrupted urban environment, thereby making the modern city more resilient.

Disclosure statement

No potential conflict of interest was reported by the author.

Notes on contributor Nune Chilingaryan

Associate

Fig. 1 .Fig. 2 .

 12 Fig.1. Extract from the Napoleonic Cadastre in 1817; the future Maurelette ensemble with the sycamore alley, the House of Senior and the Castle (in red)

Fig. 6 .Fig. 7 .

 67 Fig.6. Proposal for the reconstruction of the No. 6 sector in Paris, Le Corbusier, 1936

Fig. 9 .

 9 Fig.9. The "current squares", with typological analysis by N. Chilingaryan

Fig. 11 .

 11 Fig.11. Diagram with the position of square-types in Maurelette, with morphological analysis by N.Chilingaryan

Fig. 12 .

 12 Fig.12. The Pablo Picasso complex in Nanterre, E. Aillaud (architect), 1972 Conclusion

13 .

 13 Figs. 1-5, 8-11, 13. Tchilingayan N. Cité de la Maurelette à Marseille: Une Expérimentation du Logement Social dans les Années 60, Mémoire de Fin de CEAA, Ecole d'Architecture de Marseille-Luminy, 1994, with drawings from Les Matérieux du Projet de la Maurelette, Archives Départementales de Marseille.

Fig. 6 .

 6 Fig. 6. Corbu, https://fr.pinterest.com/martabrandao/corbu.

Fig. 7 .

 7 Fig. 7. Durousseau T. Patrimoine du XXe Siècle en PACA, Ensembles & Résidences à Marseille 1955-1975, Notices Monographiques, 1521 -La Viste, http://www.culturecommunication.gouv.fr.

Fig. 12 .

 12 Fig. 12. Dhuis J.F. L'Architecture Selon Emile Aillaud, Paris, Dunod 1983.

-Expert member, International Scientific Committee (ICOMOS) for "20th Century Heritage" -Member of the State Commission for the State Prize of the Republic of Armenia in the field of architecture and urbanism


Shlomo Barak
email: shlomo@tagapro.com

On the Essence of Electric Charge Part 1: Charge as Deformed Space

Keywords: Electric charge, Space lattice, Electromagnetism

The essence of electric charge has been a mystery. So far, no theory has been able to derive the attributes of electric charge, which are: bivalency, stability, quantization, equality of the absolute values of the bivalent charges, the electric field it creates and the radii of the bivalent charges. Our model of the electric charge and its field (Part 1) enables us (Part 2), for the first time, to derive simple equations for the radii and masses of the electron/positron muon/antimuon and that of quarks/anti-quarks. These equations contain only the constants G, c, ℏ and α (the fine structure constant). The calculated results based on these equations comply accurately with the experimental results. In this Part 1, which serves as a basis for Part 2, we define electric charge density, based on space density. This definition alone, without any phenomenology, yields the theory of Electrostatics. Together with the Lorentz Transformation it yields the entire Maxwell Electromagnetic theory.

Introduction

In our model a positive elementary electric charge is a contracted zone of space, whereas a negative elementary electric charge is a dilated zone of space. Relating to space as a lattice (cellular structure), we define Space Density  as the number of space cells per unit volume (denoted  0 for space with no deformations). Based on this we define (postulate, invent)

Electric Charge Density as: q = 1/4π • (ρ -ρ 0 )/ρ [q] =1. This charge density is positive if  >  0 and negative if  <  0 .

Electric charge, in a given zone of space τ, is then:

τ qd Q τ   [Q] = L 3 .
Our definition of electric charge density alone yields electrostatics, without any phenomenology, and together with the Lorentz Transformation -the entire Maxwell theory. This result encourages us to further pursue our idea of the essence of electric charge and, as Part 2 shows, it yields the important results presented below. Note that our definition of "charge density" is axiomatic. This approach is in the spirit of Einstein [START_REF] Barak | Einstein: Ideas and Opinions, Wings Books NY[END_REF] that: …. the axiomatic basis of theoretical physics cannot be extracted from experience but must be freely invented… We consider charge to be a deformed zone of space, and since the geometry of both deformed spaces and bent manifolds is Riemannian, we can attribute curvature to an electric charge. This new idea, of charge as curved space, enables us to use the theory of General Relativity (GR) in our derivations. These derivations, in Part 2 of this paper, yield the attributes of matter. Some of which are presented below:

The proton charge radius, r p is: r p (calculated) = 0.877428549•10 -13 cm. well within the experimental error range [START_REF] Pohl | The size of the proton[END_REF]. These results speak for themselves, and justify our axiomatic approach.

