Elie Cartan and pan-geometry of multispatial hyperspace

Chaos, Solitons and Fractals 19 (2004) 479–502 www.elsevier.com/locate/chaos

Elie Cartan and pan-geometry of multispatial hyperspace Jakub Czajko P.O. Box 700, Clayton, CA 94517-0700, USA Accepted 16 May 2003

Abstract Elie Cartan has proved that highest dimensionality of any simple geometric space is three and that an exterior differentiation of a 3D+ geometric object gives bivector, which may correspond to some two 2D surfaces as if the 3D+ geometric object comprised two 3D objects. Since one cannot increase the dimensionality of a 3D space even though more than four independently varying physical magnitudes do exist, then an expansion of dimensionality requires a multispatial hyperspace that contains many simple geometric 3D spaces. Presence of such a hyperspace prompts for an entirely new concept of vectors with an isometric operation of vector multiplication of traditional vectors (3-tuples). This new operation on 3-vectors implies presence of a 3D mass-based linear vector space and consequently thus a 9D geometric hyperspace for classical mechanics alone. Also an outline of entirely new, synthetic approach to physics and mathematics is introduced. This synthetic approach can be used to design a computer-aided knowledge extracting system, which could generate entirely new scientific knowledge. Ó 2003 Elsevier Ltd. All rights reserved.

1. Physical reasons for multispatial hyperspace Mathematical consequences of Special Theory of Relativity (STR) imply a spatial structure of time flow (SSTF) [1]. The SSTF suggests unanticipated nonradial (tangential and binormal) gravitational potentials, whose existence has been confirmed in several formerly unexplained experiments [2–4]. The nonradial potentials cause nonradial effects of gravity (NEG), which explain an inaccurate local prediction of the EinsteinÕs General Theory of Relativity (GTR) [2] in agreement with many independent experiments [5]. The SSTF simplifies mathematical physics (MP) by splitting impact of artificial motion and gravity [3]––a move deemed impossible by Einstein, who founded his GTR on the Principle of Equivalence (PE) which substitutes the effects of accelerated artificial motion with the action of an amalgamated gravitational pseudofield. Presence of NEG restricts the GTR to essentially global, radial happenings and reveals cracks in some algebraic, topological, vectorial and tensorial methods when applied to certain discrete phenomena of physics. Despite its evident inability to handle near surface phenomena [2–4], the GTR is a formidable cosmological theory. For only from the standpoint of the universe as a whole one can disregard such nuances as conservation of energy and choose to investigate an amalgamated gravitational pseudofield instead [3]. For happenings near surfaces of stars or planets, the latter must be treated as discrete objects, not as masses dissolved into the amalgamated pseudofield. For exact predictions near surfaces of masses, the GTR should include the NEG [2–4]. Just as the NewtonÕs theory the GTR did not really explain the phenomenon called gravity, but it accounts for some of its radial consequences. Even if only one single particle would have to be considered, gravity should have also some nonradial impacts besides the radial one––for differential geometry sake. The GTR emerged in mathematically relaxed German-speaking scientific community. Were it not for the fact that pure mathematics (PM) did not care about physics, the GTR might not survived

E-mail address: [email protected] (J. Czajko). 0960-0779/$ - see front matter Ó 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0960-0779(03)00254-6

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critique of its theoretical foundations in more mathematically sophisticated circles. In this and subsequent papers I will expose theoretical nonsenses that virtually concealed the necessity for both the SSTF and NEG. I must begin my critique with Descartes. His analytic geometry is a great tool, but it must not serve as a model for happenings in the physical reality, because it has terrible misgivings that have never been disclosed before. The whole nonsense concealed a part of physical reality and paved the way for some abstract algebraic alternatives to differential geometry. In order to understand that, however, I should explain the meaning of certain forgotten and/or suppressed mathematical results. It took a lot of tweaking for the PM to conceal the SSTF, which nature endorsed in many experiments [2–4]. In order to advance physics we must remove obscurity from mathematics. Yet mathematics should rely on physics, unless it would like to become an art. Unlike some cosmological theories, physical and geometrical ones should be formulated in terms of local differentials and be defined in infinitesimally small vicinities. The SSTF does not contradict the GTR, but complements it for local phenomena. However, the SSTF reveals presence of structural, essentially geometrical laws of physics, which can form deeper common mathematical infrastructure of physics [3]. The infrastructure should be the same for classical mechanics (CM), quantum mechanics (QM) and all their derivatives. The SSTF calls thus for unprecedented conceptual restructuring of the PM and MP. It needs a Pan-Geometry (PG) to deal with multispatial hyperspace (MH) [1,3]. For the very essence of the SSTF is existence of a multispatial infrastructure of our physical reality, wherein geometrical and physical dimensions are distributed among multiple 3D geometric spaces. The SSTF changes virtually everything about physics, especially gravity. For gravity operates in an abstract hyperspace that comprises at least two dual 3D linear vector spaces (LVSs), one of which is the usual length-based space (LBS) whereas the other is a pure time-based space (TBS) [3]. This new and quite unexpected interpretation has been developed in [1,3] upon El NaschieÕs idea of conjugate complex time [6–10]. Physics needs entirely new pan-geometric paradigm that would extend the single-space paradigm of former geometry onto multispatial reality. Dual spaces are well-known features of LVSs. They look like two faces of a vector space over the same topological manifold in a multispatial structure of a hyperspace, rather than in a single LVS [3]. The old single-space paradigm was just an untenable geometric oversimplification, defied by––formerly unexplained––nonradial and mixed experiments [2–4]. Those experiments challenged not just single theories, but the whole set of former mathematical and physical paradigms. From the viewpoint of differential geometry the NEG is unavoidable [2–4] and its omission produced twisted mathematics. Moreover, ‘‘undesirable’’ rules of differential geometry were set aside. That is why we must develop nonpostulative, synthetic mathematics (SM), which would supply balance checks for the PM as well as synthetic methods (complementary to analyses) to create new mathematical knowledge. Only declarations are allowed in the SM, but no existential postulates. Even the very abstract mathematical existence is not a matter of a clever definition or a smart postulate, but that of an actual construction, which requires resolution of logical contradictions. Only noncontradictory objects can actually exist. If an abstract object exists (i.e., is constructible) and yet its existence appears somewhat contradictory, then the mathematics that indicates the contradiction must be invalid and so it should be revised. Existential postulates can ‘‘create’’ nonexistent reality. The traditional postulative algebraic and topological approach to geometry and physics is inadequate. It has built almost Ptolemaic infrastructure of our physics. Mathematics must not be arbitrarily postulated, but discovered. Or else, it may create veiled nonsenses, one of which will be shown below. The nonsense is over 300 years old, but it was never contested. It stems from the art-like methods that the PM deployed in defiance of experiments and often against logic. Its elimination opens totally new approach to geometry. Such nonsenses have eventually turned against the PM and MP and put them in a theoretical limbo, loudly complaining about ‘‘weird’’ nature. However, if it is nature that sets ipso facto standards of scientific soundness, then perhaps PM and MP lost touch with the actual physical reality. There is no shred of evidence that time is really one-dimensional; neither is there a theoretical reason for that. Rate of time flow is affected in three linearly independent ways (dimensions) [3]. Once time is allowed to flow in its own 3D TBS––which is a dual LVS to the usual 3D LBS––everything begins making sense [1,3]. Evidently the nature is designed for spatial flow of time in a multispatial hyperspace. Experimental evidence confirmed it and theoretical evidence, some of which will be shown below, suggests that it is necessary. For if I can devise new, logically necessary operations, which were impossible to design before, then this fact constitutes theoretical evidence. Flow of time was already perceived as spatial, though still linear (one-dimensional). The STR has made time flow virtually two-dimensional as dependent on speed [1]. For if time flow depends on speed (i.e., inverse time or frequency or time rate), then it is essentially a 2D geometric effect. The STRÕs is still imprecise, however [1]. The GTR has not really enhanced the STR, but found an excuse for our inability to accept its consequences by shifting focus from physics to cosmology. I have extended the flow of time into third time-based dimension, as dependent also on changes to speed (i.e., accelerations) [3]. The SSTF is basically pure geometry in some time-based coordinates. However, it is physics that points to the geometric, 3D spatial flow of time. Yet the new physics that includes the SSTF needs physically meaningful mathematics and so it is imperative that we develop the SM.

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El Naschie, who investigated hierarchy of spaces, has demonstrated that discrete finite dimensionality is an expectation value of that on an infinite-dimensional Cantorian manifold [11,12]. His result shows how to make transition from a semi-structured topological manifold to fully geometric LVS. Geometric space is not a primitive notion that could be declared in an arbitrary fashion and then equated to whatever one may wish. Reasonable definition of geometric space should come out of investigation of properties of geometric objects immersed in the space. Topology is obsessed with the constituents of spaces (points and their immediate surroundings). Although true in its own right, topology does not reflect features of geometric objects, unless one could actually squeeze them to nothing. Physics and geometries are concerned with preservation or transformation of geometric objects. Physically meaningful geometry should take into account all conservation laws of physics, which must have geometric foundation and origin anyway. My goal here is to show a ‘‘kingÕs way’’ to certain consequences of some geometric results of Elie Cartan. His ideas were so difficult to grasp that even he himself did not realize what he actually obtained [1]. The allegedly exact language of PM obscures meaning of mathematical ideas. I will show that the alleged exactness may actually hide logical nonsense, which the PM silently proliferated and the MP wholeheartedly embraced. This is the main reason for the synthetic approach to both mathematics and physics. But far more important reason for the SM is reality check for the PM and MP. The SM aims at making mathematics comprehensible not only for those who may consider themselves mathematically challenged, but for professionals too. Postulates put restrictions on a part of the mathematical, and eventually also physical reality. If a postulate is made just to ease derivation, it can cut off a big chunk of the reality for no scientific reason. The SSTF was hidden behind quite unwarranted assumptions and certain arbitrary, common sense postulates. Physical reality does not have to comply with our intuitions, but it should satisfy restrictions of (manyvalued) logic. Physics is mathematics in disguise. Faulty mathematics can lead to utterly confused physics. I will try to show detailed analyses and proofs, wherever an exact proof is possible. However, some conclusions I have drawn are so advanced, that one cannot prove them from what has been achieved in the past. For they seem to defy even most liberal scientific common sense and our imagination. Euclidean method of derivation is inadequate for dealing with some facts of modern physics. We need experimental evidence, for the nature is the supreme judge of all scientific theories and abstract reasonings. Our mathematics should be operationally complete, because nature seems to operate flawlessly on such abstract structures as the multispatial hyperspace. If something can exist even though we cannot derive it from what is known, then we should look to nature for confirmation of its existence. If the nature confirms its existence, then we must rewrite not only the former physics that was unable to predict it, but also the former mathematics that ignored it. We should not define geometrical objects arbitrarily, or just postulate their properties if we could obtain them synthetically from the presumption that they must be (flawlessly) operated on. New and incomplete theories may be inconsistent at any given time and can contain apparent contradictions. The desire for unearned completeness caused misrepresentations and instigated misguided interpretations of some physical theories. The best example here is the celebrated Kaluza–Klein theory. Kaluza was right that the GTR could be expanded onto five abstract dimensions. However, when Edward Kasner proved that such an immersion needs six dimensions arranged in two triples [3,13], his result was simply ignored. Inasmuch as it was not supposed to be expansion onto single 6D space, but into two distinct orthogonal 3D LVSs, his result did not fit the old ideas of space. Perhaps the extra dimensions are not really compactified. Maybe we did not know where to look for them. An alternative to the compactification is to spread all those extra dimensions over several 3D spaces. The assumption that one would have discovered the badly needed extra physical dimensions, if these had not been ‘‘concealed’’ (i.e., compactified) has effectively derailed the search for them. I set seven objectives for this paper: to show that usual vectors in 3D space span multiple spaces and are 3D images of multispatial hypervectors, and to introduce a vector multiplication of vectors and to show that the operation is isometric (i.e., that it preserves length of its operands). For those who by now might have forgotten that PM does not know how to multiply vectors by vectors, this may come as a surprise. The long awaited operation on the usual vectors (3-tuples) can be seen as theoretical evidence for both the SM and SSTF. It also explains how gravitational forces emerge from potentials. My fifth and most difficult objective is to show that the new SM suggests an existence of a 3D mass-based vector space within 9D hyperspace. None of these objectives could be presented without the ideas, which I developed in my previous papers [1–4]. However, some topics discussed in these papers could not be finished without the results to be obtained in the present paper. My sixth objective is to show that abstract notions carry definite although often multifaceted meaning and abstract operations are physically delimited. They are not arbitrary. We may not always comprehend everything at once, but we should not operate without an understanding of their consequences. And last but not least, I must dismantle some prejudices of the former PM as we go. The reign of the old Euclidean, purely derivative mathematics is not quite over yet, but we need desperately the checks offered by new synthetic methods to make old analyses accountable. Postulating existence of abstract objects and then deriving their properties from postulates was irresponsible, because it did not

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preclude introducing of half-baked notions such as vectors seen as algebraic 3-tuples, which were then used to paint half-true picture of physical reality. Arbitrary existential postulates of former mathematics have eliminated parts of our physical reality and we are not always aware of that. The PMÕs pursuit of art-like status has created maimed and distorted image of our physical reality. From arbitrary axioms one may derive not only what is true, but also any nonsense. Abstract inferences may inadvertently extend some features of the physical reality we live in far beyond of what can exist, while transparently cutting off some other features that could exist. Because we still maintain centuries old mathematical nonsenses, we are often unable to explain physical phenomena of relatively simple types. Such nonsenses may invalidate some previous results of sciences. But their most disastrous consequence is that they severely restricted our capability to comprehend the physical world around us. Therefore we must get rid of them.

