Home Science Page Data Stream Momentum Directionals Root Beings The Experiment
So far weÕve only seen how True Root Beings of different orders behave. Let us begin our pictorial discussion of these other Root Beings with the simplest condition, when the Root Number is Two. Remember this is when all the constants equal one, giving our Root Equations their most economical expression.
The following is a ÔpictureÕ of many types of Roots Beings when the Root Number is Two. It is a ÔpictureÕ of the first two hundred iterations, with 8 different Root Beings.
Let me introduce a few of them to you. The steepest blue line is the True Root Being and reaches 16 digits of accuracy (the limits of our computer) after less than 25 iterations. Notice that each of the Root Beings except one reaches this level of accuracy before 200 iterations. While they approach at different speeds, they all approach the same limit, in this case, the Fourth Root of 2 minus One. Each of these Root Beings is either the True or Bogus Root Beings by virtue of the fact that they approach the same limit. The other Root Being which looks like a horizontal line in this ÔpictureÕ is a False Fourth Order Root Equation. While it approaches a limit, it has a distinctly different limit than the rest. From the graph, it looks like it is about one thousandth different from the True Root. We call this characteristic of the False Root Being ÔHitting the WallÕ, because it canÕt get any closer. More on this topic later.
Notice that the dimensions of the vertical axis are logarithmic and represent the distance or difference from the True Root. Notice further that each of the Root Beings approaches the Root in a relatively linear manner from this logarithmic perspective. This indicates an exponential relation between the number of iterations and the Distance from the Root, i.e. how close our Root Being is to the True Root. We discovered the equation for the slope of the Path when it represents a Square Root, but not any of the Higher Roots.
From the above exploration, we can easily see that many iterative equations of a certain type would certainly approach the Root of Two, slower than the True Equation, but steadily nevertheless. These other Root Beings still reach 16-place accuracy relatively quickly when the Number is two. This is nothing to turn oneÕs nose up at under any circumstance. Now letÕs explore what happens to these Root Beings when their Circumstances, i.e. their Root Number, is less than two.
On the following page are two ÔpicturesÕ of our Root Beings. The top ÔpictureÕ is when the Root Number equals 1.5. The bottom ÔpictureÕ is when the Root Number equals 1.25. We deliberately left out the False Root Being from the ÔpictureÕ because we wanted to focus exclusively on the Bogus Root Beings.
Notice that the slope of these Root Beings increases as the Root Number leaves two and approaches one. Under both of these Circumstances their slope is much steeper than when the Root Number was Two. Each of our Bogus Equations approaches the Root even more rapidly than before.
Specifically let us look at the progress of the slowest Root Being, the light blue line with the shallow slope. When the Root Number is two it reaches 16-place accuracy computer accuracy after nearly 200 iterations. When the Number is 1.5 it reaches computer accuracy well before 150 iterations. When the Root Number is 1.25, our Slowest Root Being reaches this same accuracy before 100 iterations.
This process continues as the Root Number approaches One. All of the Bogus Root Beings approach the Root more and more quickly. When the Root Number equals One each of the Root Beings immediately equals the Root minus one, which equals zero. This is because some relevant constants turn to zero. This is the only exact number in this scheme of things. Below is an unnecessary graph. All of the Root Beings equal zero immediately, which is the Root of one minus one.
As the Root Number decreases below one, the slopes become more and more shallow. At one the slope is infinite because it is immediately exact. When the Root Number is less than or greater than one our Root Being only approaches its goal. They will never be exact again. The further the Root Number is away from one the slower the approach. While two is the Normal Number, one is the Center Number.
On the following two pages are pictures of the Root Beings when the Root Number is less than one. When our Root Number is just under one at 0.75, the Slowest of Root Beings reaches its goal at about 100 iterations, still quicker than when the Number is Two. As the Root Number is lowered to 0.5, the approach of our Root Beings slows substantially, to about the same level as when the Number is 2. Our slowest reaches computer accuracy after about 200 iterations. When the Root Number is dropped again to 0.25, 1/4, the approach becomes even slower. The slowest of our Root Beings never reaches the mythical 16-place accuracy of the computer after 200 iterations. It only reaches 6-place accuracy. However it is visually apparent that while the approach of all the Root Beings to their Root is slower, it is still steady.
We lower the Number again to 0.1, 1/10. The True Root Being and two of the other Root Beings, that we will call Now-based, (to be explained later) still steadily approach their Destiny, reaching 16 place computer accuracy before 100 iterations. However the rest of these Root Beings, which we will call the Past-based Root Beings, (to be explained later) fall apart, seriously dissipating. There is no apparent order to their number generation. It certainly does not approach the Root.
Lowering the Root Number to zero, seriously slows down the True Root Being and the Now-based Bogus Equations. The Now based Root Beings, only reach 1 place accuracy after 200 iterations, while even the True Root Being has only reached two place accuracy. The rest of the Root Beings, the Past-based ones, are seriously flawed by their dissipated, unorganized behavior. They bounce wildly about approaching no limit.
We canÕt go under zero for computer computations for the even Roots because this is the realm of imaginary numbers, based upon the square root of negative one. Because our study is only of real numbers, we ceased explorations at this point.
When the Root Number is One our Root Beings, True and Bogus, immediately equal zero. As the Root Number moves away from One, our Center Number, in positive and negative directions, the Root Beings approach their Destiny more and more slowly. When our Root Number is somewhere between 1/4 and 1/10, the Past-based Root Beings, dissipate into irregularity. The True Root Being and the Now-based Root Beings, while approaching their goal much more slowly, still move steadily toward their Root all the way to and including when our Root Number is zero. The Past Based Root Beings just dissipate further. Our explorations have shown that all of our Root Beings approach their destiny when the Root Number is between 2 and 1/4. However the Past-based Root Beings dissipate by 1/10.