1. The Geometry and Physics of the Multidimensional World
Alexander Egoyan
University Geomedi, Georgia

2. According to Immanuel Kant “a science of all these possible kinds of space [the higher dimensional ones] would undoubtedly be the highest enterprise which a finite understanding could undertake in the field of geometry... If it is possible that there could be regions with other dimensions, it is very likely that God has somewhere brought them into being.” (I. Kant idem, p. 12.)

3. Leonhard Euler stressed that “only removing the constraint of a definite dimension and the possibility of such a formulation of the nature laws in which the dimension manifests itself as something random ensure that we have achieved their whole mathematical understanding.”

4. Most physical and mathematical theories such as Special and General Relativity theories, Quantum Mechanics and String theory are multidimensional, the number of dimensions in these theories being not fixed. The same thing may be said about geometrical and topological theories. All these theories do not require the number of dimensions to be equal to three. At the same time all these theories do not explain how spaces with different numbers of dimensions are embedded one into another, how the points of the embedded space differ from the points of the higher dimensional space and how we can define a metric on these spaces.

5. A new multidimensional geometry is based on the idea introduced firstly in the smooth infinitesimal analysis according to which lines are composed of infinitesimally small linear segments. These segments can be thought of as being long enough to have a definite direction, but not long enough to be curved.

6. In the present work we propose geometry which postulates that instead of being composed of points, spaces and surfaces are composed of dualistic objects - “point-connections”: connections connect the points of the surface and create a metric – the distance between two points is the minimal number of connections needed to connect these points. One-dimensional connections are closed lines, n-dimensional connections are compositions of (n-1) - dimensional connections. All the connections with the same number of dimensions are identical and form isotropic and homogeneous spaces. For the external observers they are composed of lines formed by infinitesimal segments, while for the internal observers they are sets of points equipped by a metric.

8. The theory is based on the idea introduced firstly in the smooth infinitesimal analysis according to which lines are composed of infinitesimally small linear segments. Smooth infinitesimal analysis is a mathematically rigorous reformulation of the calculus in terms of infinitesimals. It views all functions as being continuous and incapable of being expressed in terms of discrete entities. The nilsquare or nilpotent infinitesimals are numbers ε where ε² = 0 is true, but ε = 0 need not be true at the same time.

9. In SIA every function whose domain is R, the real numbers extended by infinitesimals, is continuous and infinitely differentiable. Intuitively, smooth infinitesimal analysis can be interpreted as describing a world in which lines are made out of infinitesimally small segments, not out of points. These segments can be thought of as being long enough to have a definite direction, but not long enough to be curved.

10. Obviously, we can assign two different metrics to the points of the embedded manifold – internal and external: the former measured from the position of the internal observer of the manifold and the latter from the position of the external observer from the higher dimensional space.

11. The infinitesimal segments which form connections of the embedded manifold may be stretched along the infinitesimal segments of the connections of the higher dimensional space.

12. This means that the embedded manifold can change its external metric without changing the internal one. It may be stretched, squeezed or curved without changing its internal metric. In other words, such geometry is an elastic one.

13. This geometry has an intuitive and simple physical interpretation: manifolds may be associated with elastic membranes. Elastic membranes look different for internal and external observers: for the internal observers they are space-times we are used to, while from the point of view of the external observers they are material objects - surfaces with homogeneous material properties. When interacting with elementary particles and other membranes, a membrane will transform their energy into its elastic energy (a new form of energy): the energy of stretching of the infinitesimal segments.

14. Each elastic membrane is an indivisible wholeness: it cannot be torn or pierced, but it can be squeezed, stretched or curved.

15. For example, living organisms play the role of internal observers of the Universe, and at the same time they serve as external observers for two-dimensional (2D) membranes embedded into our Universe.

16. Elementary particles are true multidimensional objects and may have different number of dimensions depending on the dimensionality of the membrane they live in.

17. Elementary particles from the higher dimensional space cannot directly interact with particles living in the embedded elastic membrane; they interact with the embedded elastic membrane by exchanging the elastic energy with it. The particle passing through the elastic membrane of the Universe will not have material properties in the directions tangent to the membrane representing our Universe and, therefore, cannot be detected by 3D observers.

18. In this model gravity is explained as transformation of material properties of the surface of the membrane caused by elementary particles. From the point of view of the internal observer gravitation is a consequence of the curvature of space-time governing the motion of inertial objects. From the point of view of the external observer gravitation may be understood as a result of internal elastic deformation of the elastic membrane.

