Number Theory

Consider the following chart:

On the left we have the column of sequential integers. Let this column represent, say, time, or, say, initial members of an organisation. The column on the right represents the rate of expansion over time, or per each different original group member. The left column reads up and down, and the right column reads side to side. Here is how to read them: begin with square one, expansion rate two. Next in sequence would be to start with two, and measure the expansion rate four, etc. The chart goes up to integer five, rate of expansion thirty two, but of course you can see how applying this method works.

The three middle columns show some relationships we can draw across the board between sequential integers and expansion rates. 1+1=2, 2+2=4 and 3+3=6 represent the initial integers and expansion rates. The difference between each initial integer and the next across is the corresponding row on the left column. The difference between each integer sum and the next across corresponds to the expansion rate column on the right.

Basically to read across a row, you count the difference between the sums by the factor to the right. Thus: (2,4,6)=2; (32,64,96)=32. The expansion rate in the right column is the same as that for the 1+1=2 column because this entire table represents only multiples of two. 1+1=2 and 2+2=4 establish the base two system as a root function. The rest of the chart is based on multiplying each sum by two.

So, in short, this chart represents the corner-stone of the Base-Two system. There are similar charts that can be constructed for all the multiples, and it will show that, for each multiplicative step by its base number system (3,4,5,6... etc.) it will increase in the same fashion: As the sequence of integers expands "arithmetically," the column on the right, the expanson rate, will increase "exponentially."

on the tables of the eighth and the ninth

The tables of the eighth and the ninth tell us many things about the physical construction of our universe on a purely mathematical level, the level which serves as the bridge between the external formation of the material world and our relatively internal domain of consciousness. My arrangement of these tables therefore allows for the application of numerology, or the realization of relationships between the digits en-soi, or in themselves. This existentialist approach to computation is as ancient as the human practice of collecting objects in countable sets, and constitutes an esoteric equivalency to the exoteric fact of pure traditional mathematics. Regardless of the excuse of its origins, this practiceąs credentials are frowned upon in the light of pure mathematics as an escape from the understanding provided by methodological calculation. They are, quite to the contrary, no more than a reapplication of methodological computation on an entirely other level, self-contained and non-threatening to the approach of exoteric mathematica. Thus, while their examination may be seen as untraditional, it is at least not unacceptable.

One of the fundamental insights of the tables is provided by comparison between the two. Their primary similarity is the repetition of pattern; their primary difference being the nature of these patterns. As the eighth demonstrates quantifiable decline in sequence, so the ninth yields first exact self-replication, and then increasing self-referential sequentialism. These patterns are apparent and undeniable. If the calculations are repeated in any other setting the conclusions will be exactly the same, and therefore the patterns displayed in their relationships will be identical. Numbers do not lie. They lack that motivation for symmetry.

It is possible to see these sequences of both as related to one another dimensionally. The perpetual numerological decline of the eighth and the perpetual numerological duplication and factorial increase of the ninth may be seen as ascent through the first three dimensions respectively.

The diminishing table of eights constitutes collapse into a singular point in space. Naturally one can question how there can be any differentiation at all in the measurement of a point, but the answer is a simple one indeed, for without it there would be no scale-correspondent maps.

The repetitive aspect in the nines resembles the extension of a line in space. At all points along the line two of its dimensions are canceled, leaving only the third behind to mark its position. This is quite obvious in the graphing of a straight vertical or horizontal line in a two dimensional Cartesian coordinate system, where either the x or the y coordinate pairing remains undefined. It may be less obvious in a diagonal, where every point on the line has a defined x and y coordinate pair that differs from every other coordinate pairing of a point along the line. However the distance formula shows that any two points on a graphed straight diagonal will cancel one another out leaving only one integer behind, that being the value of the line itself. This may seem trivial now, but it is essential for understanding the next comparison of the ninth table to dimensionality. One need only to consider that a line in a two dimensional coordinate system, straight or diagonal, is equivalent to a plane in a three dimensional coordinate system to begin to apprehend why.

Although it is a confirmable fact that the integer sums of the multiplicative quantities produce a doubling of results along factorially related lines throughout the entire ninth table, this only becomes really evident after the process has entered its third repetition, when the quantities involved are of such an amount that their sums render divergent factors. What this process is in fact describing is the event of entrance into the third dimension from the second. At first, as the plane is defined as two lines of value nine, the factorial sums elevate rapidly through the established sequence. As the shape becomes clearer while the coordinate system is rotated the number of quantities between the factorial transitions becomes greater, allowing, as it were, more time to pass between phase shifts. The key to understanding the shape that is described is contained within the non-numerological pattern of table nine.