2. Problems with single-space approach to geometry Since DesCartes has successfully algebraized geometry, mathematicians replaced some of their thinking with operating [14]. Comprehension of differential geometry coupled with complex numbers was so difficult, that the trend gained momentum and resulted in topologization of mathematics. Topological treatment of manifolds made of indistinguishable points is very meaningful, but any reduction of geometry to topology must eventually fail, because methods of topology are not rich enough to represent all differential structures of geometry. Differential geometric structures can be pictured or analyzed, but not truly modeled by topology alone, wherein spaces and sets became synonymous [15]. It seems as if space, which is actually a construct (structural object) [1,3] and also a method of handling geometric objects, is identified with its substance (a point-set manifold). Topology defines space as a manifold that is still tied to motion [16] and presumed as differentiable [17]. However, simply postulating differomorphism did not really enhance topology. If the notion of space includes vector space, then it is a set over a set-manifold [18] (superset). Hence linear vector space is a very compound abstract structure of a different kind than the underlying manifold. Most of geometric and physical spaces are basically vector spaces. This fact calls for more geometric, rather than just a topological approach to such spaces. Topology may provide a representation for geometric spaces, which can be reduced to topological ones, but not a model for handling geometric objects. Since HilbertÕs sixth problem was simply stated as ‘‘mathematical treatment of axioms of physics’’ [19], physics was supposed to get axiomatized so that mathematics would solve all its problems at once. However, when real logical problems with motion emerged, mathematicians said that there is no compelling reason to believe that mathematical description of motion is still physically meaningful in infinitesimally small vicinity [20]. In other words: physics should not count on topology. Since physical and geometric objects should not vanish or be destroyed by abstract operations, they must persist or be transformed into different images (representations) of these objects. In topology, however, one can always squeeze and distort geometric objects far more than physics could ever tolerate. Neither topology nor algebra assures conservation of physically meaningful vectors under abstract operations. A force vector could not vanish without a physical reason––i.e., without an opposite force vector. Only another vector can nullify physical vector. In plain algebraic approach to geometry, vectors are viewed as n-tuples of numbers [21] and LVSs are then defined in terms of group operations over (mathematical) number fields [22], where numbers and points of abstract fields are interchangeable. In more abstract algebraic approach to geometry, vector spaces are defined in terms of transformations, but still over abstract mathematical number fields [18], where the (real) numbersÕ field is seen as a topological manifold. Yet topological manifold itself is a construct whose points are determined by family of open sets (patches) [23]. Hence LVS, as an algebraic construct over a topological construct, is a higher-level abstract construct, structurally different than the manifold. Metric space is a set with metric [24]. In set-theoretical setting one can equate coordinatization with parametrization [25] and define duality without any reference to them [26], or assign dimensions to spaces spanned by n-tuples [27]. Such ideas work in topology, but they paint wrong picture of LVSs. This may be caused by ambiguous language [28] and by lack of balance checks. PM is plagued by arbitrary postulates, axioms and definitions. It does not model any reality, but makes derivations, some of which are then hailed as proofs of existence. The SM aims at modeling, not just deriving attributes of the physical reality. Speaking of algebraic n-tuples as points of abstract n-dimensional spaces can be misleading. Between algebraic 2tuples (complex numbers), 4-tuples (quaternions) and 8-tuples (octonions) no self-contained 3-tuples, 5-tuples, 6-tuples or 7-tuples exist. Nevertheless, we have 3D geometric spaces, but a self-contained 2D space is impossible to construct, and self-contained 4D or 5D geometric spaces cannot exist either, for a double-3D hyperspace must embrace them [3,13]. Such a 6D hyperspace is a structure composed of two dual spaces, however. One can talk about algebraic or topological nD space, which is an abstract topological manifold, but there is no way to construct them as physically meaningful geometric vector spaces. Although the terms ÔspaceÕ, ÔsetÕ and ÔmanifoldÕ are used interchangeably, the latter

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refers to a superset, i.e., set of subsets (patches) in topology. Hence geometric vector space could be called supersuperset in topological and set-theoretical terms. By topological space is meant an arbitrary set together with a family of its subsets [29]. A differentiable manifold is just a topological space on which derivatives can be defined [30]. Riemann was fully aware of the fact that his ideas on geometry in manifolds border on philosophy [31]. Topological manifold covered by domains of d-dimensional coordinate systems is called d-dimensional [32]. This follows the ingenious Poincar
eÕs idea of inductive dimensionality [33], which just assumes unlimited dimensionality by default or induction, if you will. The problem is that we actually have three distinct theories of dimension in topology [34] and no one of them is addressing the needs of physics. One can entertain supermanifolds (with its supervector spaces that can be without basis [35]), on which it is problematic even to define tangent vectors [36]. However, tangent vector space is necessary to establish differential properties of objects independently of local coordinates [37]. Moreover, the process of reduction of vector fields by applying Lie transformation is not unique [38]. Hence the cherished topological ideas of geometric space and dimensionality cannot stand the scrutiny of logic––they induce contradictions. It is not good enough just to postulate a differentiable manifold. One has to construct it. Poincar
e is right that one can enumerate dimensions, but he is not associating them in a unique way with physically meaningful properties of constructible geometric objects. Yet he is wrong in his silent assumption that all dimensions should belong to the same space. The assumption was quite self-evident and remained uncontested under the former paradigm of single-space reality, but now it is time to reevaluate it. I will recall Elie CartanÕs forgotten proof that highest-dimensional simple geometric space has at most three distinct dimensions. If so then a 6D space must be abstract geometric structure of an entirely different kind. What is then wrong with the whole mathematical picture? Perhaps nothing. Yet we were unable to interpret it before. The SM should help us discover deeper logical foundations of both the abstract and the physical reality. It requires thinking in objectoriented terms, however. Such dissimilarities between geometrical and algebraic pictures of our physical reality are not accidental. Physical reality is quite different from what we thought it is. The analytic methods of former PM are suitable for disassembling objects, whereas the SM should help us also with assembling and instantiating physical objects. Mathematics is not an art where one can project our preconceived concepts on the physical reality and create simple though surreal picture of the latter.

3. Elie Cartan vs. Riemann or geometry vs. algebra Elie Cartan has proved that highest possible dimensionality of any simple geometric space is three (see [1,3,39,40] and Refs. therein). Yet Riemann had argued that no natural limit seems to prevent extension of the notion of geometric dimensionality onto an abstract, essentially algebraic one [41]. Though his arguments are valid, both of them can be right only if the higher than three dimensions belong to distinct 3D spaces. The Elie CartanÕs result excludes the algebraic approach to dimensionality that had been proposed by Grassmann, Riemann and embraced by Clifford, as well as the idea of unlimited inductive dimensionality that has been postulated by Poincar
e. If geometric dimensionality would be as inductive as Poincar
e has envisioned it, this Elie CartanÕs result would not be possible. Equating differential and algebraic methods of geometry was a mistake. The PM misinterpreted the notion of geometric space, for it treats spaces as sets rather than as structural objects, even though it was clear that spaces are also methods for handling objects immersed within them. Hence geometric spaces are objects too. Note that methods are objects in object-oriented parlance, just as steering systems are object-methods for handling the movements of cars. Even the physical space of motion resembles multispatial structure of hyperspace [3]. Hence it must not be treated as a simple 3D or 4D space, as former PM and MP used to do. I will show elsewhere that some images (representations) of an algebraic bispatial 6D hyperspace resemble relativistic 4D space–time, but this does not mean that 4D algebraic space–time is simple geometric space. Elie Cartan has not challenged RiemannÕs views. However, if one cannot extend dimensionality of single 3D geometric space, then a family of simple spaces must supply the higher than third dimensions, since dimensions are attributes of spaces. Since one can find more than four quite independently varying magnitudes, then the RiemannÕs intuition may be correct, although not the easy way he used to assume. Evidently the geometric dimensionality must be somehow expandable, but the postulative algebraic and topological approach is not the way to go. If the geometric dimensionality is properly attributed to spaces, then a set of coupled simple 3D spaces should form an abstract multispatial hyperspace, which is an operational hyperstructure of a different kind than simple spaces. Mixing types of abstract structures could invite back the old set-theoretical paradoxes that once plagued mathematics. We used to think that all representations of the same object are equivalent, whether formulated in algebraic or geometrical terms. However, this is not always true. In fact, algebraic and geometric models can be very different. This was the main problem with former exact sciences. If we want to move forward, we must change our old paradigms and

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then restructure the whole conceptual environment we have inherited. For most self-evident paradigms of former geometry are merely simplistic abstractions of ancient postulates. They appear broken and so are former theories of exact sciences. Defective paradigms must never be mended. They should be abandoned as quickly as possible, for they paint distorted pictures of the physical reality we live in. Although philosophically true, the claim that physics determines geometry is not factually valid; neither our incomplete physics nor our still unfinished geometry can provide model for each other as of now. Ideally they should, but this has not materialized yet. According to an ingenious interpretation of the GTR, geometry instructs matter on how to move and matter in turn instructs geometry on what to conserve [42], for matter and geometry are tied together. What I am saying is that there exists an additional geometric structure, which also influences matter. The extra structure is affected by nonphysical structural laws, and is distinct from the metric structure, which is ruled by known operational laws. Does it mean that, contrary to the GTR [42, p. 431], a prior geometry exists? No. The extra geometry is not prior. It has always coexisted with matter. Yet I have discovered it only recently. From the premise that no prior geometry exists one cannot conclude that the GTR got everything right. Physics supports geometry, but is not a substitute for the latter. In this paper I will formulate new abstract operational law, which fulfils the dream about vector multiplication of vectors. As great as the GTR is, it crowned unfinished classical physics by means of incomplete mathematics. The NEG exposed incompleteness of the, hardly aspiring for completeness, QM and the allegedly final and completely understood CM. Before one could explain logical discrepancies between mathematical and physical pictures of reality, they may be attributed to a prior geometry. The presence of multispatial hyperspace may appear as prior geometry, because former physics has not developed the mathematical ideas that prompted me to introduce hyperspace. However, I am not introducing any prior geometry here, but am admitting quite new geometrical features, which the previously ignored nonradial and mixed experiments implicitly indicated [2]. The GTR is large-scale cosmological theory of an amalgamated gravitational field. It provides thus only general, almost statistical account of gravity that is valid wherever all massive bodies can be seen as dissolved into the amalgamated pseudofield. Since such a condition is unacceptable for local phenomena, the GTR has to be appended in local settings [2]. Neither the GTR nor the STR can account for certain rotational phenomena such as the Sagnac effect [3,43] or mixed and nonradial effects of gravity [44]. We must not ignore experimental data or theoretical facts. The single-space picture of physical reality never really existed outside of some pure-mathematical minds. Former geometry is incomplete and so is former physics. In a strict sense, gravitational field is not a separate entity generated by mass, but it is always present wherever mass exists. Both are properties of matter and affect each other. Yet the radial linear curvature of space–time alone was insufficient to account for all the effects of gravity that are known today. Hence one must consider also nonradial (tangential and binormal) aspects of the curvature, which were regarded as nonessential at the time when the GTR was created [2,3]. These aspects imply presence of the NEG, which was unknown back then. They expand the linearly acting radial gravity into three dimensions. Even travel along equipotential surfaces is not toll-free as we used to think, for gravity affects energy also in nonradial directions [2]. Laws of former physics have been formulated algebraically, with disregard for differential geometry and only with lip service for such abstract discrete geometrical structures as vector spaces. Vectors were used for visualization of physical effects, but their hidden consequences were simply ignored. The aforesaid result of Elie Cartan was considered so insignificant that it was almost totally dismissed. Yet in conjunction with another his result, which will be discussed below, it could raise the question of whether algebras and geometries are really in sync. I would not care about synchronizing of any algebraic and geometric pictures of the physical reality, were it not for the fact that if these stay out of sync, then we are unable to decide which one of them is true. The synchronization is thus not a goal in itself, but just a way to ascertain proper understanding of their results, which must be assured by experimental evidence. Physics cannot judge between algebra and geometry, but it should provide experimental confirmation giving us ‘‘second opinion’’. Space and time are abstract methods by which we handle material objects. The methods are affected not only by properties of matter alone. Yet the way an object is embedded in surrounding it space depends on the objectÕs properties as well as on some properties of the space, which is our method here, in the object-oriented parlance of computer sciences (wherein method means a formal, legitimate way of handling an object). Distribution of mass alone is insufficient for representing gravitational phenomena [2]. This was confirmed by formerly unexplained experiments [1–5]. I am not saying that physics––or the GTR in particular––is totally wrong, but that it is logically incomplete without the NEG [2–4]. The state of incompleteness is the most stable feature of all theories. Hence no theory should be treated as a sacred cow, no matter how great was its origin. Physics is manifestation of pure logic or pregeometry ([42, p. 1212]), but we did not arrive yet at the point where the pregeometry has been manifested, and former physics is far from being complete. The pregeometry is still insufficient, however, for we need the pan-geometry of multispatial hyperspace to explain nonradial and mixed experiments [1–4]. If geometry could be reduced to physics or algebra, then the latter two would account for some geometric aspects such as duality that are not rooted in either of them, but are peculiar to