19. Internal elastic deformation (stress) doesn’t change the external metric of the membrane. These two interpretations of gravitation are complementary: the second interpretation requires a preferred space-like foliation of space-time with space-time curvature replaced by the internal elastic deformation of the membrane.

20. Reality may be considered as the process of time evolution of holistic macro objects - elastic membranes. An embedded membrane in this multidimensional world will look different for the external and internal observers: from the outside it will look like a material object with smooth infinitesimal geometry, while from the inside our Universe-like space-time fabric. According to the proposed model, reality has a hierarchical structure where lower dimensional universes are embedded into higher dimensional universes. The hierarchy may have an infinite or finite number of levels. An infinite hierarchy of universes is a logically complete structure with an infinite number of levels. On the contrary, a finite hierarchy has a finite number of levels finally embedded into the infinite dimensional Absolute space-time with different physical laws.

21. Another advantage of multidimensional elastic geometry is its possibility to represent infinite surfaces as finite ones with the squeezing point. In other words, the infinite space-time we experience from the point of view of the internal observer is just a finite elastic membrane with closed topology from the point of view of the external observer.

22. Obviously, when approaching the squeezing point on the finite elastic membrane the membrane’s surface energy will grow and become an infinite number in the squeezing point. This infinity may be compensated if there is an infinite amount of matter in the squeezing point. The internal singularity will produce the internal elastic deformation of the membrane which will suppress the growth of the surface energy of the membrane when approaching the squeezing point.

23. This cosmological model exploits the idea that the Big Bang had more complicated nature and the explosion didn’t happen at ones. In the beginning only a part of the singularity exploded and the remaining part played the role of the squeezing point in the expanding universe. There may be other Big Bangs after the first Big Bang if the singularity will explode part by part. The model has common features with cyclic universe model proposed by Sir Roger Penrose. The accelerating expansion of the universe that we currently observe is explained as the result of falling of massive objects planets and stars towards the squeezing point under the influence of gravity.

24. This Escher's drawing gives some impression of what is going on in Lobachevski geometry - as you walk from the center out to the edge, you have to imagine that, because of the way the picture of the geometry has had to be distorted, the actual geometry there is exactly the same as it is in the middle, so that the geometry all about you remains the same no matter how you move.

25. Remarkably, in this model points which seem to be very far from each other from the point of view of internal observer may be very close to each other from the point of view of the external observer. This happens because of two reasons: 1.) squeezing towards the squeezing point makes a very distant point closer when viewed from the higher dimensional space, 2.) external elastic deformations may change the external metric of the membrane, so that points which seem to be unreachable from the point of view of internal observer will appear to be very close to each other in the higher dimensional bulk space-time.

26. According to the theory the probabilistic character of Quantum Mechanics is a manifestation of higher dimensional elastic vibrations of the elastic membrane we live in (the Universe). Such vibrations supposed to be responsible for the outcome of the interaction between two elementary particles. The theory is in agreement with experimentally proved phenomenon of quantum entanglement which points to the existence of now and the discovery of Higg’s boson which tells us that there is some underlying reality beyond space-time. If Higgs boson gives mass to particles, then we have to bring back the terms of 'absolute motion' and 'absolute speed'.

27. This approach may help us to use dimensional analogy and build a science consistent with our perceptions and imagination. 2D elastic membranes arise from such properties of qualia as: 1. Indivisibility: qualia cannot be separated into pieces; 2. Continuity: qualia are continuous; 3. Permanent character (elasticity): the character of perceptions doesn’t change during the process of perception; 4. Two-dimensionality: we cannot perceive two different signals in the same direction; 5. Macroscopic size: our perceptions also are macroscopic.

28. Our perception may be considered as the result of elastic oscillations of two dimensional (2D) elastic membranes with closed topology embedded in our bodies. Only one elastic membrane responsible for its perceptions will correspond to the selected organism, but there may be other membranes, even at the cell level.

29. The geometry and physics of the living world should be holistic, multidimensional and elastic, while the science of the inanimate world is reductionist, three-dimensional and rigid

30. References:
A. Egoyan, Historical Reasons and Possible Ways of the New Scientific Synthesis, XXIII International Congress of History of Science and Technology, 28 July - 2 August 2009, Budapest, Hungary, p. 649.
A. Egoyan, M. Mirtskhulava, D. Chitashvili, The Role of Physics in Science Integration, Albert Einstein Century International Conference, June, 2005, UNESCO, Paris.
A. Egoyan, Elastic Membrane Based Model of Human Perception, Toward a Science of Consciousness 2011 International Conference, Stockholm, p. 130.