Chaotic Fractal Tiling for the Missing Dark Energy and Veneziano Model
Affiliation(s)
Technical School Center, Maribor, Slovenia. epartment of Physics, Faculty of Science, University of Alexandria, Alexandria, Egypt.
Technical School Center, Maribor, Slovenia. epartment of Physics, Faculty of Science, University of Alexandria, Alexandria, Egypt.
ABSTRACT
The formula for the quantum amplitude of the Veneziano dual resonance model is shown to be formally analogous to the dimensionality of a K-theoretical fractal quotient manifold of the non-commutative geometrical type. Subsequently this analogy is used to deduce the ordinary energy of the quantum particle and the dark energy of the quantum wave. The results agree completely with cosmological measurements. Even more surprisingly the sum of both energy expressions turned out to be exactly equal to Einstein’s iconic formula E = mc2. Consequently Einstein’s formula makes no distinction between ordinary and dark energy.
KEYWORDS
Hausdorff Dimension; Cantor Set; Dark Energy; Kähler Manifold; Quantum Entanglement; Veneziano Model
Hausdorff Dimension; Cantor Set; Dark Energy; Kähler Manifold; Quantum Entanglement; Veneziano Model
Cite this paper
L. Marek-Crnjac and M. Naschie, "Chaotic Fractal Tiling for the Missing Dark Energy and Veneziano Model," Applied Mathematics, Vol. 4 No. 11, 2013, pp. 22-29. doi: 10.4236/am.2013.411A2005.
L. Marek-Crnjac and M. Naschie, "Chaotic Fractal Tiling for the Missing Dark Energy and Veneziano Model," Applied Mathematics, Vol. 4 No. 11, 2013, pp. 22-29. doi: 10.4236/am.2013.411A2005.
References
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http://dx.doi.org/10.1007/978-1-4612-0543-2 [17] J.-H. He, “Hilbert Cube Model for Fractal Space-Time,” Chaos, Solitons & Fractals, Vol. 42, No. 5, 2009, pp. 2754-2759. http://dx.doi.org/10.1016/j.chaos.2009.03.182 [18] J.-H. He, “Twenty Six Dimensional Polytope and High Energy Spacetime Physics,” Chaos, Solitons & Fractals, Vol. 33, No. 1, 2007, pp. 5-13.
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http://dx.doi.org/10.4236/jqis.2013.32011 [23] M. S. El Naschie, “The Missing Dark Energy of the Cosmos From Light Cone Topological Velocity and Scaling the Planck Scale,” Open Journal of Microphysics, Vol. 3, No. 3, 2013, pp. 64-70.
http://dx.doi.org/10.4236/ojm.2013.33012 [24] M. S. El Naschie and A. Helal, “Dark Energy Explained via the Hawking-Hartle Quantum Wave and the Topology of Cosmic Crystallography,” International Journal of Astronomy and Astrophysics, Vol. 3, No. 3, 2013, pp. 318343. [25] M. S. El Naschie, “The Quantum Gravity Immirzi Parameter—A General Physical and Topological Interpretation,” Gravitation and Cosmology, Vol. 19, No. 3, 2013, pp. 151-155.
http://dx.doi.org/10.1134/S0202289313030031 [26] M. S. El Naschie, “What Is the Missing Dark Energy in a Nutshell and the Hawking-Hartle Quantum Wave Collapse,” International Journal of Astronomy and Astrophysics, Vol. 3, No. 3, 2013, pp. 205-211.
