Generating the Simple Bi-directional Wave

(1) The Simple Forward Wave                                                           [skip to next]

The process by which the 384 number data set is generated begins with the King Wen sequence of I-Ching hexagrams (a listing of which appears in the Time Explorer manual, pp. 58-59), which is believed to be the earliest arrangement of hexagrams. McKenna chose to examine [4] the number of lines that change (yin to yang, and yang to yin) as one moves from hexagram to hexagram, beginning at hexagram 1 and proceeding to hexagram 64, and he called this quantity the First Order of Difference (FOD). The FOD number set that is generated as one moves from hexagram 1 to hexagram 64 contains 63 elements; a 64th element is determined by recording the FOD as one moves from hexagram 64 "wrapping" back to hexagram 1, thus establishing a closed system with periodic waveform. This FOD number set can be computed mathematically by treating each hexagram as a binary number as reported by Meyer [5]  but in this case I simply recorded each number manually in DeltaGraph [6] and Excel [7] spreadsheets.

The FOD number set, which I will now call the Simple Forward Wave [8] , is graphed in Fig. 1 with straight line segments connecting the individual FOD data points. The x-axis of this graph shows the hexagram transition number, where transition n is defined as the transition from hexagram n to hexagram n+1; transition n=0 is simply an x-axis wrap of transition 64, and is thus defined as the transition from hexagram 64 to hexagram 1. The inclusion of the zero transition data point is a way of graphically illustrating the "wrap-around" nature of this number set, or possibly a way of mapping a 3-dimemsional cylindrical surface onto a 2-dimensional plane. For clarity, let us define this feature of the FOD number set:

The y-axis values shown in Fig. 1 are the actual FOD transition values, integers that would normally be shown as points along the transition axis. In Fig. 1, however, these points are connected by straight line segments, which establishes the piecewise linear nature of this number set, generating non-integer values and creating a general function that is defined at every point in its domain (all possible x values in the domain ). Generating this function requires the acceptance of a general principle, which will now be defined for clarity: This FOD function is viewed as having a forward flowing, or +x directed nature, and it is the basic or simplest number set in the TimeWave development process. Thus it is called the simple forward flowing wave, or just the Simple Forward Wave.


Figure 1


(2) The Simple Reverse Wave

In order to clarify the process of simple bi-directional wave generation, and the production of the Simple Reverse Wave, let us first define another general principle:

This is an important statement, for without it there is neither reason nor unambiguous path for the construction of the bi-directional wave function, nor is the proper form of wave closure obvious. Once this principle has been established, however, it is then possible to proceed with a step-by-step process of reverse wave, and bi-directional wave generation. Figures 2a-2f illustrate this process of generating the Simple Reverse Wave, followed by a closure with the Simple Forward Wave to form the Simple Bi-directional Wave. Fig. 2a shows Step 1 in the process of Simple Reverse Wave generation - a 180° rotation of the Simple Forward Wave about the x, y axes origin (0,0).

This rotation operation can be visualized by observing that the Simple Forward Wave, shown in quadrant I (upper right hand corner of Fig. 2a) is fixed relative to the x, y axes (red lines). The axes are then spun counter-clockwise 180° around their origin (intersection point), carrying the wave with them. The mathematical formulas for this rotation are expressed as:


Where:  is the rotated  value

is the rotated  value

is the angle of rotation in degrees

With 180° as the rotation angle, these equations reduce to:



Equation [3] and [4] show that this 180° rotation operation results in a simple sign change of the original forward wave x, y pair data set. The rotation places the developing reverse wave in quadrant III of the graph, shown as the solid blue line-plot. The dotted blue line-plot shows the position of the parent Simple Forward Wave.



Step 2 of the reverse wave generation process is shown in Fig. 2b, and involves the translation of the rotated forward wave in the  direction. This operation is expressed by the following translation equation:


Where:  is the translated value of  of equation [3]

is the magnitude of the translation in the  direction

Since this translation must x-align the endpoints of the forward and reverse waves, the magnitude of the translation, , is +64. This positions the developing reverse wave in quadrant IV as shown in Fig. 2b.



Figure 2b


Fig. 2c shows Step 3 of the reverse wave generation process, and is defined as the  translation of the x-translated wave of Fig. 2b. This translation is performed so that the forward and reverse waves will be in position to achieve closure at the Index 1 and Indices 62, 63, 64 endpoints as specified by definition 3, once the next and final step is performed. The translation equation for this step of the process is express as follows:

Where:  is the translated value of  as expressed in equation [4]

is the magnitude of the translation required to position the reverse wave for proper closure with the forward wave

In this case the  positioning for proper wave closure requires a  value of +9. Fig. 2c shows the reverse wave position that results from this translation, and also shows that the forward and reverse waves are offset, and have not yet achieved endpoint closure. The next and final step is performed using a different type of mathematical operation called the "shift", which can be understood by using the following analogue:

Figure 2c


This final step, the -1 -shift, is shown in Fig 2d, where the dotted blue line-plot is the pre-shifted reverse wave position, and the solid blue line-plot is the -1 x-shifted reverse wave position. The larger plot at the top shows the shift operation for the overall wave pair, whereas the two smaller plots at the bottom of Fig. 2d are magnifications showing the closure process at the beginning and end sections of the wave pair. The mathematics for this operation can be expressed as a two step process as follows:



Where  is defined:



such that:  is the y value of the -1 x-shifted wave at x

and:  is the y value of the pre-shifted reverse wave at x

There are two features of Fig. 2d that should be noted here. First notice that in the small graphs at the bottom, closure between the forward and reverse waves occurs at four transition axis points (excluding zero). These points are x = 1, x = 62, x = 63, and x = 64, so that wave closure occurs at one initial point (x = 1) and three terminal points. Point zero is excluded since it is simply a "wrap", or duplicate, of point 64 and will eventually be discarded. Secondly, the two smaller graphs at the bottom of Fig. 2d show the process of endpoint shift, or transferring the "marble/line segment" that was initially in slot 0 into the vacated end slot 64. The green arrow line runs from line segment 1 in the graph at the left, to line segment 64 in the graph at the right, and shows that segment 1 is being transferred to segment 64 as the -1 x-shift is performed. The figure shows that this is not a simple translation operation as in the previous two steps, but a definite shift - much like the operation of a shift register in digital electronics. Note that if a simple -x translation were performed, line segment 1 would be translated into the negative x -domain to the left of the y-axis, and there would be no line segment 64.

With the performance of the -1 x-shift operation, the production of the Simple Bi-directional Wave is now complete. We have thus created a forward and reverse flowing waveform, which is closed at either end, something like nodes on a standing wave. Although this is the correct procedural process for generating the reverse wave from the forward wave, and for producing endpoint closure, the relationship between forward and reverse waves can be expressed simply by the following equations:



Where  is defined:



and where:  is the y value of the reverse wave at x

is the y value of the forward wave at (63 -x)



Figure 2d


Equations [9] and [10] are good examples of mathematics that do the job, but fail to give one a visual image or sense of what is really going on in the process. This type of math is actually quite useful, nonetheless, for computer generation of the reverse wave number set.

We have thus created a simple bi-directional waveform, having the properties of directed flow and endpoint closure, and which can be characterized as a piecewise linear function - a function we have yet to define over its non-integer domain. That will be our next step in the formalized development of the TimeWave data set.

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