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geometry. The duality is not a sensu stricto geometric feature. It belongs to the external shell of PG that encompasses geometry, even though it manifests itself in geometries. Nothing in geometry requires duality, which appears as superfluous feature. Mathematical ideas of Elie Cartan are very difficult to explore. Synthetic approach makes them comprehensible [1,3]. They virtually revolutionized mathematics without us realizing that for over half a century. He stirred a Copernican revolution when it comes to its scope and physical implications. Its significance for physics is enormous, though still underestimated. Its mathematical consequences remain unexplored. Even he himself was not aware of the extent of his impact on sciences. In hindsight, however, neither his ideas nor the former ones are inherently right or wrong, I suppose. It is the domain where they apply that counts. He has found the limits beyond which our inborn ideas and common sense imagination may not apply. Yet that was just a prelude to his more stunning achievement. Elie Cartan might have asked himself the following question: Since the highest possible dimension of any simple geometric space is three, then what would a 3D+ geometric object look like? I am not saying 4D, because this would imply that the 4D object exists, in which case one may get any answer if it could not exist, but am saying any higher than 3D object. We must not postulate existence of anything. Since Stokes theorem provides switch between any 2D and 3D geometric object, it was natural to try to upgrade it to a 3D–4D switch. And that is exactly what Elie Cartan did. His approach is so subtle, however, that no (pure-) mathematical mind––himself included––was able to interpret it. But it is the unbelievable implication of his result that tacitly made it anathema. For he would have to admit that perhaps some properties of objects depend on the way they are represented. No respected scientist could dear to issue such a ‘‘truth in research’’ kind of disclosure, of course, and none was required. It was natural thus to tweak and twist both physical and mathematical formulas in order to show that nature is not acting crazy, despite the fact that QM provided some compelling evidence to the contrary. It is not up to mathematics, however, to defend the common sense, which is indefensible anyway. Eventually we must face the actual reality just as it is. We must not define the reality by postulating its properties, but we should explore the conditions of its existence and discover what it should look like. Twisted mathematics is difficult to discern and I will show it later. Physics is much more transparent, however, and therefore its faults are easy to see. Nonetheless, the tweaking and twisting was unnecessary. We got classical examples that could provide some logical templates for the allegedly strange phenomena of QM. Although I would not consider it strange that water, for example, rushes through all open floodgates at once, we do have conceptual problem with picturing single photon that goes through two slits at once, in two-slit experiment. It is because, if we always think of every photon as of a point-like particle, we tend to apply the very logic that is appropriate for point-like objects: a massive ball does not go through all open slots at once. However, the concept of multispatial hyperspace has an inherent duality of representations included, for a point in one space may look like a line or surface in another space [3]. When a photon knocks out electron of a solid surface, it surely behaves just as a point-like object would, but this particular behavior does not fix its mode of existence, so to say. Is mixing of objects, methods, properties and behaviors improper only for computer programmers who design object-oriented applications? If all physical and mathematical theories were devised with geometric objects in mind, like most objects on my computer screen, maybe we would not face so many daunting problems. One can imagine a set of paradigms in which the seemingly strange world of QM appears normal. Let us find them and explore their consequences. When I discovered the SSTF it was a shock and unbelief at first. Yet when I asked myself what could it mean, I realized that it implied just a multispatial structure of our physical reality, whose consequences have been suggested in some formerly unexplained experiments and observations [1–4]. Instead of asking why is the nature acting weird, I prefer to ask questions we could try to answer, at least in principle. If I do not see any rationale behind the natureÕs behavior, then what would I need in order to see it? What should the allegedly weird physical reality look like to appear rational? The idea of hyperspace originated from synthesis of requirements posted by few strange results of formerly quite unexplained experiments and observations. It has not been arbitrarily postulated; neither is it supposed to be axiomatized. I would not prevent an axiomatization for the sake of a comparison, however. Carefully crafted axiomatic system could be very helpful for certain analytic purposes, of course. Nevertheless, the idea that a set of axioms could serve as a container of true information about objects seems repulsive to me. Another stumbling block of the QM––nonlocality––is also quite legitimate and very natural in the hyperspace [3]. There is no problem with the nature. We may have problems with former paradigms that influence our thinking. Sometimes we are virtually paralyzed by our paradigms and cannot finish the (logical) sentence that we have started. The ingenious generalization of Stokes theorem by Elie Cartan is the best example of that. It is tremendous breakthrough in mathematical thinking, though unrecognized at first. It is a pinnacle of differential geometry and it opens the door to pan-geometry [1], for Elie Cartan has realized that there is very profound conceptual difference between interior and exterior differentiation, where the latter can be thought of as referring to some encapsulating geometrical objects. The idea behind it is simple, but some of its consequences are mindboggling. Going up and down the hierarchy of structural objects is not exactly the same experience.

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The most astonishing, although misunderstood, experiment in mathematics was performed by Elie Cartan. The essence of his ingenious experiment can be summarized as follows: Even though I cannot see what kind of animal is the higher than 3D geometric object, if I would skin it, I could get an idea of what it looks like. For linear differentiation in mathematics is like peeling a tomato. We can ‘‘differentiate’’ an abstract tomato without really ‘‘seeing’’ it, simply by keeping the analytic ‘‘knife’’ always tangent to its surface, as if using touch instead of sight, so to say. I am not trying to be too exact here, of course, but am conveying the message that in some instances one could simulate an operation on an object without being able to define the object, just by using an analogy, for analogy is the very essence of mathematical thinking [45]. It is analogy that really matters for comprehension of most scientific ideas. Formal derivations are merely comments, or post factum documentaries. Even a computer program can select derivation trees for proofs, without any understanding of the subject matter being proved. Said that, we could let computer programs generate quite new knowledge. A fairly sophisticated computer program should also be able to discover all that which I did, perhaps in a matter of hours rather than years, even though it may take years to design such a program. Nonetheless, the really amazing part is that some of the most unanticipated discoveries might have happened during the programÕs debugging and testing phases. The search for logical gaps and inconsistencies in scientific notions can be performed by certain computer-aided knowledge extracting system (CAKES) that would query a knowledgebase and generate leads to be evaluated by experts. A knowledge generator program could also offer alternatives and options based on such extracted leads. The formal synthesis of sets of alternatives could then be accomplished automatically. Evaluation of its results is up to human mind, of course, for the system would replace just the brain, not the mind. Once designed, the actual synthesis would require mostly mindless computations.

4. Slam-dunk: Elie Cartan refines the Stokes theorem The often misunderstood result of Elie CartanÕs generalization of the old Stokes theorem is this: He actually asked what would be the differential of a higher than 3D geometrical object, and obtained a bivector (as the value of crossproduct of two vectors) [39,40]. One could expect that a simpler 3D object (or a triple vector product, for that matter) should result, if geometric dimensionality would be inductive as topology assumed after Poincar
e (see [1,3,33,46] and Refs. therein). This unexpected result threatened the former geometric paradigm of single-space structure of our physical reality. Since each of the two vectors of the bivector could correspond to a surface, then the lowering of geometric dimensionality of a 3D+ object by differentiation gives two tied surfaces, as if the 3D+ object was actually composed of two 3D objects in two separate geometric spaces. This generalization of Stokes theorem reveals a multispatial structure rather than single 4D space [1]. It suggests existence of two 3D spaces arranged in a 6D bispatial hyperspace. Presumably he did not grasp broad consequences of his result, for he never challenged the single-space paradigm of former geometries. He virtually discovered, or perhaps at least sighted, an entirely unanticipated abstract mathematical infrastructure of the physical world we live in. Unfortunately, he probably dismissed the sighting as an abstract theoretical mirage. The two results of Elie Cartan uphold each other. They provide theoretical evidence for actual existence of the abstract hyperspace. The multiplicative character of bivector in conjunction with rules for differentiation of scalar composite functions strongly suggest compound hierarchical structure of the multispatial hyperspace he stumbled upon. Synthesizing the essence of his results I have attained thus new insight into abstract geometry, without any references to physics. The fact that physics also points to that [1] makes me even more confident that my conclusion is correct. For I have devised quite new physical predictions––based on some abstract structural (mathematical) requirements for the hyperspace––and found them experimentally confirmed [2–4]. Several formerly unexplained experiments and apparently confusing observations point to presence of the multispatial hyperspace as their cause. As a matter of fact, Ed Kasner has already showed [3,13] that an algebraic approach to Einstein equations suggests that dimensionality should not grow inductively as 1, 2, 3, 4, 5, 6,. . ., but in a discrete sequence, namely as 1, 2, 3, 6. His result was also overlooked, because it was quite incompatible with the predominant then pure-mathematical mindset of single-space geometry [3]. PM used to water down the Elie CartanÕs results to the level of algebraic geometry, trading off integrity for continuity of mathematical thinking since antiquity. In PM the bivector is treated merely as its value, which equals to the area between its two vector components, rather than as really geometric object. To paraphrase the situation: The PM does not answer the question why were apples packed in two separate bags, even though one single bag was supposed to be large enough to hold them all, but instead it tells us how large area would the apples cover, which––even though quite correct––was not really the question asked there. The PM is very evasive when it comes to dealing with any reality. The basically algebraic approach of Grassmann, Riemann, Gibbs, Clifford, Ricci and Levi-Civita was very inspirational, but geometrically off target (I shall discuss Hamilton and Cayley elsewhere). In the history of development of

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mathematical ideas they had played important roles [47]. However, we must not discard the breathtaking achievements of Elie Cartan and Ed Kasner, which were too subtle even for the PM. It is not just geometry vs. algebra, for GaloisÕ group theory was ignored for almost a century too. It is really about arbitrary postulates vs. abstract experiments. The supreme reign of derivations from postulated/assumed notions is over. The overlooked results challenged former physics. Instead of investigating the shocking consequences of these results and changing its attitude towards geometry, the PM just tried to suppress them. However, I have resolved the apparent contradiction between the highest-dimensional simple space (3D) and the obvious need for 3D+ geometrical objects by conjecturing existence of multispatial structures [1,3] and investigating experimental evidence in favor of their physical existence [1–4]. This conclusion cannot be proved directly, because it does not follow from any conceivable set of axioms of former geometries, for nobody foresaw multispatial hyperspace. Axioms were designed so that they tend to affirm contemporary paradigms. Hence one could not derive from them anything really new that would contradict the assumptions of the former paradigms, upon which they were built. In fact, the G€ odelÕs undecidability theorem [48] supports this assertion. Technically the PMÕs approach to mathematics is not at fault. However, it is actually misrepresenting mathematics and physics by creating the illusion that they have no outstanding foundational problems. Henri Cartan and the other giants of twentieth century French mathematics have formalized some axioms and compiled them into coherent theories, published under the name of long gone mathematician Nicolas Bourbaki. They did really marvelous though incomplete job. They have showed that one can present PM as if it was already quite complete [49], while tacitly omitting all its really tough issues. Yet by ignoring difficult questions they were actually building new mathematical Almagest, a splendid abstract mathematical model of physical reality that is conceptually out of sync with the very reality it was supposed to model. To match the elegance of BourbakiÕs PM presentation with totally new ideas and conjectures would be a formidable job. Nonetheless, the SM should rather trade elegance for truth, which was never fashionable anyway. The PMÕs old concept of proof as a derivation from primitive notions and axioms is inadequate for issues of existence. If something follows from an assumption, this does not mean that it can be constructed and consequently exist. Hence synthetic reasonings should aid proofs in order to preserve the integrity of conclusions derived from axioms and primitives. If the derived object exists or is constructible, and yet it creates a real or apparent logical conflict, then one should question the validity of its set of premises. These cannot be both valid and lead to mandatory though invalid consequences. If a resolution of contradictions makes an existential claim necessary, this fact may indicate existence. Yet it is the construction that assures the possibility of actual existence. The PM has built axiomatic systems, based upon which one could prove or disprove almost anything by using formal derivations. However, no derivation from axioms and primitives can guarantee that what seems to follow could be constructed and consequently that it can exist. The ancient Euclidean methodology of mathematics was already dead on arrival. Old perceptions of the mathematical infrastructure of physical reality may not count. They have been purposely oversimplified during development of physical conjectures––and that was all right. Yet to claim that our physical reality should be built upon those old, simplistic mathematical assumptions would be arrogant at best. Mathematics should be founded upon physics and then entirely new physics should be created upon the new mathematics. Mathematics cannot afford to ignore experimental feedback from physics. Mathematics defines the shape of physical reality. We should deduce true mathematics from logical constraints placed on the physical reality. After Poincar
e, the PM used to count dimensions consecutively, which is quite admissible, but its subsequent claim that they can be added one by one was unwarranted. We must not impose our ideas on nature. We can always enumerate dimensions consecutively, but one should not deny the fact that they seem to come in triples [3]. The way we count them does not count. The generalization of the Stokes theorem by Elie Cartan was not deficient. Yet it produced results incompatible with the old, single-space paradigm of geometry. Hence the single-space paradigm is entirely wrong. Although not incorrect, the usual, formal algebraic presentation of his abstract results concealed their true meaning. Unwilling to recognize its own achievements, the PM became math-magic, a sort of. The pursuit of consistency created virtual censure of mathematical thinking. Science is an ongoing work in progress and therefore its theories could be inconsistent at any given time. Yet their controversies should be openly discussed. This is a vivid example where synthetic approach to mathematics reveals quite unexpected features, which were hidden behind the arbitrary postulative formalism of the PM. If open questions are the door to unknown mathematical features of physical reality, then by expelling such questions from mathematical writings the PM has effectively shut the door to its own and the MPÕs future progress. Though unquestionable, the results obtained by Elie Cartan run contrary to our traditional perception of geometry, and of the physical world around us. His unanticipated geometric results are tantamount to the discovery of quite new world where physical reality blends with an ingenious design that was beyond our wildest imagination. Suddenly we got answers to questions that no one would dear to ask before. From as abstract as why the matter always moves, to as direct as how come the sun survived its differential rotation [2]. Elie Cartan saw mathematics as handmaid for other

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sciences––or as a tool to justify their theoretical framework, I suppose. For otherwise he would have realized how strange a message is really conveyed by his results. Once one deploys definite mathematical model, a concrete foundation is being laid for the subject to be modeled. After choosing a mathematical model, physics is not much more than an experimental comment. We should bear this in mind when making our choices. Physical reality is almost fixed by its underlying geometrical infrastructure. That is why PM needs feedback from physics.

