20th March, 2011.
E-infinity Communication No. 82
The small world of ‘t Hooft-Susskind holographic boundary – An E-infinity view
The diffeomorphic kinship between the Penrose fractal tiling universe and the compactified Klein modular curve is well known from El Naschie’s work on the holographic boundary theory. Following this theory a fundamental equation was established stating that the total numbers of state-like particles or massless gauge bosons may be taken to be the dimension of E8E8, namely 496 and that this number must be equal to certain isometries and dimensions pertinent to all elementary particles living on the holographic surface of the 496 dimensional bulk as well as pure gravity and finally electromagnetism. Since in four dimensional Einstein gravity as well as eight dimensional pure gravity the number of the corresponding independent components of the Riemannian tensor and the number of isometries is the same, namely 20, then it follows that 496 must be equal to 20 plus electromagnetism plus particle physics. Following E-infinity theory the number of particles on the holographic boundary are equal to the number of isometries of the classical Klein modular curve, namely 336 plus the compactification effect taking the boundary to infinity as in projective hyperbolic geometry, namely 3. Thus the total number of particle-like isometries is 336 + 3 = 339. Using the equation the inverse coupling of electromagnetism 137 is found exactly and given a topological meaning, namely a dimension of an electromagnetic manifold determined by the fundamental equation 496 – 20 – 339 = 137.
There is something even more astonishing about this holographic boundary which relates it to the theory of small world in an unexpected way which upon reflection should have been expected. The so called isometric length of E-infinity theory applied to the holographic boundary is given by half of the E-infinity core Hausdorff dimension which is half of the famous value 4.2360679, that is to say 4 plus the golden mean to the power of 3. Half of that is exactly 2.118033989. Now there is an approximate value to the Hausdorff dimension found using the classical continuous gamma distribution which was given long ago by El Naschie as well as S. Al Athel, namely 2 divided by the natural logarithm of the inverse golden mean which leads to 4.156173841. Dividing this by two we find an approximation to the isometric length, namely the (inverse) natural logarithm of 1.618033989 which means the isometric length is equal to the inverse of the natural logarithm of the inverse golden mean. There are two important points which we have to consider at this point. First the isometric length is the distance which we have to maximally travel in order to find our surroundings replicated almost exactly as if we had not moved at all. That means that our compactified holographic boundary which describes an infinite universe in all directions is still a finite and in fact small world-like universe. Let us call it small world-like holographic and Penrose universe. Second a small world non-transfinite ordinary network is typically characterized by a distance given also by a logarithmic value. Nor N nodes the distance between the two randomly chosen nodes is proportional to the logarithm of N, namely Log N. This logarithmic relation is behind the relation between social networks like Facebook and Twitter and the transfinite neural network behind quantum mechanics such as E-infinity transfinite networks as proposed for the first time by Mohamed El Naschie and his student Dr. Mahrous Ahmed as well as several other members of the E-infinity Group.