The Egyptian engineering scientist and theoretical physicist Mohamed El Naschie has found a definite resolution to the missing dark energy of the cosmos based on a revision of the theory of Relativity. Einstein’s equation of special relativityE = m_{0}c^{2}, wherem_{0}is the controversial rest mass and c is the velocity of light developed in smooth 4D space-time was transferred by El Naschie to a rugged Calabi-Yau and K3 fuzzy Kahler manifold. The result is an accurate, effective quantum gravity energy-mass relation which correctly predicts that 95.4915028% of the energy in the cosmos is the missing hypothetical dark energy. The agreement with WMAP and supernova measurements is astounding. Different theories are used by El Naschie to check the calculations and all lead to the same quantitative result. Thus the theories of varying speed of light, scale relativity, E-infinity theory, M-theory, Heterotic super strings, quantum field in curved space-time, Veneziano’s dual resonance model and Nash’s Euclidean embedding all reinforce, without any reservation, the above mentioned theoretical result of El Naschie which in turn is in total agreement with the most sophisticated cosmological measurement. Incidentally these experimental measurements and analysis were

Abstract:
Penrose fractal tiling is one of the simplest generic examples for a noncommutative space. In the present work, we determine the Hausdorff dimension corresponding to a four-dimensional analogue of the so-calledPenrose Universe and show how it could be used in resolving various fundamental problems in high energy physics and cosmology.

Abstract:
Penrose fractal tiling is one of the simplest generic examples for a noncommutative space. In the present work, we determine the Hausdorff dimension corresponding to a four-dimensional analogue of the so-calledPenrose Universe and show how it could be used in resolving various fundamental problems in high energy physics and cosmology. 1. Introduction As explained in detail in Connes’ [1], Penrose fractal tiling constitutes mathematically a quotient space . Using this fact A. Connes following earlier work due to von Neumann deduced a dimensional function which we generalize to a simple formula function linking both the Menger-Urysohn topological dimension and the corresponding Hausdorff dimension. The present work is subdivided into three main parts. First, we show an explicit application and generalization of the Connes’ dimensional function. Second, we derive the Hausdorff dimension of the Hilbert space which X represents. Finally, we show the relevance of these results in high energy physics and cosmology. 2. The Dimensional Function and the Hilbert Space Let us start from the Connes’ dimensional function for the Penrose universe [1]: Writing and using the Fibonacci sequence, it is easy to see that, starting from the seed and , we obtain the following dimensional hierarchy: By complete induction, one finds We obtain an exceptional Fibonacci sequence : The classical Fibonacci sequence is defined by the recurrence relation where , , and . The first few Fibonacci numbers of the classical Fibonacci sequence are given . The th Fibonacci number is given by the formula which is called the Binet form, named after Jaques Binet, where and are the solutions of the quadratic equation : The Binet form of the th Fibonacci number of the sequence can be expressed similar to the classical Fibonacci sequence: The Fibonacci sequence can be presented as an infinite geometric sequence: The Golden Section principle that connects the adjacent powers of the golden mean is seen from the infinite geometric sequence. The formula for the th Fibonacci number is clearly identical to the bijection formula of E-infinity algebra and rings, namely [2, 3], Here, is the Menger-Urysohn topological dimension which should not be confused with the embedding dimension and is the Hausdorff dimension whose topological dimension is . To see that this extends in a simple fashion to negative dimensions [4], we set and find that the empty set is structured and possesses a finite Hausdorff dimension equal to because Now, we claim that is effectively a random Hilbert space and is four dimensional

In this paper Nottale’s acclaimed scale relativity theory is given a transfinite Occam’s razor leading to exact predictions of the missing dark energy [1,2] of the cosmos. It is found that 95.4915% of the energy in the cosmos according to Einstein’s prediction must be dark energy or not there at all. This percentage is in almost complete agreement with actual measurements.

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 formulaE = mc^{2}. Consequently Einstein’s formula makes no distinction
between ordinary and dark energy.

In this short survey, we give a complete
list of the most important results obtained by El Naschie’s E-infinity
Cantorian space-time theory in the realm of quantum physics and cosmology.
Special attention is paid to his recent result on dark energy and revising
Einstein’s famous formula.

At its most basic level physics starts with space-time topology and geometry. On the other hand topology’s and geometry’s simplest and most basic elements are random Cantor sets. It follows then that nonlinear dynamics i.e. deterministic chaos and fractal geometry is the best mathematical theory to apply to the problems of high energy particle physics and cosmology. In the present work we give a short survey of some recent achievements of applying nonlinear dynamics to notoriously difficult subjects such as quantum entanglement as well as the origin and true nature of dark energy, negative absolute temperature and the fractal meaning of the constancy of the speed of light.

Abstract:
This tutorial review is dedicated to the work of the outstanding Egyptian theoretical physicist and engineering scientist Prof. Mohamed El Naschie. Every physics student knows the well-known Einstein’s mass-energy equation, E=mc^{2}, but unfortunately for physics, few know El Naschie’s modification, E(O)=mc^{2}/22, and El Naschie’s dark energy equation E(D)=mc^{2}(21/22) although this new insight has truly far reaching implications. This paper gives a short tutorial review of El Naschie’s fractal-Cantorian space-time as well as dark energy. Emphasis is put on the fundamental concept of Cantor set, fractal dimensions, zero set, empty set, and Casimir effect.

Abstract:
Modern advances in pure mathematics and particularly in transfinite set theory have introduced into the fundamentals of theoretical physics many novel concepts and devices such as fractal quasi manifolds with non-integer (Hausdorff) dimension for its geometry as well as infinite dimensional wild topology and non classical fuzzy logic. In the present work transfinite fractal sets and fuzzy logic are combined to enable the introduction of a new theory termed fractal logic to the foundation of high energy particle physics. This leads naturally to a new look at quantum gravity. In particular we will show that to understand and develop quantum gravity we have to bring various fields together, particularly fractals and nonlinear dynamics as well as sphere packing, fuzzy set theory, number theory and quantum entanglement and irrationally q-deformed algebra.

The paper is a condensed but accurate account of El
Naschie’s theory of Cantorian space-time which was used by him to clarify some
major problems in theoretical physics and cosmology. In particular El Naschie’s
revision and completion of relativity theory and demystification of dark energy
are destined to be two milestones in the history of theoretical physics.