5. Multispatial structure of the hyperspace Exterior derivative of an encapsulating geometric object (in the sense of Elie Cartan) may appear as lowerdimensional object [1,3,39,40]. However, if the exterior object should comprise both the primary geometric object and its dual [1,3], then geometric dimensionality is not really strictly inductive set (1, 2, 3, 4, 5, 6, . . .) as topology postulated. Since all three distinct directional derivatives should exist in each LVS, then the exterior object should possess at least six quite distinct directions (hence also six dimensions), where each triple must belong to one of the two dual 3D LVSs. Evidently the geometric dimensionality itself seems quantized [1,3]. All geometric dimensions come packaged in triples (3, 6, 9, . . .) within 3D LVSs [3]. One just cannot add one or two geometric dimensions to a space. Only whole 3D LVS can be added to the multispatial structure of the hyperspace that holds many 3D LVSs as its components [1,3]. Geometric dimensions are localizers akin to abstract symmetries and as such they may form abstract algebraic groups. Evidently geometric dimension always comes as a certain class property of some class of abstract objects called (geometric) LVSs––never by itself. However, dual space is just another abstract spatial structure over the same manifold. It is a distinct image, but not a separate object. Dual space is just another face of the same ‘‘coin’’ (manifold). This feat creates apparent physical nonlocality. The duality of LVSs is based on inverse transpose [3]. Hence what seems nonlocal in primary space can still be local in its dual one. Such an apparent nonlocality would not involve any interaction at a distance in the dual space, even though in the primary space two twin particles may be separated. That is why one particle can almost instantaneously adjust its state to a change of its twin particle state. It does it in the dual space wherein they still appear as staying together. The mechanism of the adjustment is not quite clear to me; neither is it in the former QM. However, the multispatial hyperspace does replace former magic with merely a wonder. The design of hyperspace is a wonderful feature, without the logical problems of an action at a distance or an instantaneous interaction. The almost instantaneous adjustment of spins, for example, is like a switch that turns on one light bulb while turning off another (coupled to it) light bulb at the same time. The single switch causes that both these coupled events are triggered in parallel, not consecutively. Theoretically the 3D LVS cannot exist as a standalone space [3]. Since it evidently exists, it should be paired with its 3D dual counterpart. This is a known fact in theory of LVSs. For every simple 3D LVS comes hand in hand with its dual 3D LVS, and the two LVSs are joined like two faces of the same coin. Together, they create a (3 + 3)D ¼ 6D hyperspace over single 3D topological manifold [3]. We have six dimensions to consider, but it is still a 3D manifold with two different and quite distinct vector bases that are mutually exclusive. As it stands now, however, even such a pair of LVSs forms an operationally incomplete spatial structure and so it is theoretically unstable. For duality of LVSs requires an inverse transpose [3], which may pose problems with zeros. To attain both geometric and algebraic stability, even the hyperspace must be somehow codetermined by an abstract extra structural symmetry. I will show elsewhere that this is indeed the case. The 6D multispatial hyperspace changes not only geometries, but also algebras. The aforementioned issues are very sensitive and I am (pain-) fully aware of that. They have been solved in a way that probably no one ever expected (myself included), and will be published shortly. Nonetheless, we have arrived at the point where I can enhance the notion of traditional 3D vector (3-tuple). However, I could not define something before it is fully explored. We do not need an exact definition of vectors and numbers to communicate our ideas, as PM used to claim. If the reader would not have an operational idea of what vectors and numbers are, then she/he would not be reading this paper, I suppose. The idea that one has to define something and then try to extract its properties from that definition is just another conceptual absurd, which the PM carelessly propagated. Complete definitions of objects should result from investigation of stability of objects, not only from some abstract operations performed on them. Though an initial definition is necessary, it should not be treated as container of proven knowledge about the defined object [50]. Definitions may then evolve with increasing knowledge. Former PM usually defined algebraic linear vector spaces by the following specification of operations on vectors X, Y, Z and real numbers r and s [51]: Commutativity : X þ Y ¼ Y þ X

ð1aÞ

Associativity of addition : ðX þ YÞ þ Z ¼ X þ ðY þ ZÞ

ð1bÞ

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Additive identity : X þ 0 ¼ 0 þ X ¼ X

ð1cÞ

Existence of additive inverse for any X : X þ ð
XÞ ¼ 0

ð1dÞ

Associativity of scalar multiplication : rðsXÞ ¼ ðrsÞX

ð1eÞ

Distributivity of scalar sums : ðr þ sÞX ¼ rX þ sX

ð1fÞ

Distributivity of vector sums : rðX þ YÞ ¼ rX þ rY

ð1gÞ

Existence of scalar multiplication identity : 1X ¼ X

ð1hÞ n

with vectors treated as n-tuples of scalars that belong to a real number field R . The number of dimensions was coupled to the number of independent linear combinations of the LVSÕs base vectors [52]. Algebraically speaking the definition is perfect, but when it comes to object-like approach to vectors it sounds like a definition of certain coordinate-based, scalar representation of geometrical vectors. None of such standard sets of definitions of vectors and LVS is adequate for the synthetic approach to mathematics and physics, however. Let an abstract algebraic 3-tuple be called traditional 3D vector for now, and then by investigating their representations and by operating on such 3-tuples we will learn the actual meaning of physically meaningful geometrical vectors seen as objects. Although valid, the axioms (1a)–(1h) are insufficient, because they do not support all object-like attributes of vectors. MH seems somewhat incompatible with the operations on vectors that are known thus far. However, the very way vectors were presented by the MP is artificial. Not everything you have learned about them is wrong, but there are many things missing and their former reckless interpretation left much to be desired. Vectors are not necessarily what the MP would like them to be. They could be only what the nature can operate on. We have to discover the actual meaning of vectors, if these should reflect physical reality. With no preconceived ideas about what vectors are, I will do whatever nature and logic forces me to do. Then I will show that the notion of vectors can lead to operations we surely dreamed about, but were unable to devise. The fact that the conclusions lead to consistent new operations on vectors constitutes virtual theoretical evidence that my conclusions are correct. In this and few subsequent papers I will show that nature too performs such operations, for we need both theoretical and experimental physical evidence for existential claims. Mathematics needs truth. Beauty and elegance are less essential. The demand for extra 3D spaces has emerged from mathematical analysis of physical phenomena, but the statutory lack of interest in physical reality that was maintained by the PM, has stagnated its development. The PM did not investigate its most fundamental questions, because it was not really interested even in its own abstract mathematical reality. In order to create new physics, one must first resolve mathematical inconsistencies and logical conflicts, whether actual or apparent. For even in certain well-established formulas of such allegedly complete branches of classical mathematics as analytic geometry and vector calculus, we will see unbelievable misgivings and undisclosed disastrous logical inconsistencies. We must eliminate them in order to proceed with advanced investigations of physical phenomena.

6. Vectors in multispatial structure of hyperspace Cartesian representation of a 2D plane in 3D Euclidean space is written as: ax þ by þ cz þ d ¼ V þ d ¼ 0

ð2Þ

where the coefficients a, b, c, d can be normalized by the common factor: ða2 þ b2 þ c2 Þ1=2 ¼ ðV  VÞ1=2

ð3aÞ

so that the value of the vector n normal to the plane (2) equals to: jnj ¼ jdj=ðV  VÞ1=2 ¼ d

ð3bÞ

which gives the planeÕs distance to the local center of coordinates [53]. V is the algebraic stack vector [54] that represents the stack of parallel planes (2). Nonetheless, those types of vectors are not their inherent attributes. Types of vectors are not specified among the aforementioned properties of vector spaces, but depend on the vectors particular representations, just as a straight line can be defined in terms of points or in dual terms of intersecting planes. Eq. (2) is logically ‘‘dubious’’, however. For it raises extremely serious conceptual problem in a 3D coordinate system. For if its three ‘‘regular’’ coefficients a, b, c are related to coordinates, then where is the coefficient ÔdÕ represented in the whole system? Since it is evidently not a local coordinate and moreover, it counterbalances (nullifies) the

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traditional 3D local vector V, then what kind of magnitude does the ÔdÕ really represent in this setting? Only another vector can nullify a vector, if these two should be physically meaningful. Of course, one could enclose the vector in vertical bars (as in the Eq. (3b)) and bar any mention of the apparent contradiction, as the PM used to do. However, I want to resolve this conflict. Difficult issues should not be seen as an embarrassment, but as an opportunity for an enhancement. If I would replace all Latin letters in the Eq. (2) with corresponding to them Greek ones, then I can rewrite its (normalized and fixed) representation with usual directional coordinate versors (i.e., unit base vectors: j, k, l) as follows: ja þ kb þ lv þ d ¼ W þ d ¼ 0:

ð4Þ

where fixing the equation hides the variables x, y, z for the sake of simplicity. Since presence of varying coordinates x, y, z would not affect our reasoning here, I will omit them in order not to needlessly obscure the whole picture. Except for literal substitution nothing else has changed there. But now the logically dubious algebraic Eq. (2) turned into geometrically incorrect Eq. (4), for the coefficient d is nowhere represented within the Cartesian coordinate system ðj; k; lÞ, even though the coefficient d has not been distinguished from the coordinates a, b, v; neither is d a coordinate in another coordinate system. Yet the pseudoscalar d somehow nullifies the fixed traditional 3D vector W, which should not be possible for physically meaningful object-like vectors. Eq. (2) appears somewhat invalid in geometry. The Cartesian analytic geometry is actually an algebraic approach to geometry, not really geometry per se, for it deals with coordinates of geometric objects rather than with the objects themselves. In other words, analytic geometry is actually algebraic representation of the Euclidean geometry. Since substitution of variables in the formerly uncontested algebraic formula (2) produced wrong geometric formula (4), both of them must be somehow logically deficient. Yet the PM has never disclosed presence of this peculiar controversy, as far as I can tell. What is then hidden behind the conspicuous disparity of former algebraic and geometric representations of the same geometric object? Acceptance by algebra does not preclude an equation from being incorrect in geometrical context. Algebra is devoid of many rules that are enforced in geometries. What would be a transgression of geometric laws is not being enforced in algebra, because algebra has not enacted such laws. Hence no algebra, or topology for that matter, should impose its relaxed methods on no-nonsense geometries. Algebras are too simplistic for geometries [1–3]. Nevertheless, all geometric and algebraic representations of the same object should be consistent. If simple translation from an algebraic language into a geometric one gives incorrect representation, then something went terribly wrong with the former PM. Postulative algebraic methods turned operations on numbers and variables into manipulations of notions, ideas and minds. It is not inadmissible to devise some logically inconsistent abstract equations, but mathematical foundations for physics should be more reliable than that. Poncelet emphasized that, in geometry, the imaginary unit ÔiÕ designates such variables that cannot be represented in a given geometric system, if you will [1,55]. The imaginary unit indicates foreign, or somehow incompatible, variables. Even the position of the coordinate center itself should be quite independently determined from outside of the coordinate system. The three coordinates that are relative to the given system are obviously insufficient to determine an object in 3D space. They determine it in its coordinate system alone. Standalone simple 3D geometric space is unstable. If Eqs. (4) and (2) should be geometrically correct too, then I must write them symbolically as: W þ FormVectðdjiBÞ ¼ 0 ) ifWg þ ConvBasefFormVectðdjiBÞgjA ¼ 0

ð5aÞ

V þ FormVectðdjiBÞ ¼ 0 ) ifVg þ ConvBasefFormVectðdjiBÞgjA ¼ 0

ð5bÞ

where

fWg ¼ fja þ kb þ lvg ¼ fa; b; vgjA;

and FormVectðdjiBÞ ¼ id when

B ¼ Null

with A ¼ fj; k; lg

ð5cÞ ð5dÞ

Here the algebraic 3-tuple fWg ¼ fa; b; vg corresponds to the whole vector W represented in the base A. The imaginary unit makes any tuple basically direction-independent. FormVect is an abstract operator that splits a scalar into a foreign vector with an imaginary base B. ConvBase is an operator that changes base of a tuple from B to A. For the terms ÔdÕ and ÔdÕ should represent certain foreign traditional vectors (3-tuples) in a foreign 3D space, if they are supposed to nullify the primary vectors W and V, respectively, as Eqs. (2), (4) and (5) seem to indicate. And vice versa, the primary vectors should denote offsets of these foreign vectors in the foreign space where d and d are represented. This issue will be discussed in more detail elsewhere, because we do not have all elements in place yet. Nonetheless, I can show that this approach retrodicts results of CM and predicts quite unexpected features. They will also be elaborated elsewhere, because I must develop theoretical apparatus needed for that. Eq. (5d) is just an abstract restatement of the aforementioned PonceletÕs thesis when the foreign vector base B is Null (or nonexistent) and so the ÔidÕ is a directionless and ‘‘baseless’’ quasi-vector.

