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There is only one last topic that we would like to handle before bowing out. This is the topic of the Root Being as a Dynamic Being rather than as a static equation. Each static Root Being is uniquely defined by four elements, i.e. Starting Point, Events, Coefficients, and Circumstance. If each of these elements is specified then we can generate an exact and unique graph for any Root Being.
These factors are illustrated for the General Fourth Order Root Equation, shown below. First we show the Givens. Note that small ÔcÕ is a function of our Root Number, a/b. It defines the Circumstance.
The numbers, D, E, & G are the Coefficients which determine if our Being is balanced or not. The numbers, A, B, & C are the Events from the ÔPastÕ that our Being uses to make Decisions. The Starting Point or ÔseedÕ is the exterior point that our Being starts from, i.e. the numbers that get our iterative equation going. For this 4th order equation it will take 3 numbers. If all of these numbers are fixed then our Root Being has a predestined course of action.
However if we imagine a Being existing in Time, where these factors can be modified according to experience then we have a Dynamic Being rather than a Static Being. Remember that our mathematical Root Beings are static because their existence is predestined according to preset constants. As a class we will call the predestined Beings, Statics and the others, Dynamics.
Earlier in this paper, we demonstrated that the Seed had no appreciable effect upon our True Root Beings. Thus we will ignore the Seed in the discussion to come. We will focus primarily upon Circumstance, Events, & Coefficients, which we will refer to as Pattern. These three factors have a tremendous influence upon the Destination or Limit of our equations.
Before proceeding forth let us speak a little more about inverse dimensionality. Remember that in ordinary 3 dimensional reality all elements are determined by position relative to a point of origin. Even the dynamics of ordinary reality are based upon the content of a moment, i.e. velocity, acceleration, thrust, etc. Given all the dynamical and fixed measures for any point and one can predict where it will be at any moment in the future or where it was in the past, if there were no other forces present. Well in the dimensional reality of Root Beings everything is determined by context rather than content. With each dimension comes another contextual layer. We call these contextual layers inverted dimensions. Hence with each dimension comes another layer of feedback.
The first inverted dimension is the same as the first ordinary dimension. It is static and one-dimensional. There is no layer of feedback. However the second inverted dimension is quite different from the second ordinary dimension. While the second ordinary dimension is just the static position on a plane, the second inverted dimension is based upon one layer of feedback.
Now let us imagine a Dynamic born into this inverted dimension. Through experience our Dynamic will add inverted dimensions by adding more layers of feedback to her existence. The First Dimension if Self: the second is Other. The two are balanced through constant feedback between the two dimensions. The Third Dimension breaks Family off from Other to form another Dimension. Then Friends are broken off to form a Fourth Dimension. Family is broken into Parents and Siblings to add a Fifth Dimension. School is eventually broken off from other to add a 6th Dimension. And so forth.
Thus as our Dynamic matures more layers of contextual feedback are added to increase dimensionality and complexity in our DynamicÕs existence. As each layer of feedback is added it becomes more and more complex to balance the elements. When there was just Self and Other it was much less complicated to Balance. However with the increasing complexity the Pattern becomes unbalanced. As the Pattern becomes unbalanced the Dynamic veers off the Path. Remember that in this world the Path is everything. Thus one of the tasks of our Dynamic is to Balance the Pattern, the Coefficients. This might be the most important task because without Balance our Dynamic will Hit the Wall.
Before going on, let us point out that while the increase in dimensionality speeds the approach to the Limit, the Destiny, of our Dynamic, that it simultaneously adds a layer of feedback, which increases the complexity of appropriate response. It is easier to unbalance the Pattern. Remembering that the Pattern leads inexorably towards the Root under Normal Circumstances, we value maintaining the Pattern as one of the first virtues. Because of the stress of additional dimensions on the Pattern, we mentioned earlier in this paper that it would be better to handle two dimensions well rather than multiple dimensions poorly. This is under the assumption that being on the Path is everything.
It would seem then the best strategy is to have as few dimensions as possible in order to do it well. This is true if we were playing a Game. But each of us must choose the number of dimensions that we can handle in order to test our processing system and not get bored. Thus our Dynamic is naturally led to the edge of Chaos by inner mechanisms, which seek to challenge the System for spiritual and emotional growth. Thus limiting dimensions is a good strategy but is limited by the personal potentials of our Dynamic. The Boredom Principle is the limiting factor on the bottom of the polarity of increasing and reducing dimensions. Thus Dynamics increase dimensionality as a natural process to stimulate growth and transformation. The goal is to increase dimensionality to the capacity of the Dynamic to handle it. Too much dimensionality leads to imbalance, while not enough dimensionality can lead to stagnation with its own problems. But this is a separate topic and dynamic from the thrust of this paper. This dynamic is in no ways inherent to Root Beings, and is only mentioned to put things into perspective.
