Home Science Page Data Stream Momentum Directionals Root Beings The Experiment

Before leaving the Binomial Expansion we must show another bizarre manifestation. This time we will look at the underlying elements of the number of elements in an Unraveling. The two elemental units represent two halves of an ordered pair. One element represents the first half of the ordered pair, the second element represents the other half of the ordered pair. The exponent of the units yields the numbers of the ordered pairs. When the two elements are multiplied together they yield an ordered pair. If there is only one element, it is assumed that the other element has an exponent of zero. When the ordered pair is placed in this function, ¶, it determines the number of elements in Q unravelings with N elements, or vice versa as we showed before.

Below are some examples of the combination of these odd partners. It shows how the coefficients of the Binomial expansion are the number of elements in the corresponding ordered pair.

Using the notation above yields the concise, economical, beautiful results listed below. This kind of thing turns us on – sends chills down our spines, bows our head before the glory and majesty of it all. Thank you, Mother Nature, for lifting your skirt for us. We drink your sweet nectar and call it good.

Again this was just shown as another weird system generated by parallel systems analysis based upon the Binomial Expansion, Pascal's Triangle. For those engineers among you, this has not yielded anything new. For the mathematicians among you, we hope that it thrills you with another way of looking at the Binomial Expansion.

Before continuing, let us get a few things straight, curved or fractal, fuzzy or fluffy. In the blind moment of inspiration we rushed through to a conclusion, which confirmed many theories and led to some unusual concepts. But the exact steps are unclear in our many minds. So the following and preceding are an attempt to put some shaky formalism upon some flashes of insight. The insights have been confirmed on many levels, but no formal proof is offered here, only an insight into the reasoning that went into the conclusions that were confirmed experimentally on multiple levels. Hence the preceding and following are based upon intuitive pattern recognition that is tied to experimental result. We leave the task of tightening up the proofs to better mathematicians. We are only trying to roughly show the paths that were taken. If the symbolic approach gives the appearance of an attempt at formalism, it is only an illusion, a fractional attempt, for we are only fractional scientists, traveling from many dimensions, attempting to be real on your plane.

We have, however, tied our derivations to concrete results of two kinds. First, our equations fit the patterns generated on the lower levels that they were based upon. This verification covers only a small finite number of cases and is circular in nature. Second, our infinite series, for both Ravelings and Directionals add up to one, as they should. Not every infinite series adds up to one, totality. These series are complex and diverse. Finally our equations have ended up beautifully. Terms have magically dropped out and reappeared. There is an economy of result that feels good, a reflection of our experience of existence.

"Yaaahhhhh!!?" we hear the dimensional scientists scream.

"Is there something wrong?"

"Emotions and science don't mix!" is their frantic response.

"Ah, but the art, the beauty of its simplicity, in which is contained so much."

"Hasn't been proved, beyond a shadow of a doubt."

"We've seen enough and must move on. We've got the tiger by the tail, you know, and can't let go."

"Let go. Come back to Dimensionality."

"Sorry, we've gone Fractal and can't come back."

The correctness of the results is attested to experimentally and artistically, we'll leave it at that.

In binomializing the Seed Equation, we found that Raveled numbers behave like positive integers. When they are multiplied externally in the binomial expansion, they are added internally, similar to logarithms but not. Looking closely at a Raveled number, we find a number that is constructed of all the data of the Stream and more. A Raveled number contains traces of all that went before and then some. R= 0 is Now, XN. R= 1 is one step into the past, XN-1. Therefore the larger R grows the further into the past one travels. One finds that when R reaches N, the number of elements in the Data Stream, that the total potential impact has not yet been reached. All the elements of the Data Stream after beginning existence are not enough?! Right?! So to achieve totality one must extend infinitely into the void before existence began. So Raveled numbers are derivatives based upon all the elements of a Data Stream back to the beginning of non-existence. Practically speaking the potential impact of non-existence fades out rapidly as existence fades in. But theoretically the potential impact of non-existence never disappears completely. Furthermore the greater the number of Ravelings the greater the potential impact of non-existence. {See Potential Impact Graphs}

These weird numbers, which extend from existence back into non-existence, are also dimensional in nature. When Raveled Numbers are combined binomially, they add Dimensions, Q, and degrees of Separation, R, simultaneously. Q and R are so integrally linked that they turn Raveled numbers into unitary numbers. Raveled numbers when multiplied externally, add internally. When Raveled numbers are taken to a power externally, it is like multiplying them internally by the power. However Raveled numbers cannot be added externally without quite a bit of algebraic mumbo jumbo.

Below is an expanded notation for the square terminology. The exponent represents which X is the Now. The zero exponent means that XN is Now. An exponent of one means that XN-1 is Now. An exponent of minus one means that XN+1 is Now. We are sticking with the R style terminology, i.e. R=0 is Now, while R=1 is one step into the past and R=-1 is one step into the future.