On Our Electromagnetism (EM) 1.1

In this paper we derive only "Electrostatics", which is needed for Part 2. The entire theory of electromagnetism and the extension of the General Relativity field equation to incorporate the contribution of charge to curvature will appear as separate papers.

Our EM utilizes the dimensions of length L (cm) and time T (sec) only. Thus, each physical quantity of our EM has a dimensionality, which is L x T y . Using General Relativity we can establish quantitative equivalence between our system and the conventional system of units.

By relating the field energy density to charge density our EM becomes non-linear, which is its specificity. Note that QED is also a non-linear theory.

The issues of structure, stability and quantization of an elementary charge are discussed in Part 2. Note that our EM is applicable for both a single elementary charge and an ensemble of elementary charges. Note also that EM waves, in our theory, are simply space transversal vibrations.

Space as a Lattice 1.2

Our definition of Electric Charge Density is based on the concept of space density. Space density is related to the cellular structure of space; relating it to a continuum is not prohibitive but problematic. Attributing a cellular structure (a lattice) to space explains its Hubble expansion, its elasticity (see 1.3) and introduces a cut-off in the wavelength of the vacuumstate spectrum of vibrations. Without this cut-off, infinite energy densities arise. The need for a cut-off is addressed by Sakharov [4] and Misner et al [5]. The Bekenstein Bound [6] sets a limit to the information available about the other side of the horizon of a black hole. Smolin [START_REF] Smolin | Three Roads to Quantum Gravity[END_REF] argues that: "There is no way to reconcile this with the view that space is continuous for that implies that each finite volume can contain an infinite amount of information". A review, relevant to our discussion, appears in a paper by Amelino-Camelia [START_REF] Amelino-Camelia | [END_REF].

The Elastic Space 1.3

We relate to space not as a passive static arena for fields and particles but as an active elastic entity. Physicists have different, sometimes conflicting, ideas about the physical meaning of the mathematical objects in their models. The mathematical objects of General Relativity, as an example, are n-dimensional manifolds in hyper-spaces with more dimensions than n. These are not necessarily the physical objects that General Relativity accounts for and n-dimensional manifolds can be equivalent to n-dimensional elastic spaces. This equivalence allows us to use General Relativity, and also relate to our own space as an elastic 3D space. Rindler [START_REF] Rindler | Relativity Oxford[END_REF] uses this equivalence to enable visualization of bent manifolds, whereas Steane [START_REF] Steane | Relativity made relatively easy[END_REF] considers this equivalence to be a real option for a presentation of reality. Callahan [START_REF] Callahan | The Geometry of Spacetime Springer[END_REF], being very clear about this equivalence, declares: "…in physics we associate curvature with stretching rather than bending". After all, in General Relativity gravitational waves [START_REF] Abbott | Observation of gravitational waves from a binary black hole merger[END_REF] are space waves and the attribution of elasticity to space is thus a must.

The deformation of space is the change in size of its cells. The terms positive deformation and negative deformation, around a point in space, are used to indicate that space cells grow or shrink, respectively, from this point outwards. Positive deformation is equivalent to positive curving and negative deformation to negative curving. This is discussed in Part 2.

Recent Papers on Electric Charge 1.4

"Nonlinear models of electric charge and magnetic moment" [13] (2015)

"The enigmatic electron" [14] (2013) "Singularity-free model of electric charge in physical vacuum: Non-zero spatial extent and mass generation" [15] (2013) "Duality and 'particle' democracy" [START_REF] Castellani | Studies in History and Philosophy of Modern Physics[END_REF] (2016) Although these papers relate to different aspects of our subject, none present a similar idea to ours.

Electric Charge Electric Charge Density 2.1

We define the electric charge density as:

      0 4 1 q [q] = 1 (1)
The factor 1/4π is introduced for no other reason, than to ensure resemblance to the Gaussian system.

The charge density is positive if  >  0 .

The charge density is negative if  <  0 .

