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Derivation of the Euler characteristic and the curvature of Cantorian-fractal spacetime using Nash Euclidean embedding and the universal Menger sponge

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  • El Naschie, M.S.

Abstract

The present work gives an analytical derivation of the curvature K of fractal spacetime at the point of total unification of all fundamental forces which is marked by an inverse coupling constant equal α¯gs=26.18033989. To do this we need to first find the exact dimensionality of spacetime. This turned out to be n=4 for the topological dimension and ∼〈n〉=4+ϕ3=4.236067977 for the intrinsic Hausdorff dimension. Second we need to find the Euler characteristic of our fractal spacetime manifold. Since E-infinity Cantorian spacetime is accurately modelled by a fuzzy K3 Kähler manifold, we just need to extend the well known value χ=24 of a crisp K3 to the case of a fuzzy K3. This leads then to χ(fuzzy)=26+k=α¯gs. The final quite surprising result is that at the point of unification of our resolution dependent fractal-Cantorian spacetime manifold we encounter a Coincidencia Egregreium, namelyK=χ=D=α¯gs=26+k=26.18033989.Finally we look for some indirect experimental evidence for the correctness of our result using the COBE measurement in conjunction with Nash embedding of the universal Menger sponge.

Suggested Citation

  • El Naschie, M.S., 2009. "Derivation of the Euler characteristic and the curvature of Cantorian-fractal spacetime using Nash Euclidean embedding and the universal Menger sponge," Chaos, Solitons & Fractals, Elsevier, vol. 41(5), pages 2394-2398.
    • Handle: RePEc:eee:chsofr:v:41:y:2009:i:5:p:2394-2398
    DOI: 10.1016/j.chaos.2008.09.021
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    References listed on IDEAS

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    1. Elokaby, Ayman, 2009. "Knot wormholes and the dimensional invariant of exceptional Lie groups and Stein space hierarchies," Chaos, Solitons & Fractals, Elsevier, vol. 41(4), pages 1616-1618.
    2. El Naschie, M.S., 2008. "Transfinite harmonization by taking the dissonance out of the quantum field symphony," Chaos, Solitons & Fractals, Elsevier, vol. 36(4), pages 781-786.
    3. El Naschie, M.S., 2006. "Elementary prerequisites for E-infinity," Chaos, Solitons & Fractals, Elsevier, vol. 30(3), pages 579-605.
    4. El Naschie, M.S., 2009. "Yang–Mills action as an eigenvalue problem in the theory of elasticity and some measure theoretical interpretation of ‘tHooft’s instanton," Chaos, Solitons & Fractals, Elsevier, vol. 41(4), pages 1854-1856.
    5. El Naschie, M.S., 2009. "Kac–Moody exceptional E12 from simplictic tiling," Chaos, Solitons & Fractals, Elsevier, vol. 41(4), pages 1569-1571.
    6. El Naschie, M.S., 2008. "Quantum golden field theory – Ten theorems and various conjectures," Chaos, Solitons & Fractals, Elsevier, vol. 36(5), pages 1121-1125.
    7. El Naschie, M.S., 2008. "Fuzzy knot theory interpretation of Yang–Mills instantons and Witten’s 5-Brane model," Chaos, Solitons & Fractals, Elsevier, vol. 38(5), pages 1349-1354.
    8. El Naschie, M.S., 2009. "Curvature, Lagrangian and holonomy of Cantorian-fractal spacetime," Chaos, Solitons & Fractals, Elsevier, vol. 41(4), pages 2163-2167.
    9. El Naschie, M.S., 2008. "An energy balance Eigenvalue equation for determining super strings dimensional hierarchy and coupling constants," Chaos, Solitons & Fractals, Elsevier, vol. 38(5), pages 1283-1285.
    10. Munroe, Ray, 2009. "Symplectic tiling, hypercolour and hyperflavor E12," Chaos, Solitons & Fractals, Elsevier, vol. 41(4), pages 2135-2138.
    11. Munroe, Ray, 2009. "The MSSM, E8, Hyperflavor E12 and E∞ TOE’s compared and contrasted," Chaos, Solitons & Fractals, Elsevier, vol. 41(3), pages 1557-1560.
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