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The vector W, for instance, denotes local vector in the coordinate system defined by the versors ðj; k; lÞ, whereas the symbol fWg denotes the vector W as 3-tuple (i.e., a set of its coordinates). However, the coordinate versors themselves are imaginary. Therefore if I must compare the vectors W and V with their foreign counterparts FormVectðdjiBÞ and FormVectðdjiBÞ, then I should consider the vectors also as tuples with imaginary unit i, just as the foreign vectors formed from scalars in the Eqs. (5). For otherwise I would inadvertently create complex vectors. Algebraic vectors written as tuples of coordinates (without versors) must be appended with imaginary unit. Hence in order to compare vector V with d, I must write them as imaginary tuples ifVg and ifFormVectðdjiBÞg. This is similar to treating a local vector-tuple irrespective of the local directions, which are irrelevant to any foreign space anyway. To compare any primary and foreign vector requires treating them both algebraically as some imaginary tuples, regardless of their directions in their respective spaces. The imaginary unit ÔiÕ is just direction-independent abstract versor or a compound 3-versor that points to outside of the primary space, geometrically speaking. Algebraically the i, j, k, l act alike. With two 3-tuples Eqs. (5) make both algebraic and geometric sense. Two equal tuples have corresponding components equal, if represented in exactly the same base. When seen as a traditional algebraic vector, the term fdg is just an abstract algebraic tuple: fFormVectðdjiBÞg ¼ fd1 ; d2 ; d3 gjiB. Now we can merge together the results of Poncelet and Elie Cartan. If I would reformulate the normal vector in Eq. (3), then the offset will equal to: jnj ¼ jidj=½ðifVgÞ  ðifVgÞ1=2

ð6Þ

The dot product of imaginary vectors, that corresponds to a pseudoscalar in the Eq. (3) I will call henceforth biscalar. Taken together, however, the two sets of Eqs. (5) and (6) also suggest that one may also write the biscalar as follows: jFj ¼ s½ifVg  ifVg

ð7Þ

where s is a constant. Eqs. (5) indicate that the pseudoscalars d and d actually represent (foreign) veiled imaginary quasi-vectors, which would require an extra 3D vector space, if we insist that they too should be housed somewhere. Since the offset can be measured (or evaluated), but cannot be represented within the primary 3D space, then even the Cartesian 2D plane in the Eq. (2) is actually represented within two distinct 3D geometric spaces (primary and a foreign one) at once, both erected over the same manifold. The rationale for my reasoning is simple: From the apparent contradiction that the offset cannot be represented in the primary space and that it must be a vector (since it counterbalances a vector), it easily follows that there must exist another space coupled with the primary one, for the contradiction to disappear. This logical synthesis is so simple when it comes to its abstract associations, that even fairly sophisticated computer program should be able to come up with one like it. Computers can generate and examine trees of notions with associations between various classes of such notions and then offer entirely new concepts based on formal logical synthesis of conceptual associations between them. This process of automatic conceptual synthesis is technically feasible and it should work for exact and semi-exact sciences. From logical analysis I synthesized very intriguing geometric conclusions. These conclusions are not tailored to what we knew, but to the overriding principle of logical consistency and the feasibility of flawless operations. Multispatiality is necessary for fully analytic representation of 2D plane is a 3D space. Single standalone 3D geometric space is thus underdetermined (unstable). Hence multispatiality is unavoidable. For if the traditional 3D vector V in the Eq. (2) is nullified by a number d, then the number d must somehow acquire a vector status, although perhaps not in the primary space that houses the vector V. Even the traditional 3D vectors must somehow span multiple 3D spaces. This conclusion was not simply postulated, but synthesized from the logical contingency for validity of all Eqs. (2)–(5). I have obtained the properties and character of the coefficients d and d from preconditions for validity of the equations in which they are featured. By operating on objects I probed their behavior. Since Eqs. (2), (4) and (5), (6) should represent essentially the same geometrical object, then only Eqs. (5) and (6) are acceptable, for both geometric and algebraic representations of the same geometrical object should be in sync. The multispatial hyperspace sits thus at the highest level in the hierarchy of abstract spaces known thus far. Although Eqs. (5) look like quaternions, geometric implementation of a structure with two triples of independent parameters needs a 6D hyperspace composed of two 3D single linear vector spaces [3]. The imaginary unit ÔiÕ turns the formerly incomplete Eqs. (2) and (4) into valid Eqs. (5) and (6), which are both algebraically and geometrically correct. Since the other scalars a, b, v are represented on coordinate axes, then the scalar d should be either included in these or distinguished from them. We have no other choice here. However, the actual meaning of the distinction has profound consequences for physics and mathematics. The fourth parameter defines an offset with respect to the primary coordinate systemÕs center. But if it also creates symmetry, as it does, then with respect to what? A space is equated to another space of the same kind. Geometric objects can only be equated to (or counterbalanced by) objects of the same type. Henceforth I will call the 4-tuple (composed of the traditional 3D vector coupled with a biscalar) a hypervector, because it obviously spans distinct 3D spaces within the multispatial hyperspace.

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Traditional vectors are parts of hypervectors that span mutually dual LVSs. Writing equations is about balancing. The former paradigm of single-space reality disregarded the very fact that equations with geometric objects do not equate them to zero, null, or nothing, but to objects of the same type so that the difference is zero. Hence even the old analytic geometry virtually needs the multispatial hyperspace in order to give consistent representations of 3D vectors, lines, planes and surfaces. Algebraic representations of geometric objects must be unambiguous. Compliance of algebra with geometry should be on our target, for we just cannot always deduce valid conclusions from inconsistent premises. If it takes multispatiality to attain consistency, so be it. Multispatiality emerges from all three requirements: logical, algebraic and geometrical. Single-space reality paradigm was an oversimplification. It is indefensible even at the level of Cartesian analytic geometry, which by the way seems to finally agree with the abstract theory of geometric LVSs.

7. Isometry defines vector multiplication of vectors Isometric transformations must preserve length [56,57]. This should apply to physically meaningful vectors too. If abstract isometry is taken seriously, then all admissible operations on object-like quantities should comply with it. Isometry of operations should be the overriding principle for physically meaningful geometries. The length of the resulting object-vector produced by physically meaningful product should equal to the product of the lengths of its operands. Yet the very concept of LVSs sustains a lame duck kind of theory, for vector multiplication of vectors was never defined therein and no consistent explanation of what it could mean was ever given. Only scalar multiplication of vector by real number was consistently defined (see [58] and p. 42 in [56]) in addition to two standard products of vectors, namely the dot and cross vector products. One can check that for traditional 3D vectors (d ¼ 0) the following abstract operation of vector multiplication could exist: S  T ¼ S  T
jS  Tj

ð8Þ

It could mean actual vector multiplication of vectors whose result would be isometric (i.e. length preserving), were it not for the fact that the dot product ðS  TÞ is just a pseudoscalar, whereas the cross product ðS  TÞ is evidently a 3D traditional vector. Eq. (8) is certainly formally and algebraically valid, but it is not operationally isometric. Is operational isometry really too much to ask for, if the Eq. (8) is supposed to be also physically meaningful? Eq. (8) becomes operationally correct, once it is rewritten as follows: S  T ¼ S  T þ iðS  TÞ

ð9Þ

which means that vector multiplication of two 3D vectors comprises vector and dot products and gives geometric sum of bivector and biscalar (hence a hypervector in a multispatial hyperspace). The result looks like a complex vectorrelated object, because the imaginary unit is superimposed on the unit versors ðj; k; lÞ present in the dot product. Although the vectors S and T are local, their dot product (as a pseudoscalar) is a foreign (not local) geometric object, according to Poncelet. Eq. (9) is evidently isometric, for we get: jS  Tj ¼ ½jSj2 jTj2 sin2 ðS; TÞ þ i2 ½
jSj2 jTj2 cos2 ðS; TÞ1=2 ¼ jSjjTj

ð10Þ

The mathematics in hyperspace is almost perfect, for even algebra complies with geometry. Note that bivector is identified with the plane element (area) whose value jSjjTj sinðS; TÞ is the same as the value of both cross and wedge products of the given vectors, and that bivectors were defined algebraically as ordered pairs of vectors, whereas biscalar was identified with the negative value of their dot product, namely ½
jSjjTj cosðS; TÞ. The former algebraic treatment of bivectors was not incorrect and it fits a very nice schema [59]. But both algebraic and geometrical pictures of bivectors should be in sync, which is achieved in the hyperspace. Evidently the multispatial structure of the abstract hyperspace is also indispensable for consistent theory of LVSs. The PM has already recognized the fact that the exterior product is distinct and conceptually different kind of operation from the cross product, and it has instituted wedge product as the algebraic counterpart of the exterior one. Yet geometric role of exterior product was misunderstood. For even the PM does not deny that surface integrals (where the wedge product is being used) are quite different from double integrals––see p. 662 in [53]. However, we should operate on whole structures––whether geometric or algebraic––rather than arbitrarily reduce them to merely algebraic operations on single values. This result also suggests an incompleteness of the supersymmetric approach to physics that uses Z2 -graded algebra [60], which deals with traditional 3D vectors. I will discuss this issue elsewhere, along with the purely algebraic hypersymmetric approach that is based on similarly structured Z3 -graded algebra [61,62], although still under the old single-space paradigm.

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Eq. (8) defines vector multiplication of traditional vectors and so it can expand the theory of LVSs, which will be done elsewhere. The lack of this operation marked an incompleteness of the former theory of LVSs. It is very important operation for future physics, for it allows us to explain many curiosities in former physics. Former mathematics could not come up with the abstract vector multiplication of vectors, which is very simple operation, because it has suppressed even rational critique of logical inconsistencies of its clever, though deliberately postulative formalism. The formulation of an isometric operation of vector multiplication of vectors obviously supplies us with sound theoretical evidence in favor of the multispatial hyperspace. The simple correction to Cartesian geometry that makes it both algebraically and geometrically valid opens a new mathematical world, which is consistent with the idea of multispatial hyperspace and the spatial flow of time. I will show elsewhere that the star product defined by the Eq. (9) is valid also for hypervectors (as 4-tuples). The star product does look good on paper, but is it implemented in physically meaningful operations performed by nature?

8. No existential proofs in exact sciences Axioms and definitions should not be regarded as containers of truth about geometric objects. They are more like conveyors of what we now think the truth is, given the known circumstances. Hence no abstract derivations from them could really constitute a valid proof of existence. They may be viewed as consistent leads. Having devised several such leads in various formally independent and different representations, the actual proof can emerge from resolving contradictions that crop up from comparisons of consequences of those representations. Though still circumstantial, such proof is the best one can get at any given time. The idea of proof as the process from which an abstract eternal truth emerges is a wishful thinking. Proofs may evolve as our knowledge increases. This is the essence of the synthetic approach to strict derivations. PM must admit flexibility in order to became truly exact science. Mathematical proofs must be amendable as new discoveries reveal phenomena that were unanticipated before. They may also be expandable because of the new meaning that its old premises could acquire, not because these were somewhat incorrect. Exact sciences are subject to aging too. Physically meaningful vectors should comprise also pseudoscalar as their fourth component in order to be geometrically and operationally consistent. This is what state vectors in QM sometimes evaluate to. This interpretation of vectors is needed for mathematics as well as for physics. Former purely algebraic interpretation of vectors as just ntuples is logically untenable. To be fully operational vectors must live in a multispatial hyperspace rather than inhabit only single spaces. Even the old Cartesian analytic geometry requires a hyperspace. Without the synthetic approach, the old PM and MP cannot properly handle physically meaningful geometrical objects. Nothing was inherently wrong with the algebraic approach to vectors. But algebraic n-tuples cannot replace physically meaningful geometric vectors. Algebras and topologies are not the problem. It was the tendency to reduce geometry to algebraic or topological sets that did upset the balance between these different approaches. Yet the––a priori assumed––equivalence of all mathematical representations created even more serious problems. One of them is the apparent ability to design negative proofs that could disprove existential claims by assumptions disguised as contradictions, which are not intrinsic, but are set in arbitrary, postulative definitions. With the exception of logical contradiction, negative existential proof is impossible. One must not postulate existence of objects; neither should one define away something that is not logically contradictory––certainly not by using fancy definitions. Having multiple spaces in the hyperspace may introduce extra independent variables/parameters. However, John von Neumann has ‘‘proved’’ that such hidden parameters are impossible, because they contradicted an assumed definition [63]. This was the real ‘‘power’’ of PM: a clever definition could postulate away undesirable hidden parameters. A celebrity called the proof silly [64]. One can find an example of an event, which was announced with such preconditions that it could not take place as defined [65]. The QMÕs reliance on unfounded assumptions is disappointing [66]. The tweaking has turned against the PM and the MP. For it created an artificial world tailored to preconceived ideas about the physical reality we live in. David Bohm has formulated QM that admits hidden variables, but he did not find any one of them. He did not show what are those hidden variables and how could we actually measure or identify them. But he realized the possibility of their existence. His common sense approach recognized the fact that thermodynamical quantities, for instance, are merely averages of certain hidden variables that just cannot be observed by thermodynamicsÕ methods alone [67]. He suspected more fundamental set of laws [68] and defined a fluctuating quantum potential [69]. He realized that experiments suggest nonlocality [70] and insisted that the QM account is inadequate for individual systems. Yet he still concurred that if two observables do not commute, then they cannot be defined together, and that no wave function can exist that would be simultaneous function of all the operators which are significant for given physical situation, see p. 66 in [69].