The factorials diverge first after product 180. That is, their sums if taken by pure integer alone or by combined integer render different results, although both results that occur elsewhere as quantities within the initial multiplicative table. At quantity 108 the products break from the ascending sequence and linger at sum 18, or, if they are taken as a pair of paired integers, begin to decrease along the same factorial lines, beginning at 63, but skipping every other factor, such that the next result yielded is 45, rather than 54, etc. This is the case until 180 appears again as the multiplicative function. The result for 198, that immediately following 180, is peculiar, as it constitutes a different rate of change than has been previously established. Although it describes the results for all the quantities of multiplication between 181 and 240 (interrupted only by 200 * 9 = 1800), this still describes a much shorter set of numbers than are contained within the 36 factorial grouping from 241 to 390 which follows, 58 as opposed to 148 respectively. This anomaly also helps to point out that the phase shifts in sums donąt occur cleanly at factorially defined breaks, but are governed only by the dictates of the digits themselves. Although this opens up the realization of still another, more subtle pattern ‹ the difference between the clean factorial 243 and the true break of 240 for the 27 factorial set being 3, and the difference between the clean factorial 396 and the true break of 390 for the 36 factorial set being 6 ‹ it is unnecessary at this point to go into it in detail.

It is more instructional, from a dimensional emergence perspective, to examine the factorial breaks that are clean, and those are 180 and 360. These numbers are most immediately recognizable as the definitions of the circumference in degrees of a half circle (the angle measure of a straight line) and a full circle (or, if you like, the angle measure of a straight line that reverses its own direction). But they have more, deeper connotations than this.

The most fundamental polygons are the triangle, the square, and the pentagon, composed of 3, 4, and 5 angles respectively. The hexagon, with 6 interior angles, is somewhat more complex, and can actually be tallied to be the sum of 2 triangles. The sum of the angles of a triangle is 180. It is always, exactly, and only 180. The sum of the angles of a square is always 360 (90 + 90 + 90 + 90). This is true for any 4 sided rhombus, according to the formula that (n - 2)180 = sum of the interior angles for any object with n number of sides. Applying this same formula we can conclude that the sum of the angles of a pentagon is 540, or 3 * 180, 1 & 1/2 times around the circumference of the unit circle. These numbers are as old as the act of measurement itself, and they are absolute. So we can see how the clean fractional breaks in the ninth table pertain not only to angles that describe arcs, but to those describing well defined shapes, particularly the fundamental polygons, as well.

Moreover these fundamental polygons comprise the sides of the only five regular solid polygons that can exist in three dimensions. All other solid polygons, like the hexagon in two dimensions, are only combinations of these first five. The Platonic solids consist of the tetrahedron, comprised of triangles, the cube, composed of squares, the dodecahedron, comprised also of triangles, the icosahedron, also composed of triangles, and the dodecahedron, comprised of pentagons. So we have not only the description of angle measures on the unit circle, but also the sums of the angles of the faces on each of the five platonic solids, and all in a pure numerical form. It is easy to follow the progression of factorial breaks up through their ascending sequence and see how the sums constitute measures within a deepening three dimensional space, describing the unfolding of the platonic solids according to the sums of the interior angle measures of their faces multiplied by their facial sum.

It would be more probable, at this point, for a practitioner of calculatory mathematica to caution that the bad habits of a writer may be transferred to their readers, than it would be that they could provide hard evidence that this data is more conjecture than implicit organization. In either event it cannot be doubted that the dimensional bridge between function and form is crossed in the table of nines.

The bridge described herein is not altogether complicated, but is increasingly complex as more governing rules are discovered to determine each additional dimension. For example, the point may be seen as of any size under magnification; the line as definitive of angle, and implying the edge of a perfectly flat plane. Three dimensional objects are formed simultaneously of matter and energy ‹ comprised not only of charged particles, but of waves with measurable frequency related to this charge. This relativity of fundamental components and distance becomes substantial in the fourth dimension, the final to be considered here, where an object has form in both space and time, according to the measurement of intervals. All of these are descriptive of the same process, differing only in the complexity of dimension.

To understand the relationship between the table of the nines and the fourth dimension, it is necessary to lay a minimal foundation. Some formulae and models describing progression should be given, as it is by progression that time is measured. It is this process that has thus far been described, and which constitutes our bridge. Growth in biological organisms is measurable according to an exponential law for equiangular spirals that gradually approach ratio based orientation. An example of this is that governing the formation of a nautilus shell, given as r = ae kq , where r is the radius of curvature, a the area, e the natural number (2.71) found in exponential balances, k the kinetic energy, and q the polar angle of predicted curve continuation. Although it seems completely unfamiliar, this formula has been underlying our progress all along. It describes the function of the ninth table whereby the lower dimensions are evoked rapidly, and the forms of the higher dimensions more slowly. What this formula means is that the spiral of the nautilus shell is curled more tightly around its origin, rapidly forming a circularly bound core. As it continues, the arc of the shell expands to break its circular condition, and rate of growth slows. This is the same exact process as described in the numbers of the ninth table.

