Quantum Gravity 368.5: Gn = Gn-1 - Gn-2 introduces New Math, New physics
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From: OsherD on 2 May 2010 20:38 From Osher Doctorow We have seen that, in analogy with the Fibonacci Numbers F_n or Fn: 1) Fn = Fn-1 + Fn-2, n > 1 the Probable Causation/Influence Differences y - x from P(A-->B) = 1 + y - x in their sequence representation: 2) Gn = Gn-1 - Gn-2 are related to a "limiting type" result partly analogous to the Fibonacci quotient Fn+1/Fn limit phi, where phi is the Golden Mean or Golden Ratio = (1 + sqrt(5))/2. The partial analog for Gn is: 3) x = [1 +/- sqrt(3) i ]/2 The exactly same procedure that produces phi from Fn, namely dividing both sides of (1) by Fn-1 and taking the limit as n --> infinity, yields (3) from (2) except that the limit in a strict sense does not exist (not surprising since it contains the imaginary i while Gn are real-valued). Since (3) arguably represents Unification of the 4 Fundamental Interactions as explained in the recent post here, it involves "new mathematics" and "new physics". There are some interesting consequences of putting various initial values G1, G2 into (2) and then calculating the resulting sequences of Gn, although unlike Fn where F1 = 1 = F2, the Gn sequences are quite different: 4) If G2 - G1 is different from both G1 and G2 and neither is 0 (and G2 - G1 is not 0), then Gn takes on 6 different values in sequence which keep repeating, namely G1, G2, G2 - G1, and their negatives. 5) If G2 - G1 equals one of G1 or G2 (usually G1 in that case) and neither is 0 and they are unequal, then Gn takes on 4 different values in sequence, namely G1, G2, and their negatives. If one of G1 or G2 is 0 or they are equal, then Gn has fewer than 4 different values. The 6 in (4) is 3! = 3 times 2 times 1, which again arguably relates back to quarks (the number of ways of selecting one quark from 3 or one edge from a triangle). The 4 in (5) appears to relate to the "degeneration" of the triangle from 3 to 2 sides. For example, if we select G1 = 2 and G2 = 10, then we get: 6) G1 = 2, G2 = 10, G3 = 8, G4 = -2, G5 = -10, G2 = -8, G7 = 2, etc. (repeats). Osher Doctorow |