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http://dx.doi.org/10.4236/jqis.2013.31006 [29] M. S. El Naschie, “Dark Energy from Kaluza-Klein Spacetime and Noether’s Theorem via Lagrangian Multiplier Method,” Journal of Modern Physics, Vol. 4, No. 6, 2013, pp. 757-760.
http://dx.doi.org/10.4236/jmp.2013.46103 [30] M. S. El Naschie, “Determining the Missing Dark Energy of the Cosmos from a Light Cone Exact Relativistic Analysis,” Journal of Modern Physics, Vol. 2, No. 2, 2013, pp. 18-23. [31] M. S. El Naschie, “Towards a General Transfinite Set Theory for Quantum Mechanics,” Fractal Space-Time and Non-Commutative Geometry in High Energy Physics, Vol. 2, No. 2, 2012, pp. 135-142. [32] R. Elwes, “Ultimate Logic,” New Scientist, Vol. 211, No. 2823, 2011, pp. 30-33.
http://dx.doi.org/10.1016/S0262-4079(11)61838-1 [33] V. Jacques, et al., “Delayed-Choice Test of Quantum Complementarity with Interfering Single Photons,” Physical Review Letters, Vol. 100, No. 22, 2008, Article ID: 220402.
http://dx.doi.org/10.1103/PhysRevLett.100.220402 [34] L. Li, N. L. Liu and Z. X. Yu, “Duality Relations in a Two Path Interferometer with an Asymmetric Beam Splitter,” Physical Review A, Vol. 85, No. 5, 2012, Article ID: 054101.
http://dx.doi.org/10.1103/PhysRevA.85.054101 [35] M. F. Schriber, “Another Step Back for Wave-Particle Duality,” Physics, Vol. 4, No. 102, 2011, Article ID: 230406. [36] J.-S. Tang et al., “Revisiting Bohr’s Principle of Complementarity Using a Quantum Device,” arXiv: 1204.5304V1[quant-ph], 24 April 2012. [37] T. Jacobson, et al., “Increase of Black Hole Entropy in Higher Curvature Gravity,” arXiv: gr-qc/9503020V1, 11 March 1995. [38] V. Vedral, “In from the Cold,” New Scientist, Vol. 216, No. 2886, 2012, pp. 33-37. [39] L. M. Krauss, “A Higgs-Saw Mechanism as a Source of Dark Energy,” arXiv:1306.3239V1[hep-ph], 13 June 2013. [40] L. Grossman, “Dark Energy May Spring from the Higgs Boson,” New Scientist, Vol. 219, No. 2931, 2013, p. 11.
http://dx.doi.org/10.1016/S0262-4079(13)62067-9 [41] D. Mermin, “Quantum Mechanics: Fixing the Shifty Split,” Physics Today, Vol. 65, No. 7, 2012, pp. 8-10.
http://dx.doi.org/10.1063/PT.3.1618 [42] M. S. El Naschie, “A Note on Quantum Gravity and Cantorian Spacetime,” Chaos, Solitons & Fractal, Vol. 8, No. 1, 1997, pp. 131-133.
http://dx.doi.org/10.1016/S0960-0779(96)00128-2 [43] M. S. El Naschie, “Complex Vacuum Fluctuation as a Chaotic ‘Limit’ Set of Any Kleinian Group Transformation and the Mass Spectrum of High Energy Particle Physics via Spontaneous Self-Organization,” Chaos, Solitons & Fractals, Vol. 17, No. 4, 2003, pp. 631-638.
http://dx.doi.org/10.1016/S0960-0779(02)00630-6 [44] M. S. El Naschie, “VAK, Vacuum Fluctuation and the Mass Spectrum of High Energy Particle Physics,” Chaos, Solitons & Fractals, Vol. 17, No. 4, 2003, pp. 797-807.
http://dx.doi.org/10.1016/S0960-0779(02)00684-7 [45] M. S. El Naschie, “The VAK of Vacuum Fluctuation, Spontaneous Self-Organization and Complexity Theory Interpretation of High Energy Particle Physics and the Mass Spectrum,” Chaos, Solitons & Fractals, Vol. 18, No. 2, 2003, pp. 401-420.