Now, what does this similarity have to do with the fourth spatial dimension? If the rate of growth of either a nautilus shell or the ninth tableąs factorial breaks were plotted as a sine wave, it would have high initial frequency and minimal oscillation, followed by lower gradual frequency and more pronounced oscillation. This chart describes the unfoldment of progression for either equally well. It is, itself, also a three dimensional shape. If it were plotted in a three coordinate system the sine wave would orbit about the x-axis, alternating positive and negative in the y-axis, with an expanding radius in the z-axis. If it were displayed with its compliment as well, numerically negative products of all the multiplicatives or geometrically the cosine wave, it would take the familiar form of the double helix of DNA.

Before we go into the implications of the gnomon described in the ninth table, let us first pause briefly and consider the construction of a sinusoidal wave. In trigonometry, the study of triangles, lies the basis of wave measurement. A wavelength, l, may be measured as two right triangles extending up to the endpoints from the base line of the standing wave, sharing either a common point, where the wave form crosses the base line, or the vertical leg, where the wave form reaches its peak or trough. These triangles are equivalent for a sine wave depicting circularly bounded progression, and isosceles ‹ possessing two equal length legs apiece. The method of creating the sine wave itself comes from a technique for deriving a right triangleąs non-right angle measurements from the measures of its legs and hypotenuse. The sine function for an angle is the opposite leg over the hypotenuse; the cosine function for the same angle being the adjacent leg divided by the hypotenuse. These two functions, as it was stated before, are complimentary. This is expressed in the relationships between sine and cosine for a triangle ABC whose right angle is C: sin A = cos B and cos A = sin B. When the numerical solutions of these functions are graphed, a sine wave appears, which is a spiral in three dimensions. As a measurement of interval, this is also a fourth dimensional shape.

Now we may consider the progress through dimensions of the described spiral as it breaks from circular boundary conditions and what form it takes after it does so. The first point that should be made is the distinction between degrees and radians. Both are measurements of the arc of a circle, or the radial expansion of a spiral bound by circular or exponential conditions, but the former is fixed and the latter is open. Radians, unlike degrees, are dependent upon the transcendental measurement of pi, an irrational integer whose value is roughly 22/7 or 3.1415926. Pi, or p, is an expression for the ratio of the circumference of a circle to its diameter. The relationship between degrees and radians is such that each pradian = 180°. It is interesting to note that a spiral column of tetrahedrons, such as Buckminster Fuller modeled for the geometry of a double helix, undergoes 1/3 full rotation while 22 of its faces are exposed. Thus p is the limiting factor for the spiral in its early stages of progression, as it crosses the threshold of the first two dimensions. As it enters into the third, the forms it describes become exponentially complex, and the time interval elapsed between factorial breaks therefore begins to widen.

The limiting factor remains a transcendental number, an irrational integer similar to p, but one that describes an open set for growth along dimensional lines. This integer is phi, f, and will be shown to occur in pure mathematics, quantum mechanics, and the helialical pattern of DNA. It is this number that governs the third dimensional forms derived from the ninth table, as well as the fourth dimensional pattern of their progress. In order to decant f from the table of nines we need only convert the products given to degree measurements, a process already implied by the clean factorial breaks occurring at the 180 and 360 multiplicatives. Once this is done a chart for the sine and cosine functions of these angles may be assembled, and, using larger, later occurring sums for these, f will be revealed. The (2 sin)2 and (2 cos)2 sums are preferred, and yield results containing both phi and phi prime, f1, that is the reciprocal of phi, or f/1. Some of the most notable results of this table are, for 9° or p/20 radians: 2 - *(f + 2), 2 + *(f +2), sin and cosine functions respectively; for 18°, p/10: f1 + 1, f + 2; for 45°, p/4: f + f1 for both; for 72°, 2p/5: f + 2, f1 + 1; and for 81°, 9p/20: 2 + *(f + 2), 2 - *(f + 2). The complimentarity of the sin and cosine functions is readily apparent here. These figures represent distance relationships between points on what is called the Golden Triangle of 36° by 72° by 72°, after f, which is itself the Golden Ratio of (1 + *5)/2, with approximated value of 1.61803. Some other properties of note possessed by f are that multiplied by its reciprocal, the value of which is -0.61803, its product is -1, and that squared it is equal to itself plus 1. The spiral described by f is thus the so-called Golden Spiral, and is exponentially bounded rather than circularly, its points being f, f2, f3, f4, f5, and so on. These constitute the bridging of dimensional gaps described by the tables of eights and nines, and the slower, more numerous three dimensional forms that arise by angle sum recurrence during the later stages. It is interesting to note that a line connecting two points on opposite sides of a third point in a pentagon forms a line that, intersected by another such line drawn from the third point, forms the Golden division. Thus, not only is f determinant of the rate of third and fourth dimensional progression, but a fundamental building block in the Platonic forms themselves.

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