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23.3D More Pictures of Root Beings

D. When the Root Number is greater than 2

True and Bogus Root Beings 3->17

Now that we have seen what happens to our Root Beings at two and under, let us see what happens when the Root Number is greater than 2. After the following explications there are 7 pages of ÔpicturesÕ of our Root Beings under a continuum of Circumstances as the Root Number rises from 3 to 17. It is nice to see our Root Beings in process.

4th Order Root Beings when the Root Number is 3, just a little slower

When the Root Number becomes three, the approach to their Goal, their Holy Grail, becomes slower. Already the most Past-based Root Being of this computer experiment doesnÕt even reach computer accuracy within 200 iterations; it only reaches about 13 digits of decimal accuracy. However it is easy to visually see that each of our Root Beings, except the False Root Being, the flat green line, steadily if slowly still approach their goal. The True Root Being still approaches most quickly. The Now-based Root Beings are just a little slower while the Past-based Root Beings are the slowest of all. The False Root Being has slightly different coefficients and is approaching a distinctly different goal. Therefore it ÔHits the WallÕ.

When the Root Number is 4, Slower Still, No More False Beings

When the Root Number becomes 4, each Being approaches their Destiny a little more slowly still. The slowest Bogus Equation only reaches 9-place accuracy after 200 iterations. The True Root Equation reaches computer accuracy quickest, after only 25 iterations.

When the Root Number is 5, the Now-based separate

When the Root Number is five, another of the Past-Based Root Beings doesnÕt reach 16-place computer accuracy after 200 iterations. All of the Root Beings approach a little more slowly. The Now-based Root Beings have finally separated from each other. The Now Being based totally upon the past moment, which we will call All-Now, the steady red line, begins to move more slowly than the Now Being who extends a little into the past, which we will call Almost-Now, the steady green line.

When the Root Number is 6, Three donÕt reach computer accuracy

When the Root Number is 6, three of our Past-based Root Beings donÕt reach computer accuracy after 200 iterations. The slowest only reaches 6-place accuracy. Note that the Past Based and the True Root Being all approach their Goal erratically, while the Now-based approaches their Goal steadily without wandering. However the True Root Being still approaches most rapidly even though its Path is erratic.

When the Root Number is 7, All-Now slows to level of Past-based

When the Root Number is increased to seven, the approach of the Root Beings to their Destiny is slower still. Note that All Now, the steady red line, approaches no more quickly than the best Past Being. This is a real come down for All Now. When the Number was under 7 the All Now Being always approached his goal more quickly than any Past Being.

When the Root Number is 8. Slowest at 4 Place accuracy

When the Root Number is raised to 8, the True Being still reaches relative perfection within 30 iterations. The others are slowing down. The worst only reaches 4-place accuracy within 200 iterations. And humiliation for All-Now. He is now slower that the Best of the Past and only reaches 14 place accuracy. He and his brother Almost-Now still approach their Goal from the Straight and Narrow, as opposed to the erratic approach of the rest of the Root Beings including the True Equation.

When the Root Number is 9, only Almost-Now reaches bottom

When the Root Number is raised to 9, the True Being is little changed, while only one of the rest, Almost-Now, achieves computer accuracy. All-Now is further humiliated by being confused with a lower level Past Being.

When the Root Number is 10, All-Now at 4 place accuracy

When the Root Number is 10, the True Being is unaffected, still reaching computer perfection after about 30 iterations. Not so with the rest. All-Now is second slowest, only reaching 4-place accuracy after 200 iterations. Note however that all these Beings are still steadily approaching their goal even though much more slowly.

When the Root Number is 11, All-Now is slowest

When the Number is raised to 11, the True Being is still unaffected. However All-Now is the slowest, only reaching 2-place accuracy. Further his brother Almost-Now, has gotten slower than Best-of-the-Past, the erratic reddish brown line. All Beings still approach their Goal although more slowly.

When the Root Number is 12, All-Now one-place accuracy

When the Root Number is 12, our high and mighty All-Now doesnÕt even reach one place accuracy, but still approaches his goal, like the rest.

When the Root Number is 13, All-Now bifurcates

13 is an unlucky number for All-Now. He is the first to fall apart, bifurcating with zero-place accuracy. He will never reach his goal and the Root Number is only 13. His straight and narrow Now-based solution worked well enough when the Number was smaller, under 4, but now that the going is getting a little rougher All-Now canÕt handle the pressure and bifurcates.

When the Root Number is 14, All-Now lost, Almost-Now losing ground

When the stakes are raised All-Now further bifurcates; he is lost. The rest are maintaining, while Almost-Now is losing ground to his Past-based sisters. He still approaches on the straight and narrow, while the rest approach erratically. The Past-based Beings still approach their Destiny but unsteadily.

When the Root Number is 15, things only get worse

Nothing new when the Root Number reaches 15. The True Root still reaches computer accuracy fairly rapidly at about 40 iterations, a little slow for her. However all of the other Root Beings except All-Now still approach the Root even though very slowly. The Best-of-the-Past only reaches 9-place accuracy after 200 iterations.

When the Root Number is 16, a little more dissipated

As the Root Number reaches 16, only two of the Past-based equations reach four places of accuracy after 200 iterations. The True Root still reaches computer accuracy before 50 iterations. However all of the Root Beings except All-Now continue to steadily approach the Root.

When the Root Number is 17, A Closer Look

Let us take a little closer look at our Root Beings by eliminating the True Being from our graph/picture. When the Root Number is 17, only two of our Bogus Equations still approach their destiny. The Rest seem to have dissipated or bifurcated. None of them reaches more than 1 place accuracy regularly over 200 iterations. All-Now is visibly bifurcating in this snap-shot.

 

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