Necessarily, only two types of electric charge exist, positive and negative. Let n be the number of space cells in a given volume V. Since n = ρ 0 V and also n = ρV' we get:

V=n/ρ 0 V'=n/ρ Hence:

(V'-V)/V = -(ρ -ρ 0 )/ρ (2) V'>V is dilation, V'<V is contraction. Electric Charge Q in a Given Volume  2.2
We define Electric Charge in a given zone of space τ as:

τ qd Q τ   Electric charge has the dimensions of volume [Q] = L 3
For clarity, in this section alone, we omit the factor 1/4π in [START_REF] Barak | Einstein: Ideas and Opinions, Wings Books NY[END_REF]. For the spherical symmetric case where dτ = 4πr 2 dr, for a given r, the radius of the charge Q, we get the result:

Q = ∫ q4πr 2 r 0 dr = 4π/3• r 3 (1- 0 /ρ). Thus  >  0 gives Q > 0 whereas  <  0 gives Q < 0.
For us, outside observers, positive charge in a given spherical zone of space with radius r, means more space cells in the zone than in an un-deformed space (contraction), whereas negative charge means less space cells in the zone than in an un-deformed space (dilation). In Note that both ρ + and ρ -are, probably, functions of r and not just constants. 1) suggests simplistic models of positive and negative charges, both as spheres of "radius" r 0+ and r 0-. The contracted space in the sphere with radius r 0+ , our Q + , contracts space around it (its field) whereas the dilated space in the sphere with radius r 0-, our Q -, dilates space around it (its field). In this model, of charge and its field, there is no physical separation between the particle and its field, and the integral of ρ over the entire space, for two bivalent elementary charges together, is zero. Note that in the field of a positive charge space is also curved positively. Similarly, in the field of a negative charge, space is curved negatively.

Hence the field equation is non-linear, as is the field equation of gravitation.

The Elastic Spatial Vector u and the Electric Field E

By relating to space as an elastic media we can use the theory of elasticity and its Elastic Displacement Vector u = r'r. In Appendix A we show that:

       0 u  (A2)
Thus, according to (1):

  u = -4q (3)
By defining the Electric field vector E as:

r ρ r 0 ρ 0 Q(-) ρ 0 r 0 Q(+) r ρ E = -Hu H = 1 [H] =T -2 [E] = LT -2 , (4)
equation ( 3) becomes the known equation:

  E = 4Hq (5)
Note that any deformation (strain) in space is related to a stress; hence the introduction of H.

E expresses, therefore, the tension in space due to a deformation in it.

For a positively charged particle, E points outwards and for a negatively charged particle inwards, as it is in the Maxwell theory of electrostatics (see Fig. 2 in Appendix A).

Coulomb's Law

Gauss's theorem is:

dσ u u         d 
For a spherical surface with radius r we get:

2 r r 4 u       dσ u and since Q 4 qd H 4 d            u  we get r r u 3 Q  or: r r E 3 HQ  Coulomb's Law (6)
5.

The Electric Field E and Scalar Potential 

Every vectorial field can be decomposed into a field that is a gradient of a scalar potential (the polar part) and a field that is a vector potential (the axial part), subject to the boundary condition 0  E at infinity. Hence:

Α E        (7)
In the simple static case for the electric field:

    E (8)
and, in case of spherical symmetry, in spherical coordinates:

  r E r r         (9)
and since:

2 r r HQ E  we get: r HQ   [] = L 2 T -2 (10)
From:

Ηq 4   E  and     E we get Hq 4      or: Hq 4    
Poisson's equation [START_REF] Callahan | The Geometry of Spacetime Springer[END_REF] In the absence of charge:

0    Laplace equation (12)
We can modify equation [START_REF] Callahan | The Geometry of Spacetime Springer[END_REF] to incorporate the charge density of the field, which is equivalent to the field energy density. This modification turns ( 11) into a non-linear equation that resembles the non-linear equation of gravitation but it is out of the scope of this paper.

The Electrostatic Force

From (3)

u   = 4q we get u     4 1 q
, or in tensor notation

j j x u 4 1 q     (Appendix B
relates u ij to the metric element g ij ). Multiplying both sides of (3) by u gives:

u u u     4 q (13)
The left-hand term multiplied by H is denoted by f. f = qHu or f i = qHu i (14) At this stage, f is just a symbol. After a few steps, it is identified as the electrostatic force density. Substituting the tensor notation for u, in Equations ( 13) and ( 14) gives:

                       j i j j j i j j i i j j i x u u x u u 4 H x u u 4 H u x u 4 H f hence:            ij 2 j i j i δ u 2 1 u u x 4
H f (15) where  ij is the Kroneker Delta defined by  ij = 1 for i = j,  ij = 0 for i  j, and

2 3 2 2 2 1 2 u u u    u
. Hence f i may be regarded as derived from a tensor:

          ij 2 j i ij 2 1 u u 4 H P u
. And indeed:

j ij i x P 4 H f    
is identified as the force per unit volume and P ij as the strain tensor.