http://dx.doi.org/10.1016/S0960-0779(03)00098-5 [46] J.-H. He, “A Note on Elementary Cobordism and Negative Space,” International Journal of Nonlinear Sciences and Numerical Simulation, Vol. 11, No. 12, 2010, pp. 1093-1095. [47] M. S. El Naschie, “Average Symmetry, Stability and Ergodicity of Multidimensional Cantor Sets,” Il Nuovo Cimento, Vol. 109, No. 2, 1994, pp. 149-157.
http://dx.doi.org/10.1007/BF02727425 [48] M. S. El Naschie, “Mathematical Foundations of E-Infinity via Coxeter and Reflection Groups,” Chaos, Solitons & Fractals, Vol. 37, No. 5, 2008, pp. 1267-1268.
http://dx.doi.org/10.1016/j.chaos.2008.02.001 [49] M. S. El Naschie, “Removing Spurious Non-Linearity in the Structure of Micro-Space-Time and Quantum Field Renormalization,” Chaos, Solitons & Fractals, Vol. 37, No. 1, 2008, pp. 60-64.
http://dx.doi.org/10.1016/j.chaos.2007.10.005 [50] M. S. El Naschie, “On ’t Hooft Dimensional Regularization in E-Infinity Space,” Chaos, Solitons & Fractals, Vol. 12, No. 5, 2001, pp. 851-858.
http://dx.doi.org/10.1016/S0960-0779(00)00138-7 [51] O. E. Rossler, et al., “Hubble Expansion in Static SpaceTime,” Chaos, Solitons & Fractals, Vol. 33, No. 3, 2007, pp. 770-775.
http://dx.doi.org/10.1016/j.chaos.2006.06.046 [52] M. Pusey, J. Barrett and T. Randolph, “On the Reality of Quantum State,” Nature Physics, Vol. 8, June 2012, pp. 475-478. [53] M. S. El Naschie, “Mohamed El Naschie Answers a Few Questions about This Month’s Emerging Research Front in the Field of Physics,” 2004.
http://esi-topics.com/erf/2004/october04-MohamedElNaschie.html [54] M. S. El Naschie, “This Month’s New Hot Paper in the Field of Engineering: On a Fuzzy Kahler-Like Manifold Which Is Consistent with the Two Slit Experiment,” International Journal of Nonlinear Sciences and Numerical Simulation, Vol. 6, No. 2, 2005, pp. 95-98.
http://esi-topics.com/nhp/2006/september-06-MohamedElNaschie.html [55] M. Persinger and C. Lavellee, “Theoretical and Experimental Evidence of Macroscopic Entanglement between Human Brain Activity and Photon Emission,” Journal of Consciousness Exploration & Research, Vol. 1, No. 7, 2010, pp. 785-807. [56] M. S. El Naschie, “COBE Satellite Measurement, Cantorian Space and Cosmic Strings,” Chaos, Solitons & Fractals, Vol. 8, No. 5, 1977, pp. 847-850.
http://dx.doi.org/10.1016/S0960-0779(97)00084-2 [57] L. Marek-Crnjac, “The Physics of Empty Sets and the Quantum,” Nonlinear Science Letters B, Vol. 1, No. 1, 2011, pp. 13-14. [58] J.-H. He, “The Importance of the Empty Set Underpinning the Foundation of Quantum Physics,” Nonlinear Science Letters B, Vol. 1, No. 1, 2011, pp. 6-7. [59] M. S. El Naschie, “Penrose Universe and Cantorian Spacetime as a Model for Noncommutative Quantum Geometry,” Chaos, Solitons & Fractals, Vol. 9, No. 6, 1998, pp. 931-933.
http://dx.doi.org/10.1016/S0960-0779(98)00077-0 [60] M. S. El Naschie, “Stress, Stability and Chaos in Structural Engineering,” McGraw Hill, London, 1990. [61] D. Horrockos and W. Johnson, “On Anticlastic Curvature with Special Reference to Plastic Bending,” The International Journal of Mechanical Sciences, Vol. 9, No. 12, 1967, pp. 835-861.
