Topological vacuum fluctuation and Dvoretzky‘s theorem – Mathematical proofs in the context of the dark energy density of the universe

  • Mohamed S. El Naschie Dept. of Physics, Faculty of Science, University of Alexandria, Egypt
  • Ji-Huan He School of Science, Xi‘an University of Architecture and Technology, Xi‘an, China
  • Leila Marek-Crnjac Technical School Center of Maribor, Maribor, Slovenia
Keywords: Fractal spacetime, E-infinity theory, Quantum vacuum fluctuation, Platonic quantum set theory, Dvoretzky’s theorem, Golden mean number system, G. Ord, L. Nottale, Pointless Geometry, Penrose fractal tiling, von Neumann continuous geometry, G. ‘tHooft-Weltman-Wilson fractal spacetime, A. Connes noncommutative geometry, E-infinty bijection.


Starting from the initial triality of physics, namely mathematical philosophy, transfinite set theory and number theory we drive the inevitability of a topological quantum vacuum fluctuation of spacetime resulting in the fundamental reality of pair creation and annihilation. Subsequently we give a simple but strong mathematical proof of Dvoretzky‘s marvellous theorem on measure concentration, thus making dark energy and accelerated cosmic expansion not only an astrophysical measurement and observational reality, but also a plausible topological-geometrical fact of a pointless Cantorian actual universe akin to the Penrose fractal tiling space. This space is described accurately via the von Neumann-Conne noncommutative geometry using their golden mean dimensional function and the corresponding bijection of E-infinity theory. The said theory was developed by the authors of the present paper and their group and is based on and starts from the pioneering efforts of the Canadian physicist G. Ord and the French astrophysicist L. Nottale.


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How to Cite
El Naschie, M. S., He, J.-H., & Marek-Crnjac, L. (2020). Topological vacuum fluctuation and Dvoretzky‘s theorem – Mathematical proofs in the context of the dark energy density of the universe. Journal of Progressive Research in Mathematics, 16(3), 2969-2985. Retrieved from