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Now that weÕve seen how and when our Now-Based Beings bifurcate, let us look a little more closely at our Past-Based Root Beings.
Before going further let us point out the difference between the Past and Now based Root Beings. The Now-Based Beings are based too heavily upon the most recent events, while the Past-based beings ignore what has just happened in favor of past events. The True Root Being extends continuously from the immediate Past backwards two events. Bear in mind that these Root Beings, while relying upon different parts of their history to regenerate themselves, all use the same binomial coefficients. Even the smallest variation in these binomial coefficients generates a different Root, always. All the rational roots are generated from the binomial coefficients. It intuitively seems that if they are not binomial coefficients, that the approached number is not a Rational Root, a patterned transcendental but not a Rational Root. There is infinitely more of these patterned transcendentals.
Referring constantly to our picto-graphs that immediately follow this explication, when our Number is 12, each of the Past-based Beings, steadily approach their Destiny. The slowest reaches two place accuracy after 200 iterations while the fastest reaches 12-place accuracy. Note that while their approach is steady, it is in no way regular. As with the True Being they make some huge advances and then fall back. This is sharply contrasted with the Now-based Beings who vibrate towards their goal.
When our Number is raised to 16, these Past-based Root Beings still approach their goal but much more slowly. Best-of-the-Past, the yellow line only reaches 7-place accuracy after 200 iterations.
Uh Oh! Now that the root Number is 20, the two slowest of these Past-based Beings dissipate, randomly shooting around the area. While they did better than All-Now, they still couldnÕt take the pressure of the higher Root Numbers.
When the Number is raised to 25 the dissipation gets even worse. Sometimes the dissipated Root Beings get within 1/100 of the Root but then they bounce away by 4 or 5. Certainly nothing useful. The other two Root Beings, while bouncing around erratically, still steadily approach their goal, both getting to within 3-place accuracy after 200 iterations.
Bumping the Root Number to 40, we see utter chaos for all the Past-based Root Being. They all bounce around erratically sometimes only 1/100 away and then nearly 10 away from the Root. It is easy to see that as the Circumstances, i.e. Root Number, increases that it is harder and harder for the Past-based Root Beings to deal with. First they just approach more slowly but then at a certain point their erratic approach dissipates, not really heading anywhere anymore.
Lest there be any diehards who might think that some of the Past-based Root Beings still approaches the Root at 40, let us see what happens when the Root Number is 100. At this level, it is unmistakable that even the Best-of-the-Past is approaching no limit, but is instead bouncing around relatively aimlessly.
Just to give a contrast, let us include the True Root Being in our picto-graph of the higher Circumstances. At 40 none of the Past-based Beings have any apparent order or Goal. When the Root Number is 100, there is even less order for these Past-based Root Beings. Both the Now-based and Past-based Root Beings approach no limit. The True Being, however, still reaches 16-place computer accuracy before 50 iterations. The True Root Beings donÕt have a problem with the larger numbers, consistently approaching a Rational Root Limit, no matter how large the number.
In summing up our study, so far, there are only a few points that need to be made. While the True Root Being always approaches the Root regardless of Circumstance, the Bogus Beings fall apart at some point as the Root Number gets higher. The Past-based Beings dissipate while the Now-based Root Beings bifurcate. The bifurcations and dissipations continue to increase as the Root Number becomes larger. However there is a point at which these Bogus Beings approach the Root relatively continually.