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Presence of hyperspace suggests that relationships between variables could be more intricate. This prompts for redefinition of the notions of functional dependency and determination. Does the Eq. (7) mean that the value of the primary vector F depends on the dot product of the foreign vectors V, which are defined in another space? No. The foreign vectors determine the vector F. It can only exist whenever they do exist and vice versa, but it depends on something that vary in the space wherein it is represented. The two foreign vectors are just another representation (image) of the vector F in a different space. If for whatever reason these two foreign vectors would change, the vector F would also change almost instantaneously, because any change to the foreign vectors is also a change to the primary one. This would appear as an almost instantaneous, apparently nonlocal change, which may happen when there is no physical interaction between the primary and these foreign vectors. Physical interactions propagate at the speed of light. Without physical interaction, however, change to one image (spatial representation) of an object will also induce change to another image of that object. Contrary to EinsteinÕs claim that gravity propagates at the speed of light, certain measurements show different speeds––depending on the particular experimental or observational setting––which fact suggests that apparent nonlocal aspects of gravity actually can propagate almost instantaneously [71], while its purely physical effects travel at the speed of light. When its propagation appears instantaneous, it is because there is no propagation of gravity at all in those cases. We observe visual changes only at the speed of light, of course, but this fact does not preclude higher speeds of propagation of abstract (i.e., nonphysical) events. Actually the question seems irrelevant to the GTR [72]. Similarly no mass-energy increase is being ‘‘performed’’ for each particular observer, but each observer sees different slice of energy spatialized in the TBS [4]. When considered in the multispatial hyperspace, even apparent action at a distance seems to make sense.

9. Biscalars in multispatial hyperspace The aforementioned fact that objects in a 3D space actually need all four independent parameters to be unambiguously and fully determined signifies the importance of parametrization. In CM the usual LBS was parametrized by time. The notion of independent variables in CM is based upon their presumed uniform and steady change. When the EinsteinÕs STR showed varying time rates, the old belief in mathematically ordered world was shattered. Nevertheless, mathematics did not crumble, even though its idea of arbitrary variability was defeated. By clever formulation of axioms and definitions, however, the PM stalled doubts about functional relationships. The variable that a function can depend upon is given only in a particular context. Geometric functional relationship relies on a spatial representation and on its parametrization, where the parameter is often viewed as basically independent linear variable, which does not belong to the spaceÕs primary system of coordinates. The space of motion, for example, was treated as the LBS with time as its independently varying parameter, but time-duration is not represented as a coordinate within the LBS and neither is mass or energy or force. To say that radial force of gravity physically depends on distance between two bodies was a grave misconception. The variability of the force is inversely proportional to the square of distance, but the gravitational force evidently depends on both masses of the bodies in question. Although no one ever contested that, we do not seem to realize what this actually means in an infinitesimally small surrounding. Since no functional relationship in physics is arbitrary and none of them should be ignored or misrepresented, then we should try to find out what does it mean. What does the inverse square law of (radial) propagation of gravity really tell us about our physical reality? Note that the inverse square law does not really refer to gravity, but to its propagation from a central source––it is the same for a light bulb. The inverse square law is not a physical law, but a structural (mathematical) law. Since topological spaces are actually manifolds, over which the LVSs are constructed [1,3], if two LVSs differ, their differences may arise either from their structures or from a topological difference between the manifolds or from both. The main result of topology was that the volume of objects does not matter [73]. Sometimes called miraculous multiplication of spheres, it was generalized into Banach-Tarski paradox, which showed that the curious ‘‘feature’’ stems from one-to-one correspondence between a set and some part of it, and so the paradoxical decomposition emerged [74]. What was a minor fault soon became a major feature of topology, which could cause a disaster when applied to certain discrete physical phenomena. Since rate of time flow, which is a parameter in LBS, is represented as 3D variable in TBS, which is parametrized by distance, the LBS and TBS are called mutually dual (symmetric) spaces. Hyperspace of motion comprises the two dual spaces. Twin spaces give spatial representations for variables other than the parameters. In this sense twin spaces could be thought of as separated by a chain of dual spaces. Hence the terms ÔdualÕ and ÔtwinÕ do not really designate spaces, but relationships between pairs of spaces. Since geometric objects seen as varying in dual and twin spaces are not directly represented in the primary space in question, their images must somehow be ‘‘encoded’’ via the fourth quasi-

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dimension. The hyperspatial geometric object is as if sliced into simple single-space images, which are coupled together by those quasi-dimensions that provide links between these spaces. Let an operator InstVect (instantiate vector) switches between two 3D spaces via a fourth quasi-dimension. Note that operators are not geometric objects, but algebraic ones. Just as the cross (outer) product of two vectors erects a new, linearly independent local vector in the very same 3D LVS, so also the dot (inner) product of two vectors generates a new foreign vector in a twin LVS, even though the productÕs vectorial status seem to change to a pseudoscalar in the same (primary) LVS, indeed. The pseudoscalar is just one representation, not really the only result of the dot operation. Both cross and dot products of vectors do generate vectors, but the dot product actually spans two LVSs. If true, this new interpretation could expand the notion of both projective and abstract duality. Eq. (7) suggests that the expansion is justifiable and actually unavoidable. Although not entirely wrong, the former postulative approach to vectors concealed this undocumented new feature. Eq. (7) has been formulated algebraically in terms of values, of course, but it suggests that its geometric treatment in terms of vectors is not impossible. We cannot be constrained to algebra. Nonetheless, I will not postulate that possibility, but show that the nature has already realized it. One cannot reduce mathematics to logic or geometry to algebra without loss of comprehension of the original subject. If vector denotes physically meaningful geometrical object, then it cannot be just dissolved by abstract operations, unless these operations would carry definite physical meaning, which they do not by their abstract definitions. Only other vector of the same class could nullify a vector. I am not saying that the dot product of vectors was defective, but that it may point to an incompleteness of former understanding of vector calculus. Treating geometrical vectors as merely algebraic n-tuples of numbers was a logical mistake and misunderstanding of geometric issues. The SM does not discard previous mathematical achievements, but it exposes their veiled contradictions. It does not try to squeeze new mathematical results in traditional conceptual structures, but it expands the structures to comply with results of physical experiments.

10. Evolution of abstract and physical duality A vector in one space could be a pseudoscalar in its twin space, and vice versa. By the same token, quaternion-like objects can be viewed as vectors defined on top of bispatial structures. Distance and time duration are dual [3]. The duality is very powerful idea. It has virtually exploded the former geometry. It indicates that 3D vectors are indispensable parts of compound abstract geometric structures, which may comprise diverse components [3]. Although duality was first observed in projective geometry, its meaning was downplayed by the PM. The projective duality principle (PDP) for plane states that in theorems about planar incidence relations, points and lines are interchangeable. For any two points determine a line and two lines intersect in a single point. Similarly for a 3D space PDP states that points and planes are interchangeable [75–77]. The PDP is an amazing mathematical curiosity [78]. There was no apparent reason for it. The PDP appears as just a fancy, superfluous geometric feature, until one casts it into an abstract multispatial hyperspace, where duality seems indispensable. The PDP suggests that the complexity of physical reality is above of, and far beyond our imagination. Although Poncelet may have already discovered duality [79], Gorgonne has formulated it as the principle we know [80]. It applies to curves [81], cylinders [82], space structures [83,84], invariance of postulates [85], all nonEuclidean geometries [86], generalized projective geometry [87,88], intersections and sums [89], generalized hyperplanes [90–92], topology [93,94], theory of LVSs [95], algebra [96] and many other branches of mathematics. Though it can be proved in various ways [97], the actual reasons for its emergence were unknown. If vectors can produce vectors and also scalars and yet not vanish themselves, then certain new, abstract generalization of the PDP is thus necessary. In hyperspace the PDP can be upgraded to form an abstract duality principle (ADP), which should state that every single point in one simple space can be represented by a vector, hence a line or even by a plane in another simple space of the hyperspace. In the sense the PDP can be obtained from the ADP by a transformation between spaces. In fact, in algebraic terms of the exterior and the wedge products, duality operator gives quite identical result as the imaginary unit ÔiÕ would (see p. 108 in [42]). This fact indicates that conversions between dual and twin spaces are very real, indeed, and that there is actually nothing imaginary about them. I shall discuss this issue in more detail elsewhere. Obviously one cannot just transform a vector into a point, but a conversion would do that. Conversion is more than just a mathematical transformation, for it involves definite representation, which depends on space. What is then the space in which vector can be represented by point and yet still keep its vector status? Any manifold can have multiple spatial representations in LVSs, for every LVS is solely determined by its base. One can change base vectors in a LBS and obtain TBS (its dual LVS) wherein distance can be measured by time-duration interval [3]. This feature is utilized in astronomy without any mention of LVSs. Although this is so natural, the PM stumbled upon this issue, because it did not distinguish between spaces and point-set manifolds or structure and substance. Substance supports the structure

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that preserves the substance. In physics substance always has to be conserved. It cannot just disappear without a physical reason. Yet the aforementioned paradoxical decomposition of spheres makes substance in topology literally immaterial and makes spatial structures ‘‘unsubstantiated’’ (unconstrained). Vector spaces unconstrained by matter (manifolds) defy laws of physics. Although physically meaningful geometry must be spatio-temporal [98], time cannot be fourth dimension in the strict geometric sense of the term [1,3,4,99]. Abstract notion of space is founded on five underlying intuitive notions: distance, vicinity, continuity, compartment (or range) and closeness (proximity) [100]. Spatial dimensionality appeared as a limitless parameter [101–104]. In topological sense dimensionality can be arbitrary large [105]. The latter is true as long as it is applied to set-point manifolds. Although set-point manifolds are often called topological spaces, their identification with geometric spaces was taken for granted because the term ÔspaceÕ was used indiscriminately and ambiguously therein. Topology did not really generalize geometry, but oversimplify the latter in general, less exact terms. Topology cannot encapsulate geometry, because most geometrical objects can possess features unrelated to constituents of their underlying manifolds.

11. Dual representation of potentials and forces Eq. (7) suggests that, with the exception of our ignorance, there is nothing that would prevent us to write the Eq. (7) in the following form: F=G ¼ InstVect½ðiVÞ  ðiVÞ ð11aÞ where G is the gravitational constant. If true, the Eq. (11a) should retrodict some already known facts of physics. If the vector F is the radial force of gravity, then the vectors V should be (collinear) radial potentials from two massive bodies M and m separated by a distance R, and so we can write: F=G ¼ InstVect½ðiVM ðRMm ÞÞ  ðiVm ðRmM ÞÞ ð11bÞ Here the local radial potentials from the two masses VM ðRMm Þ ¼
M=RMm and Vm ðRmM Þ ¼
m=RmM appear just as functions of the common distance RMm ¼
RmM ¼ R and the masses are just parameters. This form is known in the CM. Nonetheless, the two local radial gravitational potentials should be actually functions of the two masses, which are responsible for them (or generate them), namely: VM ðMÞ ¼
M=RMm and Vm ðmÞ ¼
m=RmM . Hence I can rewrite the Eq. (11b) with potentials as function of the masses: F=G ¼ InstVect½ðiVM ðMÞÞ  ðiVm ðmÞÞ

ð11cÞ

In the Eq. (11c) the radial distance R is just a parameter, for these two local potentials only vary with the distance R between the massive bodies M and m, but they depend on the masses that produce them. The CM has never considered potentials and forces as formal, mathematical functions of these masses, even though it explicitly assumed that physical dependence. Mass is thought of as being enclosed within a surface, which can be represented by a vector normal to the surface, so that there is no conceptual problem with this representation. In fact, this is its physically correct representation. We should distinguish between functional dependence and just variability. CM was successful, because it was dealing with practically constant masses. However, the Eq. (11c) implies existence of a mass-based space (MBS). If TBS exists [1,3] then MBS can also exist, for mass and energy are two faces of the same thing, since time rate is related to energy. Hence the hyperspace of CM comprises three 3D LVSs: LBS, TBS and MBS - with nine distinct dimensions. The mass-length-time (MLT) system of the CM appears as a 9D hyperspace with three 3D spaces. Hence the Eq. (11c) can be written as: F=G ¼ InstVect½
ðiMðRjMl ÞÞ  ðimðRjmm ÞÞ=R2 

ð11dÞ

with both masses M and m represented in their respective mass-based vector bases Ml and mm and parametrized by distance R treated as a scalar therein. Since the InstVect operator must not affect values, then one should keep the well-known classical relationship between gravitational and ‘‘regular’’ force: jFj ¼ GjMðRjMl ÞmðRjmm Þ=R2 j ¼ mjaj

ð11eÞ

where the ÔaÕ is the radial acceleration due to the gravitational force F. I can equate the values of dynamical (gravitational) and kinematical accelerations:

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jaj ¼ GMðRjMl Þ=R2 ¼ d2 sðtÞ=dt2

497

ð12aÞ

However, by analogy to the Eq. (12a), an entirely different expression for the value of the same acceleration follows from the Eq. (11e). Mass is actually a measure of resistance to motion, rather than that of static quantity of matter [2]. Since G ¼ const, for a uniform and linear distribution of the mass we get: jaj ¼ Go2 MðRjMl Þ=oR2

ð12bÞ

In the MBS the mass M looks like a distance s in the LBS, while the radius R looks therein like a time-duration parameter t in the LBS. If mass/energy varies in mass/energy-based coordinates of a 3D space that is parametrized by distance, the distance parameter resembles time-duration in LBS. Twin spaces in the multispatial representation of motion are literally intertwined. Eq. (12b) is a powerful reminder that there is much more to geometry and physics that the PM and MP had disclosed. The multispatial hyperspace gives consistent geometric and algebraic interpretation of traditional vectors, which seem to span multiple simple 3D spaces. The existence of the MBS predicted by the new mathematical formulations of vectors complies with the old CM. Nonetheless, it unveils different physical reality that had been hidden behind the postulative abstract formalism of the MP and PM. Hence the abstract structure of hyperspace is real and should not be disregarded. It shows that we do posses extra abstract symmetries at a very high level of the whole (not so classical) hierarchical structure of the 9D MLT hyperspace. The new interpretation of vectors shows how duality makes what appears as compactification of dimensions. Although the ‘‘extraneous’’ dimensions were not really rolled up to disappear, they are not represented in a single simple geometric space, but spread in a multispatial hyperspace. One can give thus a new meaning to the apparent compactification of dimensions, namely that three dimensions from the 3D dual linear vector space appear also in the primary 3D space disguised as pseudoscalar of the generalized hypervector. In particular the equations (12) answer the question how the distances from the usual length-based space are mapped onto the inverse square distance [106,107]. This is a cross-spatial mapping, however. This particular mapping has nothing to do with curling of the extra dimensions. It is due to dual spatial representations in hyperspace. Hence the alleged compactification of higher than four dimensions, or its some other ingenious alternatives such as branification [108], does not really put them away. The extra dimensions can be positively identified in the multispatial hyperspace, which relies on an abstract hyperspatial duality and on symmetries. Under certain conditions even QM can tolerate duality and supersymmetry [109]. The existence of multispatial hyperspace explains, or at least makes viable, some facts that were considered as curiosities in physics and mathematics. There is no way to wrap a M5 manifold around a M4 one [110]. There is no evidence for supersymmetric baryons [111] and no compelling reason for compactification [112]. In the meantime the supersymmetry that was based on a Z2 -graded algebraic structure has been upgraded to certain Z3 -graded hypersymmetry [51,52]. The hyperspace, however, shifts that old geometric paradigm of single-space physical reality to that of a multispatial structure. It allows us to formulate new structural laws for the dynamics of time itself [1–4], which is convenient for QM [113] and desirable even for cosmology [114]. Spatial flow of time enhances the old idea of DÕAlembert, who once contemplated time as a fourth dimension [115]. It suggests existence of a time–space as a quasi-dual counterpart of space– time [1,3], which could also support quantization of tachyons [116]. This will be discussed elsewhere.

