22nd March, 2011.
E-infinity Communication No. 83
On El Naschie’s small world of infinite dimensions
In the following we give a number of observations, interpretations and comments on the relationship between E-infinity theory and the theory of small world. The copyrights of the material published here, some for the first time should be respected by quoting the source, namely E-Infinity Communication Publications. We have not attempted to mention each of the sources of the information given here which is largely due to the papers as well as formal and informal talks and lectures by Prof. Mohamed El Naschie, Prof. Ji-Huan He, Prof. G. Ord, Prof. L. Marek-Crnjac and Prof. G. Iovane.
1. Ergodicity or global chaos as well as complete order may be assigned both the same zero complexity indexes. Both complete order and complete chaos are consequently structurally unstable in some vague topological sense. By contrast a KAM system which encompasses both chaos as well as order on all scales may be seen as relatively structurally stable and robust. This property of robustness to perturbation is shared by small world networks. In a sense robustness to perturbation is a substitute for friction in dissipative systems. Hamiltonian systems have no physical friction. However the irrationality of the winding number is what replaces physical friction in Hamiltonian systems such as quantum physics according to KAM theorem. That is how the golden mean comes in as being the most irrational number. From this view point E-infinity and small world theory seem to have common roots if not much more than that.
2. E-infinity’s quasi manifold is probably one of the most amazing geometrical and topological constructions which unite the un-unitable. It is infinite dimensional yet it has a resolution dependent finite expectation value for all its topological invariants including dimensions. It is infinitely large yet it is in more than one sense compact and so is its holographic boundary. It is fuzzy but within this fuzziness everything is probabilistically exact. It is completely discrete but due to the transfiniteness of its geometry it resembles the continuum. It is infinitely large but because it reproduces itself latest after moving a maximal distance equal to the isomorphic length multiplied with an arbitrary radius, it is semi-finite and resembles a semi-small world.
3. There are clear applications of both small world theory and E-infinity theory to any transfinite network such as neurons in the brain as well as complex fracture systems such as seismic fault structures relevant to earth quakes. The application in sociology may be among the most profound applications to things which may be shaping the future right this minute.
4. With regard to high energy physics Mohamed El Naschie’s theory de facto proposed the replacement of the classical lattice of the large world with the transfinite Cantorian lattices of the small world theory.
5. It is frequently argued that the six-degrees separation does not apply to a set of people alive at different times. The classical example for that is that Alexander the Great is separated from Albert Einstein by more than six-degrees. However if we take the degree of a degree into account, i.e. we take the weight of a degree (or a dimension) into account then we could still end with 6 or less degrees of separation. For instance we know that Alexander was interested in the art of knode. On the other hand Lee Smolin showed that Stuart Kauffman’s knode theory is relevant to quantum gravity. That means it is relevant to gravity and this connects Alexander to Einstein albeit it is a very weak connection.
6. Between two random people we have 24 acquaintances according to an application of Erdös-Rényi theorem. Thus we have 24 + 2 = 26. Similarly we have a world string sheet with two dimensions and when we add the 24 instantons of a Kähler to it we can simulate 2 + 24 = 26 degrees of freedom corresponding to the 26 bosonic dimensions of string theory. This is of course an extremely loose argument for the obvious but mathematically still terse connection between stringy networks and small world networks.