http://dx.doi.org/10.1016/0020-7403(67)90011-2 [62] E. Cosserat and F. Cosserat, “Theorie des Corps Deformables,” Lavoisier S.A.S., Paris, 1909. [63] M. S. El Naschie, “A Fractal Menger Sponge Spacetime Proposal to Reconcile Measurements and Theoretical Predictions of Cosmic Dark Energy,” International Journal of Modern Nonlinear Theory and Application, Vol. 2, No. 2, 2013, pp. 107-121.
http://dx.doi.org/10.4236/ijmnta.2013.22014 [64] M. S. El Naschie, “On the Philosophy of Being and Nothingness in Fundamental Physics,” Nonlinear Science Letters A, Vol. 2, No. 1, 2011, pp. 5-6. [65] M. S. El Naschie, “On the Mathematical Philosophy of Being and Nothingness in Quantum Physics,” Fractal Space-Time & Non-Commutative Geometry in Quantum and High Energy Physics, Vol. 2, No. 2, 2012, pp. 103106.
[1] L. Amendola and S. Tsujikawa, “Dark Energy, Theory and Observations,” Cambridge University Press, Cambridge, 2010. [2] B. Carr, “Universe or Multiverse?” Cambridge University Press, Cambridge, 2010. [3] Y. Baryshev and P. Teerikorpi, “Discovery of Cosmic Fractals,” World Scientific, Singapore, 2011. [4] L. Nottale, “Scale Relativity,” Imperial College Press, London, 2011. [5] G. Barenblatt, “Scaling,” Cambridge University Press, Cambridge, 2003.
http://dx.doi.org/10.1017/CBO9780511814921 [6] J. Mageuijo and L. Smolin, “Lorentz Invariance with an Invariant Energy Scale,” arXiv: hep-th/0112090V2, 18 December 2001. [7] G. Veneziano, “Ward Identities in Dual String Theories,” Physics Letters B, Vol. 167, No. 4, 1986, pp. 388-392. [8] Y. Nambu, “Quark Model and the Factorization of Veneziano Amplitude, In: R. Choud, Ed., Symmetries and Quark Models, Gordon and Breach, New York, 1970, pp. 269-278. [9] V. Vladimirov. I. Valovich and E. Zelenov, “P-Adic Analysis and Mathematical Physics,” World Scientific, Singapore, 1994. http://dx.doi.org/10.1142/1581 [10] A. Connes, “Non-Commutative Geometry,” Academic Press, San Diego, 1994. [11] R. Penrose, “The Road to Reality,” Jonathan Cape, London, 2004. [12] L. Hardy, “Non-Locality of Two Particles without Inequalities for Almost All Entangled States,” Physical Review Letters, Vol. 71, No. 11, 1993, pp. 1665-1668.
http://dx.doi.org/10.1103/PhysRevLett.71.1665 [13] C. Nash and S. Sen, “Topology and Geometry for Physicists,” Academic Press, San Diego, 1983. [14] I. Buchbinder, S. Odintsov and I. Shapiro, “Effective Action in Quantum Gravity,” Institute of Physics Publishing, Bristol, 1992. [15] M. Green, J. Schwarz and E. Witten, “Superstring Theory,” Cambridge University Press, Cambridge, 1987. [16] M. Kaku, “Introduction to Superstrings and M-Theory,” Springer, New York, 1999.
http://dx.doi.org/10.1007/978-1-4612-0543-2 [17] J.-H. He, “Hilbert Cube Model for Fractal Space-Time,” Chaos, Solitons & Fractals, Vol. 42, No. 5, 2009, pp. 2754-2759. http://dx.doi.org/10.1016/j.chaos.2009.03.182 [18] J.-H. He, “Twenty Six Dimensional Polytope and High Energy Spacetime Physics,” Chaos, Solitons & Fractals, Vol. 33, No. 1, 2007, pp. 5-13.