12. Summary and conclusions From Elie CartanÕs generalization of the Stokes theorem one can conclude that higher than three-dimensional object is composed of 3D objects. The conclusion was reinforced by his other result, which showed that the highest dimensionality of simple geometric spaces is three. Since we observe more than four independently varying physical magnitudes, the simplest general conclusion is that structure of physical reality should resemble a multispatial hyperspace, which comprises several simple 3D geometric spaces. Hence multidimensionality is a logical consequence of multispatiality in the sense that each 3D vector space of the hyperspace brings in to the hyperspace a distinct triple of dimensions, which come as proper attributes of the space. The presence of multispatial hyperspace implies a new interpretation of vectors. It suggests that traditional 3D vectors can span multiple spaces. This feature is theoretically affirmed by the fact that an isometric operation of vector multiplication of traditional vectors has been devised upon it. All these imply existence of a 3D mass-based geometric space, whose presence fully complies with well-known facts of classical mechanics. Hence having at least 9D hyperspace for the mechanics alone, one can expect 27 distinct dimensions (due to symmetries imposed by abstract operations), and

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many more if uniqueness is not enforced. Synthetic approach to mathematics and physics reveals thus unexpectedly marvelous design of our physical reality. The mathematical formalisms of both analytic geometry and the theory of linear vector spaces contain veiled contradictions that indicate presence of a hyper-reality, which shapes the infrastructure of both the mathematical and the physical reality. They also suggest existence of hierarchical multispatial structures. In some instances old respected mathematical theories virtually demand that pan-geometry of multispatial hyperspace should be developed, for the sanity of their apparently illogical consequences. When confronted with experimental evidence, the curious references to hyper-reality produced predictions that comply with results of formerly unexplained experimental data [1–4]. Formerly inconsistent abstract mathematical operations become permissible and quite meaningful within an abstract multispatial hyperspace. Some formerly unexplained results of theoretical and experimental physics make sense when cast onto the multispatial structure of the hyperspace. I was greatly influenced by abstract ideas of El Naschie who effectively deals with spatial issues on both manifolds and in purely discrete systems [117,118,11,12]. In fractal terms one can build analogies between QM and space–time [119]. I was also greatly impressed by ingenious analytic works on general relativity and related issues done by several Russian scientists ([120,121] and Refs. in [3]). Yet the common sense dogma that sees proper geometric higherdimensional spaces as impossible [122] prompted me to distinguish between geometric spaces and manifolds (topological spaces) on the foundation of a synthetic, object-oriented approach to geometric objects. The new synthetic approach distinguishes between physical and abstract interactions, where the latter may sometimes appear as almost instantaneous [123]. This distinction allows us to understand also apparent nonlocality as an actionless coupling of distinct representations (images) of a geometric object. Being represented in two quite distinct spaces at once, when one image is manipulated in one space, its other image in another coupled space may be almost instantaneously changed also, because it is actually the same object seen as two distinct images in these two separate spaces spanned over the same physical manifold. Hence change made to one image of the object also affects the other image of the very same object, and so it may appear as almost instantaneous nonlocal change of the other image. Since nonlocality cannot be simply refuted [124], it must thus have physically meaningful and presumably quite reasonable mathematical explanation of the way it works. Although there is no action at a distance, even the apparent classical action at a distance can be pictured in terms of such coupled images. One could thus represent the gravitational force as an imaginary function, which is quite feasible [125]. Without synthesis and a conceptual visualization, pure mathematics is like music that could never be performed [126]. The insane prohibition on visualization in mathematics resulted in sloppy algebraic and topological reductionism and caused veiled though uncontested nonsenses. According to Plato, philosophers (and also scientists in modern parlance) would not trade science for governing. Yet great Newton left science for administration. He did that after suffering a mental breakdown while still tackling sciences. When people see that their work can still be improved, they tend to have motivation for work. However, when they got correct results even though they know that their assumptions were not quite right, they get crazy, because they suspect that a substantial improvement could ruin their already correct results. In my personal opinion, Newton realized that there were more variables to determine gravity than his theory utilized. There was the issue of volume, which is quite irrelevant to the radial effects of gravity, which he discovered. He computed gravitational attraction forces without volumes and yet all heavenly bodies complied with his calculations. He hesitated for over 20 years before publishing his Principia in 1687. He would have to wait some 313 years more for my papers, which confirmed his doubts by discovery of nonradial effects of gravity [1,2]. Fortunately, he did not wait that long, for I would never guess them without his Principia. Although I was standing firm on the earth, I too built my theories upon the works of many giants. Some of those works provided just tools while others supplied also ideas. Having tools, anyone can erect a theoretical shack, but to build more than a shack, one needs ideas. One of such suppliers of great inspirational ideas is El Naschie. As I have mentioned above, I am painfully aware of incompleteness of this presentation and I am writing a paper that will soon complement this one. The missing idea in question here is that of an infinite-dimensional manifold (IDM), which El Naschie has proposed as fractal Cantorian space–time that may underlie physics as a whole [127]. At the fundamental level of indistinguishable points we need the IDM. Had Newton invented IDM, he could have overcome his justifiable doubts. In this paper I have demolished 300+ years old nonsense, but to incorporate the IDM I must expel even much more entrenched mathematical absurd. It is not just fancy extension of mathematics––we must have the IDM in physics. I could not move forward without it, even though I was not a fan of infinity. El Naschie is physically minded and presumably that is why he introduced his ideas gently, with all due respect to mathematics. Nevertheless, I intend to be bold and destructive, wherever necessary, and am going to merge the unfinished business of Elie Cartan on bilinear groups [128] with the IDM. The Hausdorff dimension defined on fractal space–time is a bridge between semi-structured manifold and discrete geometries [129] that yields concrete physical magnitudes [130–133], even though El Naschie

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blends space and time so that time appears spatialized [134,135]. I was so preoccupied with purely geometric aspects of physics that I did not realize how important may number theory be for both geometry and physics, until I could not finish this paper. Although the method of moving tripod is used for differential forms of all degrees [136], 3D Euclidean space is insufficient even for construction of Finsler spaces, which require a 5D space [137]. With generalized Stokes formula [138] the 27 lines for surfaces of 3rd degree [139] could suggest 27 discrete dimensions. We need the IDM as a deeper foundation for physics. In few theoretical studies on complexity the concept of abstract plects has been introduced [140]. Since the term ÔplectÕ may indicate both complexity and simplicity, it is tempting to expand it on multiplects that would convey the message of complex, multilayered hierarchical structure of those plects. Such multiplects may help visualize the multispatial structure of hyperspace.

References [1] Czajko J. On conjugate complex time I: Complex time implies existence of tangential potential that can cause some equipotential effects of gravity. Chaos Solitons & Fractals 2000;11:1983–92. [2] Czajko J. On conjugate complex time II: Equipotential effect of gravity retrodicts differential and predicts apparent anomalous rotation of the sun. Chaos Solitons & Fractals 2000;11:2001–16. [3] Czajko J. On conjugate complex time III: Superstrings and complex Lorentz transformation. Chaos Solitons & Fractals 2001;12:951–67. [4] Czajko J. Complex conjugate time in macro-experiments. Chaos Solitons & Fractals 2002;13:17–23. [5] Merat P et al. Observed deflection of light by the sun as a function of solar distance. Astron Astrophys 1974;32:471–5. [6] El Naschie MS. On the nature of complex time, diffusion and the two-slit experiment. Chaos Solitons & Fractals 1995;5: 1031–2. [7] El Naschie MS. On conjugate complex time and information in relativistic quantum theory. Chaos Solitons & Fractals 1995;5:1551–5. [8] El Naschie MS. Time symmetry breaking, duality and Cantorian space–time. Chaos Solitons & Fractals 1996;7:499–518. [9] El Naschie MS. Wick rotation, Cantorian spaces and the complex arrow of time in quantum physics. Chaos Solitons & Fractals 1996;7:1501–6. [10] El Naschie MS. A note on quantum gravity and Cantorian space–time. Chaos Solitons & Fractals 1997;8:131–3. [11] El Naschie MS. Fractal gravity and symmetry breaking in a hierarchical Cantorian space–time. Chaos Solitons & Fractals 1997;8:1865–72. [12] El Naschie MS. Hyperdimensional geometry and the nature of physical space–time. Chaos Solitons & Fractals 2000;10:155–8. [13] Kasner E. An algebraic solution of Einstein equations. Trans Am Math Soc 1925;27:101–5. [14] Neumann PM, Stoy GA, Thompson EC. Groups and geometry. Oxford: Oxford University Press; 1994. p. 116ff. [15] Hausdorff F. Mengenlehre. Berlin: 1935. p. 94. € ber mathematischen Denken und den Begriff der aktuellen Form. Berlin: 1914. p. 16. [16] Gabrilovitsch L. U [17] Laugwitz D. Zur Role der pythagoreischen Metrik in der Physik. Z Naturf 1954;9a:827–32. [18] Bourbaki N. El
ements de math
ematique: Groupes et algebres de Lie. Paris: Hermann; 1960. p. 20. [19] Fang J. HilbertÕs problems. Phil Math 1969;6:38–9. [20] Hilbert D, Bernays P. Grundlagen der Mathematik I. Berlin: Springer-Verlag; 1968. p. 16. [21] Lightstone AH. Linear algebra. New York: Appleton-Century-Crofts; 1969. p. 183ff. [22] van der Waerden BL. Algebra I. New York: Fred. Ungar; 1970. p. 61. [23] Frankel T. The geometry of physics. An introduction. Cambridge: Cambridge University Press; 1997. p. 13. [24] Wilder RL. Introduction to the foundations of mathematics. New York: Wiley; 1963. p. 177. [25] Hermann P. Vector bundles in mathematical physics. Benjamin; 1970. p. 26. [26] Halmos PR. Finite-dimensional vector spaces. Princeton: Van Nostrand; 1958. p. 23. [27] Williamson RE, Crowell RH, Trotter HF. Calculus of vector functions. Englewood Cliffs, NJ: Prentice-Hall; 1968. p. 62. [28] Bourbaki N. El
ements de math
ematique: Th
eorie des ensembles. Paris: Hermann; 1970. p. E IV.69. [29] Engelking R, Sieklucki K. Topology. A geometric approach. Berlin: Heldermann Verlag; 1992. p. 332. [30] Dodson CTJ. Categories, bundles and space–time topology. Orpington: Shiva Publishing; 1980. p. 66. [31] Scholz E. The concept of manifold 1850–1950. In: James IM, editor. History of topology. Amsterdam: Elsevier; 1999. p. 25–64. p. 26. [32] Bishop RL, Crittenden PJ. Geometry of manifolds. New York: Academic Press; 1964. p. 2. [33] Poincar
e H. Pourquoi lÕespace a trios dimensions? Rev Metaphys Morale 1812;20:483–504. [34] Engelking R. General topology. Berlin: Heldermann Verlag; 1989. p. 382. [35] De Witt B. Supermanifolds. Cambridge: Cambridge University Press; 1987. p. 17. [36] Choquet-Bruhat Y. Graded bundles and supermanifolds. Napoli: Bibliopolis; 1989. p. 74. [37] Choquet-Bruhat Y, Dewitt-Morette C, Dillard-Bleck M. Analysis, manifolds and physics. Amsterdam: North-Holland; 1977. p. 117ff. [38] Yanguas P. Lowering the dimension of polynomial vector fields in R2 and R3 . Chaos 2001;11(2):305–18.