If the x-axis is chosen parallel to a line of force at any point, then u y = u z = 0 and u x = u, and:

2 zz yy xx 8 H P P P u       .
Thus the pressure perpendicular to the surface is equal to the energy density.

From ( 4), E = -Hu, we get the expression for the energy density in the field:

2 8 1 E    ( 16
)
Since f is identified as the force per unit volume, we can return to the expression f = qHu, and recognize the electrostatic force density:

E f q  [f] = LT -2
and the electrostatic force:

E F Q  [F] = L 4 T -2

Coulomb's Force Law

Equation ( 16) expresses the field energy density of a system of charges. Hence:

      d H 8 1 U 2 E
where E is the field produced by these charges, and the integral goes over all space. Substituting E = -, U can be expressed as follows:

                   d H 8 1 d H 8 1 dτ H 8 1 U τ 2

E E E

According to Gauss's theorem, the first integral is equal to the integral of E over the surface bounding the volume of integration, but since the integral is taken over all space and since the field is zero at infinity, this integral vanishes. Substituting (5), E = 4Hq , in the second integral, gives the expression for the energy of a system of charges:

dτ q 2 1 U τ    
For a system of point charges, Q i we can replace the integral with a sum over the charges

   i i i Q 2 1 U
where i  is the potential of the field produced by all the charges, at the point where the charge i Q is located. From Coulomb's law:

    j i ij j i r HQ
where ij r is the distance between the charges Q i , Q j we get:

   j i ij j i r Q HQ 2 1 U
In particular, the energy of interaction of two charges is:

ij j i r Q HQ U 
and the force is:

2 2 1 12 r Q HQ r U F     or: F= 3 2 1 12 r Q Q H r12 Coulomb's force law (17)
8. Conclusions Our definition of electric charge density, based on the density of the elastic space lattice, enables us to relate to an elementary charge not as point-like and not as a string, which are alien to space, but as a finite zone of contracted or dilated space. Necessarily, elementary particles are also of finite size and have a structure. This understanding enables us to derive and calculate the elementary charge/particles attributes.

 

2 i i 2 i 2 2 i 2 i i i du dx ' dx ' dl dx dl du dx ' dx       by the substitution of k k i i dx x u du            above we get: l k l k i i k i i i 2 2 dx dx x x u u dx dx x u 2 dl dl'         
Since the summation is taken over both suffixes i and k in the second term on the right, we get:

k i i k k i dx dx x u x u             
In the third term, we interchange the suffixes i and l. Then dl' 2 takes the final form: where the strain tensor u ik is defined as:

                   k i l l i k k i ik x x u u x u x u 2 1 u (A1)
If u i and their derivatives are small, we can neglect the last term as being of the second order of smallness. Thus, for small deformations, the strain tensor is given by:

              i k k i ik x u x u 2 1 u
We see that it is symmetrical: u ik = u ki u ik , can be diagonalized, like any symmetrical tensor, at any given point. Thus, at any given point, we can choose coordinate axes, the principal axes of the tensor, in such a way that only the diagonal components u 11 , u 22 , u 33 of the 3D tensor u ik are different from zero. These components, the principal values of the strain tensor, are denoted by u (1) , u (2) , u (3) . We should remember that, if the tensor u ik is diagonalized at a specific point in the body, it is not, in general, diagonal at any other point.

If this strain tensor is diagonalized at a given point, the element of length near it becomes:

  r p (measured) = 0.8768(69)•10 -13 cm. Based on r p , we calculate the electron radius r e : r e = 1.409858587 •10 -13 cm. Based on this r e we derive and calculate the mass M of the electron: M(calculated) = 0.910366931•10 -27 gr. A deviation of 0.06% from the measured CODATA 2014 value: M(measured) = 0.910938356(11)•10 -27 gr. Based on this M and our model of quarks, we derive and calculate the masses M d and M u ̃ of the d and u ~ quarks: M d =4.5 MeV recent experimental value [3]: M d =4.8 +/-0.5 MeV M u ̃=2.25 MeV recent experimental value [3]: M u ̃=2.3 +/-0.8 MeV

Part 2 ,

 2 using Riemannian geometry, we relate curvature to this space deformation and open the way to the application of GR in issues related to charge. Note that the equality |Q + | = |Q -|, of the absolute values of the bivalent elementary charges means, according to the integral above, (1- 0 /ρ + ) = -(1- 0 /ρ -) and hence 2/ 0 = 1/ρ + + 1/ρ -.