http://dx.doi.org/10.1016/j.chaos.2006.10.048 [19] D. Joyce, “Compact Manifold with Special Holonom,” Oxford Press, Oxford, 2003. [20] T. Hübsch, “Calabi-Yau Manifolds,” World Scientific, Singapore, 1994. [21] E. Charpentier, A. Lesne and N. Nikolski, “Kolmogorov’s Heritage in Mathematics,” Springer, Berlin, 2007. [22] M. S. El Naschie, “Quantum Entanglement: Where Dark Energy and Negative Gravity plus Accelerated Expansion of the Universe Comes From,” Journal of Quantum Information Science, Vol. 3, No. 2, 2013, pp. 57-77.
http://dx.doi.org/10.4236/jqis.2013.32011 [23] M. S. El Naschie, “The Missing Dark Energy of the Cosmos From Light Cone Topological Velocity and Scaling the Planck Scale,” Open Journal of Microphysics, Vol. 3, No. 3, 2013, pp. 64-70.
http://dx.doi.org/10.4236/ojm.2013.33012 [24] M. S. El Naschie and A. Helal, “Dark Energy Explained via the Hawking-Hartle Quantum Wave and the Topology of Cosmic Crystallography,” International Journal of Astronomy and Astrophysics, Vol. 3, No. 3, 2013, pp. 318343. [25] M. S. El Naschie, “The Quantum Gravity Immirzi Parameter—A General Physical and Topological Interpretation,” Gravitation and Cosmology, Vol. 19, No. 3, 2013, pp. 151-155.
http://dx.doi.org/10.1134/S0202289313030031 [26] M. S. El Naschie, “What Is the Missing Dark Energy in a Nutshell and the Hawking-Hartle Quantum Wave Collapse,” International Journal of Astronomy and Astrophysics, Vol. 3, No. 3, 2013, pp. 205-211.
http://dx.doi.org/10.4236/ijaa.2013.33024 [27] L. Marek-Crnjac, “Modification of Einstein’s E = mc2 to E = mc2/22,” American Journal of Modern Physics, Vol. 2, No. 5, 2013, pp. 255-263. [28] M. S. El Naschie, “A Resolution of the Cosmic Dark Energy via a Quantum Entanglement Relativity Theory,” Journal of Quantum Information Science, Vol. 3, No. 1, 2013, pp. 23-26.
http://dx.doi.org/10.4236/jqis.2013.31006 [29] M. S. El Naschie, “Dark Energy from Kaluza-Klein Spacetime and Noether’s Theorem via Lagrangian Multiplier Method,” Journal of Modern Physics, Vol. 4, No. 6, 2013, pp. 757-760.
http://dx.doi.org/10.4236/jmp.2013.46103 [30] M. S. El Naschie, “Determining the Missing Dark Energy of the Cosmos from a Light Cone Exact Relativistic Analysis,” Journal of Modern Physics, Vol. 2, No. 2, 2013, pp. 18-23. [31] M. S. El Naschie, “Towards a General Transfinite Set Theory for Quantum Mechanics,” Fractal Space-Time and Non-Commutative Geometry in High Energy Physics, Vol. 2, No. 2, 2012, pp. 135-142. [32] R. Elwes, “Ultimate Logic,” New Scientist, Vol. 211, No. 2823, 2011, pp. 30-33.
http://dx.doi.org/10.1016/S0262-4079(11)61838-1 [33] V. Jacques, et al., “Delayed-Choice Test of Quantum Complementarity with Interfering Single Photons,” Physical Review Letters, Vol. 100, No. 22, 2008, Article ID: 220402.
http://dx.doi.org/10.1103/PhysRevLett.100.220402 [34] L. Li, N. L. Liu and Z. X. Yu, “Duality Relations in a Two Path Interferometer with an Asymmetric Beam Splitter,” Physical Review A, Vol. 85, No. 5, 2012, Article ID: 054101.