Thus first our Dynamic is attempting to Balance her Pattern in order to maintain all of her layers of Dimensional Feedback. As our Dynamic exists through Time different Circumstances are fed in from the outside, rather than being Fixed as with our Statics. If the Events that are used in Decision making are properly ordered then our Dynamic always stays on the Path. However if our Dynamic relies too heavily upon Past or Present events then she will dissipate or bifurcate under Exceptional Circumstances. This will tweak her feedback pattern throwing her seriously off the Path. Hence the second task of our Dynamic is to maintain a direct and continuous connection between Present and Past events.
Let us turn this into a human example. An individual experiences extreme emotional pain, throwing them out of the Present to avoid this Pain. They begin living in the distant Past to avoid the troubles of the Present. This works well under Normal Circumstances but is inappropriate for Exceptional Circumstances. If the Events arenÕt continuous then our Dynamic is not quick enough to respond to extreme events. If the Events chosen for decisions are too much in the Present, the Now, then the Dynamic moves too quickly, overreacting to LifeÕs Circumstances. Thus our Dynamic must be constantly aware of an unbalanced approach to Time.
In summary our Dynamics have only two tasks. The first task is to continually Balance their Dimensions through the appropriate Pattern. The second task is to Balance their response to Time and History. If they keep these two elements Balanced, i.e. Dimensionality and Historical Time, then they are automatically on the Path. Even in the face of apparent setback or enlightenment, the progress is steady towards the Limit or Destiny. Thus our Dynamics arenÕt doing anything to move towards their Destiny except focusing upon Balance. The forward motion occurs naturally with Duration if the elements of Time and Dimension are balanced.
In grand summary, after reading the History of Mathematics about 6 years ago in 1993 I was introduced to infinite repeating fractions. These were used around NewtonÕs time in the 1600s to approximate first of all square roots and then eventually transcendental numbers such as ¹. In attempting to understand these equations, just what they meant, more specifically how to evaluate their decimal value, we discovered the Denominator Series for square roots, which had some very interesting properties. We wrote two Notebooks, one regarding the Denominator Series for square roots and the second concerning the fractal representation of ¹. These papers derived and explored in some depth the properties of these Series. While there were many unanswered questions our energy was spent and we stopped the exploration.
Years later, in 1999, Jimmy York discovered us here in Santa Barbara. He gave our scientific work validation and respect from the highest scientific levels. This validation generated confidence and a scientific focus. In reviewing the two aforementioned Notebooks, we became interested in finding the F&D Series for the Higher Roots, starting, of course, with the cube root. After diligent investigation, not only was F&D Series discovered for the Cube Root, but also for fourth Root and General Root. We were then able to binomialize these equations for both the Fraction and Denominator Series. We wrote up these results in the third Notebook of this Series called appropriately enough F&D Series for the Higher Roots of Rational Numbers.
In the process of exploration a whole series of bogus equations, which we labeled Ôstressed outÕ, were discovered which had the peculiar property of working sometimes but not all the time. The exploration of these bogus equations combined with the extension of the implications of our binomial Root Equations led to this Notebook, named Root Beings.
While these papers answer many questions, they bring up many more. Following are some of the questions that we would like to see resolved. While our Beings approach the Root erratically on local levels, their approach is distinctly linear on global levels. What is the equation for this logarithmic line that our Root Beings follow to their Limit? The second question is where do the Root Beings go when they are not all real. In jumping from place to place, one gets the distinct notion that they are moving in and out of ordinary reality. We derived an unwritten equation for square roots that through it into the imaginary world for a spiraling route, but we couldnÕt for the higher roots. We would also like to explore the variations of Root Beings a little more thoroughly. We only looked briefly at one False Being, two Now beings, and 4 Past Beings. There are many variations and subtleties that could be explored. We would like to explore the boundaries of dissipations for the Past Beings with computer-generated models. Mathematically we never discovered the exact mechanisms that drove the Now based to bifurcation or the Past Beings to dissipation.
While there are myriad directions to pursue, we have spent our mathematical energies for the time being and are moving back into Chinese History, we think, for the time being at least. If that is nebulous, it is meant to be. In the midst of extreme quietude the little voice speaks and we try to listen in order to align ourselves with the Tao, the Will of Heaven, whatever. Sometimes the little voice says to do some strange things.