Fig. (

  Fig. (1) suggests simplistic models of positive and negative charges, both as spheres of

Fig. ( 1 )

 1 Fig. (1) A Charge and its Field

Acknowledgments

We would like to thank the late Professor J. Bekenstein and Professor A. Zigler of the Hebrew University of Jerusalem, and Professor Y. Silberberg of the Weizmann Institute of Science, for their careful reading and helpful comments, and Mr. Roger M. Kaye for his linguistic contribution and technical assistance.

Appendix A: Contraction and Dilation of Space, and the Strain Tensor u ij

The aim of this appendix is to prove that: [START_REF] Pohl | The size of the proton[END_REF] shows the position vector r of a spot, p, in space, with no strain. When stress is applied on space and a deformation occurs, the location of p becomes p' with a position vector r'. The vector u is the Elastic Displacement Vector (theory of Elasticity). The origin in of a strained medium. The following discussion is based on a derivation made by Landau and Lifshitz [START_REF] Landau | Theory of Elasticity[END_REF] . u = r'r can be denoted by its components:

Let dl' be the deformed distance between adjacent points, since:

We see that the expression is the sum of three independent terms. This means that the strain in any volume element may be regarded as composed of independent strains in three mutually perpendicular directions, namely those of the principal axes of the strain tensor. Each of these strains is a simple dilation, or contraction, in the corresponding direction: the length dx 1 along the first principal axis becomes

, and similarly for the other two axes.

The quantity

is consequently equal to the relative extension (dx' i -dx i )/dx i along the i th principal axis. The relative extension of the elements of length along the principal axes of the strain tensor, at a given point, is, to within higher-order quantities

, i.e., they are the principal values of the tensor u ik .

Let us consider an infinitesimal volume element dV, and find its volume dV' after a deformation. To do so, we take the principal axes of the strain tensor, at the point considered, as the coordinate's axes. Then the elements of length dx 1 , dx 2 , dx 3 along these axes become, after the deformation,

, etc. The volume dV is the product dx 1 dx 2 dx 3 , while dV' is dx' 1 dx' 2 dx' 3 . Thus

Neglecting higher-order terms, we therefore have

. The sum

of the principal values of a tensor is well known to be invariant, and is equal to the sum of the diagonal components u ii = u 11 + u 22 + u 33 in any coordinate system. Thus:

and according to (2) we get:

Appendix B: The Small Deformation Strain Tensor and the Fundamental Metric Tensor

In the peer reviewed paper [START_REF] Palacios | The Small Deformation Strain Tensor as a Fundamental Metric Tensor Journal of High Energy Physics[END_REF] the authors demonstrate that the small deformation strain tensor, see (A1), could be used as a fundamental metric tensor, instead of the usual fundamental metric tensor. We quote their Conclusion: "Through the present paper, it was possible to demonstrate that the small deformation strain tensor could be used as a fundamental metric tensor, instead of the usual fundamental metric tensor. Also, it was possible to prove that from that tensor, not only other mathematical structures could be constructed, but also another fundamental tensor was obtained; that was to say, we had constructed two of them, uμυ, and Bμυσρ. It is through these tensors that gap between pure geometry and physics is bridged. In particular, uμυ relates the observed interval ds to the mathematical coordinate specification dxμ. Also, the uμυ appear as the potentials of the inertial field {6}. Therefore, it is reasonable to assume that, in the presence of a gravitational field, the uμυ is again the potential which determines the accelerations of free bodies; in other words, the uμυ is the potential of the gravitational field. Thus, a stage has been reached at which the results obtained can be applied to the theory of gravitation {4}. However, that task that would not be repeated here was established by Albert Einstein, and finally formulated by him in 1916, as probably the most beautiful of the physical theories." Note however that space deformation is a local feature whereas curvature can also be a global feature.