http://dx.doi.org/10.1103/PhysRevA.85.054101 [35] M. F. Schriber, “Another Step Back for Wave-Particle Duality,” Physics, Vol. 4, No. 102, 2011, Article ID: 230406. [36] J.-S. Tang et al., “Revisiting Bohr’s Principle of Complementarity Using a Quantum Device,” arXiv: 1204.5304V1[quant-ph], 24 April 2012. [37] T. Jacobson, et al., “Increase of Black Hole Entropy in Higher Curvature Gravity,” arXiv: gr-qc/9503020V1, 11 March 1995. [38] V. Vedral, “In from the Cold,” New Scientist, Vol. 216, No. 2886, 2012, pp. 33-37. [39] L. M. Krauss, “A Higgs-Saw Mechanism as a Source of Dark Energy,” arXiv:1306.3239V1[hep-ph], 13 June 2013. [40] L. Grossman, “Dark Energy May Spring from the Higgs Boson,” New Scientist, Vol. 219, No. 2931, 2013, p. 11.
http://dx.doi.org/10.1016/S0262-4079(13)62067-9 [41] D. Mermin, “Quantum Mechanics: Fixing the Shifty Split,” Physics Today, Vol. 65, No. 7, 2012, pp. 8-10.
http://dx.doi.org/10.1063/PT.3.1618 [42] M. S. El Naschie, “A Note on Quantum Gravity and Cantorian Spacetime,” Chaos, Solitons & Fractal, Vol. 8, No. 1, 1997, pp. 131-133.
http://dx.doi.org/10.1016/S0960-0779(96)00128-2 [43] M. S. El Naschie, “Complex Vacuum Fluctuation as a Chaotic ‘Limit’ Set of Any Kleinian Group Transformation and the Mass Spectrum of High Energy Particle Physics via Spontaneous Self-Organization,” Chaos, Solitons & Fractals, Vol. 17, No. 4, 2003, pp. 631-638.
http://dx.doi.org/10.1016/S0960-0779(02)00630-6 [44] M. S. El Naschie, “VAK, Vacuum Fluctuation and the Mass Spectrum of High Energy Particle Physics,” Chaos, Solitons & Fractals, Vol. 17, No. 4, 2003, pp. 797-807.
http://dx.doi.org/10.1016/S0960-0779(02)00684-7 [45] M. S. El Naschie, “The VAK of Vacuum Fluctuation, Spontaneous Self-Organization and Complexity Theory Interpretation of High Energy Particle Physics and the Mass Spectrum,” Chaos, Solitons & Fractals, Vol. 18, No. 2, 2003, pp. 401-420.
http://dx.doi.org/10.1016/S0960-0779(03)00098-5 [46] J.-H. He, “A Note on Elementary Cobordism and Negative Space,” International Journal of Nonlinear Sciences and Numerical Simulation, Vol. 11, No. 12, 2010, pp. 1093-1095. [47] M. S. El Naschie, “Average Symmetry, Stability and Ergodicity of Multidimensional Cantor Sets,” Il Nuovo Cimento, Vol. 109, No. 2, 1994, pp. 149-157.
http://dx.doi.org/10.1007/BF02727425 [48] M. S. El Naschie, “Mathematical Foundations of E-Infinity via Coxeter and Reflection Groups,” Chaos, Solitons & Fractals, Vol. 37, No. 5, 2008, pp. 1267-1268.
http://dx.doi.org/10.1016/j.chaos.2008.02.001 [49] M. S. El Naschie, “Removing Spurious Non-Linearity in the Structure of Micro-Space-Time and Quantum Field Renormalization,” Chaos, Solitons & Fractals, Vol. 37, No. 1, 2008, pp. 60-64.