The same pattern is followed with one-square, below.

Using this expanded square notation we come up with the following equation.

The following is the equivalent equation in the circle notation. Do you remember this form?

This is the Decaying Average formula that started this whole mess off. Maybe it might be more recognizable in the form below.

Below is the general formula for combining different planes of Now.

Again we write the same equivalent formula in circle notation.

Note that while the circle notation is easier to read than the ordered squares, it reveals little of the underlying organization. All the Ravelings are well defined in the circle notation, but one doesn't get the sense of a line of raveling, inherent to the square notation. Basically the square notation links R and Q, integrally, inseparably. The circle notation breaks the two apart. This is indeed valid. But in each Directional the only Ravelings that are used are ones where the two variables, R & Q, are inextricably linked. This is what the square notation reveals. The only way of moving thru Directional lines of Raveled numbers is through the algebraic contortions shown above.

Below are some Raveling and Directional lines. Any Directional associated with the Now, the XN's along the borders, consists only of Ravelings along the indicated lines.

The Raveled Numbers that make up a Directional are all in a single line. Also each Raveled Number is contained in only one line of Directionals. The only way of jumping Directional or Raveling planes is through the Decaying Average formula above. The internal addition and multiplication of number-squares only occurs on these lines of Ravelings. These lines of Ravelings only interact with each other but not with the Ravelings on other lines. Thus the exponent of the squares reveals which line of Ravelings that the particular Raveling exists on, i.e., it reveals which XN is Now. When Raveled Numbers are combined binomially, they add Dimensions, Q, and degrees of Separation, R, simultaneously. Q and R are so integrally linked that only a single line of Ravelings goes into making each Directional. Each individual Raveling connects up with one and only one Directional. This is shown in the diagram above.

Another feature of Raveled numbers is that they are unitary numbers. What does it mean to be unitary? Our Source, which seems to move through Time, is also unitary. Unitary!? Anything that cannot be fractionalized is unitary. Humans are unitary; Beings are unitary; Ravelings are unitary. They cannot be fractionalized, no matter what. A being is a being, no matter how big or small. In common thought, an animal becomes unitary when it leaves the womb to operate separately from its Source, its Mother. Sometimes humans don't treat themselves in a unitary fashion. - Send the collective peasants off to war, to fight for the unitary wealthy - However in essence after the baby leaves the womb it is now unitary. As the abortion rights, animal rights, and the people rights movements testify to, there is a greater tendency to treat each creature as unitary. Every Being is one, not more, not less.

Plants are not necessarily unitary. Trees tend to be unitary with only one root system, but not necessarily. The Mandrake tree can create its own forest. Humans do not treat plants in a unitary way. We mow the lawn, cut down the forest. Running through the grass. But animals big and small are unitary.

Humans cannot be broken into fractional creatures. There are no 2/3 humans or 3/2 humans, at least where I come from. Being mobile has to do with being unitary. Being able to move through and being somewhat separate from the environment, conveys a unitary quality. Some plants are individually transplanted, but grass is treated as a group. Theoretically a blade of grass could exist independently, while practically speaking grass exists as a group of plants. Lawns are unitary. They cannot be fractionalized. There are big lawns and small lawns, but no ½ lawns. The concept of ÔlawnÕ is unitary. Any concept that cannot be fractionalized is unitary. Piles are unitary. Forests are unitary. Flocks are unitary. Rocks are even unitary. There is no such thing as half a rock.

Pies and cakes are not unitary. Anything that can be fractionalized is not unitary. Loaves are not unitary. The elements of the complex number line are not unitary. We can take half of any number. While numbers are not unitary, every measure is unitary. We can't take half an average. We can take an average of half the set, one quarter, or one third of the members of the set, but never half an average or half a Deviation. A measure can be made of unitary parts, but is still unitary itself. A Z score or The Impact is made of an Average, a Deviation, and a new Data byte, each of which is unitary. Numbers can be fractionalized while measures can't. Both/And & Either/Or logic applies to unitary things, while gray, continuous logic applies to fractional things.

Raveled numbers, as mentioned, are also unitary numbers. Each Raveled number that goes into making a Directional has the exact same potential impact. Each Raveled number is a solitary unit with the same potential. This is not true of many equations of the traditional sort. For instance, in the equation, Y2 + X = Z, X and Y have a very different and unequal impact upon Z. Small changes in Y have a greater effect upon Z than small changes in X. It also depends upon what X and Y are as to their potential impact upon Z. If Y is very large then X must even be larger to have a significant potential impact on Z. Each level of Raveling has the same level of potential impact.