500

J. Czajko / Chaos, Solitons and Fractals 19 (2004) 479–502

[39] Cartan E. La m
ethode du repere mobile, la th
eorie des groupes continues et les espaces g
en
eralis
es. Paris: Hermann; 1935. p. 39ff. [40] Cartan E. Les systemes diff
erentiels ext
erieurs et leurs applications g
eom
etriques. Paris: Hermann; 1945. p. 34ff. [41] Riemann B. Ueber die Hypothesen, welche der Geometrie zu Grunde liegen. In: Weber H, editor. Bernhard RiemannÕs gesammelte mathematische Werke. New York: Dover; 1953. p. 272–82. see, p. 272ff. [42] Misner CW, Thorne KS, Wheeler JA. Gravitation. New York: Freeman; 1973. p. 369. [43] Kupryaev NV. The sagnac vortex optical effect. Rus Phys J 2001;44:858–62. [44] Szekeres G. Effect of gravitation on frequency. Nature 1968;220:1116–8. [45] Gonseth F. Les math
ematiques et la r
ealit
e. Essai sur la m
ethode axiomatique. Paris: 1936 [p. 323]. [46] See p. 392 in [34]. [47] Diek A, Kantowski R. Some clifford algebra history. Available at: http://www.nhn.ou.edu/~ski/papers/Clifford/history.ps. [48] G€ odel K. On formally undecidable propositions of principia mathematica and related systems I. New York: Dover; 1992. p. 1, 40. [49] Cartan HP. Notion de topologie. Espaces topologiques, espaces m
etriques. Available at: http://chronomath.irem.univ-mrs.fr/ Anx3/TopoMetrique.html. [50] Janiszewski Z. Sur le r
ealisme et lÕid
ealisme en math
ematiques. In: Janiszewski Z, editor. Oeuvres choisies. Warszawa: PWN; 1962. p. 309–17. [51] Vector Space: http://mathworld.wolfram.com/VectorSpace.html. [52] Espace vectoriel: http://folium.eu.org/algebre/struct/espace_vectoriel/espace_vectoriel.html. [53] Rodin B. Calculus with analytic geometry. Englewood Cliffs, NJ: Prentice-Hall; 1970. p. 364ff. [54] Weinreich G. Geometrical vectors. Chicago: The Chicago University Press; 1998. p. 15. [55] Poncelet JV. Trait
e des propri
et
es projectives des figures. Paris: 1822. p. 28. [56] Banach S. Th
eorie des op
erations lin
eaires. In: Banach S, editor. Oeuvres II. PWN: Warszawa; 1979. p. 13–219. see, p. 154. [57] Banach S. Die Theorie der Operationen und ihre Bedeutung f€ ur die Analysis. p. 434–41 in: [56], see p. 436. [58] Banach S. Th
eorie des op
erations dans les ensembles abstraits et leur application aux equations int
egrales. p. 305–48 in: [56], see p. 306. [59] Smith T. Associative triangles, cross products, and ‘‘reflexivity’’ of octonions. Available at: http://www.innerx.net/personal/ tsmith/clcrotc.html. [60] Freed DS. Five lectures on supersymmetry. Providence, RI: Am Math Soc; 1999. p. 5ff. [61] Abramov V, Kerner R, LeRoy B. Hypersymmetry: A Z3 -graded generalization of supersymmetry. J Math Phys 1997;38: 1650–69. [62] Kerner R. Z3 -graded algebras and the cubic root of the supersymmetry translations. J Math Phys 1992;33:403–11. [63] von Neumann J. Mathematical foundations of quantum mechanics. Princeton: Princeton University Press; 1996. p. 324. [64] Gardner M. The colossal book of mathematics. New York: Norton; 2001. p. 568ff. [65] Bell JS. Speakable and unspeakable in quantum mechanics. Cambridge: Cambridge University Press; 1989. p. 15ff. [66] Friedrichs KO. Unobserved observables and unobserved causality. Comm Pure Appl Math 1981;34:273–83. [67] Bohm D. Quantum theory. Mineola, NY: Dover Publ; 1989. p. 29. [68] Bohm D. Causality and chance in modern physics. Philadelphia: University of Pennsylvania Press; 1957. p. 59. [69] Bohm D. Wholeness and the implicate order. London: Routledge; 1999. p. 76ff. [70] Bohm D, Hiley BJ. The undivided universe. London: Routledge; 1999. p. 144ff. [71] Van Flandern T. The speed of gravity––what the experiments say. Phys Lett 1998;A250:1–11. [72] Carlip S. Aberration and the speed of gravity. Phys Lett 2000;A267:81–7. [73] Hausdorff F. Grundz€ uge der Mengenlehre. Leipzig: 1914. p. 469. [74] Czyz J. Paradoxes of measures and dimensions originating in Felix HausdorffÕs ideas. Singapore: World Scientific; 1994. p. 51. [75] Verdina J. Projective geometry and point transformations. Boston: Allyn and Bacon; 1971. p. 24ff. [76] Hopkins EJ, Hails JS. An introduction to plane projective geometry. Oxford: At The Clarendon Press; 1953. p. 5. [77] Blattner JW. Projective plane geometry. San Francisco: Holden Day; 1968. p. 25. [78] Busemann H, Kelly P. Projective geometry and projective metrics. New York: Academic Press; 1953. p. 20. [79] Boyer CB. A History of mathematics. New York: Wiley; 1991. p. 536. [80] Gerhardt CI. Geschichte der Mathematik in Deutschland. M€ unchen. 1877. p. 275. [81] Pl€ ucker J. Ueber ein neues Prinzip der Geometrie und den Gebrauch allgemeiner Symbole und bestimmter Coefficienten. In: Schoenflies A, editor. Julius Pl€ uckers gesammelte mathematische Abhandlungen. Leipzig 1895. pp. 159-177 [see, p. 174ff]. [82] Weeks JR. The shape of space. New York: Marcel Dekker; 1985. p. 232. [83] Loeb AL. Space structures. Their harmony and counterpoint. Reading, MA: Addison-Wesley; 1976. p. 42. [84] Stolfi J. Oriented projective geometry. San Diego: Academic Press; 1991. p. 92. [85] von Neumann J. Continuous geometry. Princeton: Princeton University Press; 1960. p. 3. [86] Simon M. Nichteuklidische Geometrie in elementarer Behandlung. Leipzig, 1925. p. 32. [87] Sprague AP. Dual-linear, linear diagrams. In: Johnson NL, Kallaher MJ, Long CT, editors. Finite geometries. New York: Marcel Dekker; 1983. p. 413–8. [88] Goodstein RL, Primrose EJF. Axiomatic projective geometry. Leicester: Leicester University Press; 1962. p. 2ff.

J. Czajko / Chaos, Solitons and Fractals 19 (2004) 479–502 [89] [90] [91] [92] [93] [94] [95] [96] [97] [98] [99] [100] [101] [102] [103] [104] [105] [106] [107] [108] [109] [110] [111] [112] [113] [114] [115] [116] [117] [118] [119] [120] [121] [122] [123] [124] [125] [126] [127] [128] [129] [130] [131] [132] [133] [134] [135] [136]

501

Baer R. Linear algebra and projective geometry. New York: Academic Press; 1952. p. 96. Bumcrot RJ. Modern projective geometry. New York: Holt, Rinehart and Winston; 1969. p. 75. Couturat L. Les principes des math
ematiques. Hildesheim: Georg Olms Verlagsbuchhandlung; 1965. p. 134ff. van der Waerden BL. Einf€ uhrung in die algebraische Geometrie. Berlin: 1939. p. 7ff. Bourbaki N. El
ements de math
ematique VI: Int
egration. Paris: Hermann; 1956. p. 55. Whitehead JHC. Duality in topology. In: Whitehead James IM, editor. Mathematical works of JHC Whitehead, vol. 4. New York: Macmillan; 1963. p. 163–77 [see, p. 165ff]. Narici L, Beckenstein E. Topological vector spaces. New York: Marcel Dekker; 1985. p. 187. MacLane S. Duality for groups. In: Eilenberg S, MacLane S, editors. Collected works. Orlando: Academic Press; 1986. p. 163–94. p. 173ff. Young JW. Projective geometry. Chicago: Open Court; 1930. p. 25. Chudinov EM. Mathematical problems of physics. Moscow: 1981. p. 7 [in Russian]. Bell ET. Mathematics. Queen and servant of science. McGraw-Hill; 1951. p. 136. Lukyanets VS. Physical–mathematical spaces and reality. Kiev: 1971. p. 25 [in Russian]. Sieben K. Dreidimensionale Wirklichkeit. Stuttgart [p. 5]. € ber dimensionserh€ Hurewicz W. U ohende stetige Abbildungen. J Reine Angew Math 1933;71:169. € ber die Gleichverteilung von Zahlen mod. Eins Math Ann 1916;77:313. Weyl H. U Montgomery H. Space. Philos J 1969;6:127–40. Cantor G. Une contribution a la th
eorie des ensembles. Acta Math 1883;2:311–28. Witten E. Duality, space–time and quantum mechanics. Phys Today 1997;(May):28–33. Coon SA, Holstein BR. Anomalies in quantum mechanics: The 1=r2 potential. Am J Phys 2002;70:513–9. Giddings S. ‘‘Branification:’’ an alternative to compactification. Available at: http://www.phys.lsu.edu/mog/mog15/node11.html. Simon D. Duality and supersymmetric quantum mechanics. J Phys A: Math Gen 2002;35:4143–50. Duff MJ. Ten to eleven: It is not too late. In: Wess J, Akulov VP, editors. Supersymmetry and quantum field theory. Berlin: Springer-Verlag; 1998. p. 59–63. see, p. 62. Albuquerque IF et al. Search for light supersymmetric baryons. Phys Rev Lett 1977;78:3252–6. Freund PGO. Higher-dimensional unification. Physica 1986;15D:263–9. Blanchard P, Jadczyk A. Events and piecewise deterministic dynamics in event-enhanced quantum theory. Phys Lett 1995;A203:260–6. Chaichian M, Kobakhlidze AB. Mass hierarchy and localization of gravity in extra time. Phys Lett 2000;B488:117–22. Yaglom IM. Felix Klein and Sophus Lie. Boston: Birkh€auser; 1988. p. 73. Pant MC et al. Dirac spinors and Tachyon quantization. Can J Phys 2000;78:303–15. El Naschie MS. Radioactive decay and the structure of Eð1Þ quantum space–time. Chaos, Solitons & Fractals 1999;10:17–23. El Naschie MS. On the irreducibility of spatial ambiguity in quantum physics. Chaos, Solitons & Fractals 1998;9:913–9. Ord GN. Fractal space–time: A geometric analogue to relativistic quantum mechanics. J Phys A: Math Gen 1983;16:1869–84. Vlasov AA et al. Gravitational effects in the field theory of gravitation. Theor Math Phys 1980;43:375–401. Logunov AA, Loskutov YuM. Nonuniqueness of the predictions of the general theory of relativity. The relativistic theory of gravitation. Sov J Part Nucl 1987;18:179–87. Mirman R. The reality and dimension of space and the complexity of quantum mechanics. Int J Theor Phys 1988;27: 1257–76. Kogbetliantz E. Sur la vitesse de propagation de la gravitation. Ann Phys (Paris) 1931;16:71. Schafir RL. Comment on an alleged refutation of non-locality. ArXiv:quant-ph/0201065 v.1. 16-Jan-2002. Nahin PJ. An imaginary tale. Princeton, NJ: Princeton University Press; 1998. p. 115. Needham T. Visual complex analysis. Oxford: Clarendon Press; 2001. p. vii. El Naschie MS. Modular groups in Cantorian Eð1Þ high energy physics. Chaos, Solitons & Fractals 2003;16:353–66. Cartan E. Groupes Bilin
eaires. In: Elie Cartan: Oeuvres Completes. T. 1 p. 2, Paris: Gauthier-Villars; 1953. p. 7–105. El Naschie MS. On John NashÕs crumpled surface. Chaos, Solitons & Fractals, in press. El Naschie MS. VAK, Vacuum fluctuation and the mass spectrum of high energy physics. Chaos, Solitons & Fractals 2003;17:797–807. El Naschie MS. Kleinian groups in Eð1Þ and their connection to particle physics and cosmology. Chaos, Solitons & Fractals 2003;16:637–49. da Cruz W. The Hausdorff dimension of fractal sets and fractional quantum hall effect. Chaos, Solitons & Fractals 2003;17: 975–9. Marek-Crnjac L. On the mass spectrum of the elementary particles of the standard model using El NaschieÕs golden field theory. Chaos, Solitons & Fractals 2003;15:611–8. El Naschie MS. The Cantorian interpretation of high energy physics and the mass spectrum of elementary particles. Chaos, Solitons & Fractals 2003;17:989–1001. El Naschie MS. Complexity theory interpretation of high energy physics and the elementary particles mass spectrum [manuscript]. Cartan E. Sur les Formes Diff
erentielles en G
eom
etrie. In: Elie Cartan: Oeuvres Completes. T. 1 p. 2, Paris: Gauthier-Villars; 1955. p. 905–07.

502

J. Czajko / Chaos, Solitons and Fractals 19 (2004) 479–502

[137] de Rham G. La Th
eorie des Formes Diff
erentielles Ext
erieures et lÕHomologie des vari
et
es Diff
erentiables. In: De Rham G, editor. Oeuvres Math
ematiques. Geneve: LÕenseignement Math
ematique Universit
e de Geneve, p. 475. [138] Cartan E. Expos
es de G
eom
etrie. Paris: Hermann; 1971. p. 15. [139] Cartan E. Le Principe de Dualit
e et la Th
eorie des Groupes Simples et Semi-simples. In: Elie Cartan: Oeuvres Completes. P.1, Paris: CNRS; 1984. p. 555–68. [140] Gell-Mann M. LetÕs Call It Plectics. Complexity 1995–96;1(5)––also on http://www.santafe.edu/sfi/People/mgm/plectics.html.