http://dx.doi.org/10.1016/j.chaos.2007.10.005 [50] M. S. El Naschie, “On ’t Hooft Dimensional Regularization in E-Infinity Space,” Chaos, Solitons & Fractals, Vol. 12, No. 5, 2001, pp. 851-858.
http://dx.doi.org/10.1016/S0960-0779(00)00138-7 [51] O. E. Rossler, et al., “Hubble Expansion in Static SpaceTime,” Chaos, Solitons & Fractals, Vol. 33, No. 3, 2007, pp. 770-775.
http://dx.doi.org/10.1016/j.chaos.2006.06.046 [52] M. Pusey, J. Barrett and T. Randolph, “On the Reality of Quantum State,” Nature Physics, Vol. 8, June 2012, pp. 475-478. [53] M. S. El Naschie, “Mohamed El Naschie Answers a Few Questions about This Month’s Emerging Research Front in the Field of Physics,” 2004.
http://esi-topics.com/erf/2004/october04-MohamedElNaschie.html [54] M. S. El Naschie, “This Month’s New Hot Paper in the Field of Engineering: On a Fuzzy Kahler-Like Manifold Which Is Consistent with the Two Slit Experiment,” International Journal of Nonlinear Sciences and Numerical Simulation, Vol. 6, No. 2, 2005, pp. 95-98.
http://esi-topics.com/nhp/2006/september-06-MohamedElNaschie.html [55] M. Persinger and C. Lavellee, “Theoretical and Experimental Evidence of Macroscopic Entanglement between Human Brain Activity and Photon Emission,” Journal of Consciousness Exploration & Research, Vol. 1, No. 7, 2010, pp. 785-807. [56] M. S. El Naschie, “COBE Satellite Measurement, Cantorian Space and Cosmic Strings,” Chaos, Solitons & Fractals, Vol. 8, No. 5, 1977, pp. 847-850.
http://dx.doi.org/10.1016/S0960-0779(97)00084-2 [57] L. Marek-Crnjac, “The Physics of Empty Sets and the Quantum,” Nonlinear Science Letters B, Vol. 1, No. 1, 2011, pp. 13-14. [58] J.-H. He, “The Importance of the Empty Set Underpinning the Foundation of Quantum Physics,” Nonlinear Science Letters B, Vol. 1, No. 1, 2011, pp. 6-7. [59] M. S. El Naschie, “Penrose Universe and Cantorian Spacetime as a Model for Noncommutative Quantum Geometry,” Chaos, Solitons & Fractals, Vol. 9, No. 6, 1998, pp. 931-933.
http://dx.doi.org/10.1016/S0960-0779(98)00077-0 [60] M. S. El Naschie, “Stress, Stability and Chaos in Structural Engineering,” McGraw Hill, London, 1990. [61] D. Horrockos and W. Johnson, “On Anticlastic Curvature with Special Reference to Plastic Bending,” The International Journal of Mechanical Sciences, Vol. 9, No. 12, 1967, pp. 835-861.
http://dx.doi.org/10.1016/0020-7403(67)90011-2 [62] E. Cosserat and F. Cosserat, “Theorie des Corps Deformables,” Lavoisier S.A.S., Paris, 1909. [63] M. S. El Naschie, “A Fractal Menger Sponge Spacetime Proposal to Reconcile Measurements and Theoretical Predictions of Cosmic Dark Energy,” International Journal of Modern Nonlinear Theory and Application, Vol. 2, No. 2, 2013, pp. 107-121.
http://dx.doi.org/10.4236/ijmnta.2013.22014 [64] M. S. El Naschie, “On the Philosophy of Being and Nothingness in Fundamental Physics,” Nonlinear Science Letters A, Vol. 2, No. 1, 2011, pp. 5-6. [65] M. S. El Naschie, “On the Mathematical Philosophy of Being and Nothingness in Quantum Physics,” Fractal Space-Time & Non-Commutative Geometry in Quantum and High Energy Physics, Vol. 2, No. 2, 2012, pp. 103106.