| WEN_GRPH: Software for Generating Sets of 384 Numbers | |
|---|---|
| by Peter Meyer | |
Preface
In early November 1997 McKenna announced a "bombshell" on his web site: a claim that mathematician John Sheliak had succeeded in formalizing mathematically, using vector algebra, precisely this construction, so that the 384 numbers could be generated by a single (very complex) mathematical formula (see John Sheliak's "Mathematics of the TimeWave"). Matthew Watkins had done the same thing, but this time it was claimed that the process of the construction could be expressed from beginning to end in terms of vector algebra.
As a by-product of this work it was announced that the set of 384 numbers given in Appendix II in the original edition of The Invisible Landscape had embodied an error, namely, it had incorporated a step for which there was no justification (as Matthew Watkins himself had pointed out quite clearly). This was the "half-twist", whose existence I had revealed in the original version of the documentation (published in 1994) in a footnote in the DOS software manual.
The DOS Timewave Zero software was designed to accept any set of 384 numbers (satisfying certain minimal conditions) and to generate "a" timewave from them, via the process known as "fractal transformation". Thus it would take this "revised" set of numbers as easily as the "standard" or "original" set, and generate a timewave. This timewave, the "revised" timewave, was now said to be the "correct" timewave (and it was claimed that the revised timewave had a much better fit with history).
In late November 1997 I received indirectly from John Sheliak the revised set of numbers, and was asked to check them. I took the occasion to modify the WEN_GRPH program so that it would, upon command, include or exclude the "half-twist" from the construction. This program is included with the Fractal Time software, obtainable by purchase of a user license for the latter (see below).
So I ran the modified WEN_GRPH program on the King Wen Sequence, with the half-twist excluded from the construction, to generate the "alternate" set of 384 numbers. Since, John Sheliak had claimed, this half-twist had no place in the formalization using vector algebra, it had been discarded. A new set of 384 numbers resulted from Sheliak's work, the Sheliak numbers. These numbers are not the same as the alternate set of 384 numbers, those which result from the omission of the half-twist. Thus the construction formalized by John Sheliak seems to have involved more than the omission of the half-twist.
More about the various sets of 384 numbers which can be used to generate a timewave will be said in the section below entitled The Huang Ti Numbers.
Introduction
The process whereby the (or a) timewave is derived from the King Wen Sequence is a 2-step derivation: (i) The King Wen Sequence is the starting point for the derivation of a set of 384 numbers. (ii) These 384 numbers become input to the mathematical formula for the definition of the fractal timewave, and for the computational generation of the graphical timewave.The standard timewave is built upon the 384 numbers listed in the original edition of The Invisible Landscape. The Mathematical Definition of the Timewave states how the 384 numbers enter into the construction of each point of the timewave. In this article I shall explain, part (i) of the process, how the 384 numbers are derived from the King Wen Sequence. Actually I shall explain how a set of 384 numbers can be derived from any sequence of hexagrams satisfying certain conditions. Since the King Wen Sequence satisfies these conditions this will also constitute an explanation of how the 384 numbers of the standard timewave are obtained from that sequence. Different sequences of hexagrams generate different sets of 384 numbers.
The King Wen Sequence is discussed at length in
Part Two of The Invisible Landscape.[1]
The hexagrams in the sequence are numbered from 1 to 64, with Hexagram #1
consisting of six yang lines and Hexagram #2 consisting of six yin lines.
It is well known that hexagrams in the King Wen Sequence occur in pairs. The second member of each pair is [usually] obtained by inverting the first [i.e. by turning it upside-down]. In any sequence of the sixty-four hexagrams [however] there are eight hexagrams that remain unchanged when inverted. In the King Wen Sequence, these eight hexagrams are paired with hexagrams in which each line of the first hexagram has become its opposite (yang changed to yin and vice versa).[2]In other words, the King Wen Sequence is a sequence of 64 distinct hexagrams comprised of pairs of hexagrams such that the second hexagram in each pair is the inverse of the first if inversion produces a different hexagram, otherwise the second hexagram is the complement of the first. Two such hexagrams are said to be paired.
Each hexagram consists of six lines which are some arrangement of yin lines and yang lines. When any two adjacent hexagrams in a hexagram sequence are compared, the corresponding lines in the hexagrams will be found to be of the same polarity (yin & yin or yang & yang) or different (yin & yang or yang & yin). The number of lines of different polarity is a number between 1 and 6; this number is called the first order of difference for the two hexagrams, and a transition equal to that number is said to be made from one hexagram to the next in the sequence. For each sequence of hexagrams there is thus a sequence of 64 transitions, assuming that the sequence wraps around from Hexagram #64 to Hexagram #1. In the King Wen Sequence that there are no transitions of 5.
If we count the number of odd transitions and the number of even transitions we shall usually find that the ratio of even to odd is not an integer. In the King Wen Sequence it is, namely, 3.
We shall now define three increasingly strong concepts of a Wen sequence in terms of these properties and some additional ones, to be explained in the next section.
A Wen sequence of type 1 is a sequence of 64 distinct I Ching hexagrams, beginning with Hexagram #1 (six yang lines) and consisting of 32 pairs of paired hexagrams[3].
A Wen sequence of type 2 is a Wen sequence of type 1 with the following properties:
- There is no transition of 5.
- The ratio of even transitions to odd transitions is exactly 3.
- The sequence has a degree of closure of at least 3 which occurs with closure offset of 1.
- No internal segment overlap occurs.
A Wen sequence of type 3 is a Wen sequence of type 2 which has a degree of closure of exactly 3 which occurs with a closure sum of 9. The King Wen sequence is a Wen sequence of type 3.
The terms degree of closure, closure offset, closure sum and internal segment overlap are best explained by reference to the use of the WEN_GRPH program, to which I now turn.
Use of the WEN_GRPH program
The input to this program is a representation of a sequence of 64 hexagrams (this data is contained in a file). Each hexagram is represented by its position in the King Wen Sequence. Thus the King Wen Sequence itself is represented by the sequence of numbers:
|
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20 21, 22, 23, 24, 25, 26, 27, 27, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60 61, 62, 63, 64 |
A random sequence of hexagrams would begin, for example:
|
13, 5, 19, 34, 42, 10, 39, 62, 21, 9, 49, 11, 53, 38, 17, ... |
When WEN_GRPH is run with the command line parameter /h a reminder of usage is displayed:
WEN_GRPH.EXE, Version 2.1, Copyright 1994,1998 Peter Meyer Use: WEN_GRPH Z hexagram_sequence_file or: WEN_GRPH X hexagram_sequence_file or: WEN_GRPH Z or: WEN_GRPH X or: WEN_GRPH or: WEN_GRPH /h If no hexagram file is specified then the hexagram sequence file is assumed to be "KING_WEN.SEQ". Z means include the "half-twist", X means exclude it.
The first (optional) command line parameter tells the program whether to include the half-twist or not (Z = include, X = exclude).
The second (optional) command line parameter, hexagram_sequence_file, is the name of the file containing the hexagram sequence from which that the program will take its input. There is a file (contained in WEN_GRPH.ZIP) called KING_WEN.SEQ which contains the above representation of the King Wen Sequence. (A different file with a different hexagram sequence may also be used.)
When WEN_GRPH is run with some hexagram sequence it first checks whether the sequence represents a Wen sequence of type 1. If not, the program prints a message as to why the sequence was not of this type.
The program then displays the sequence with additional information. In the case of the King Wen Sequence the display is as follows:
Hexagrams and transition values:
1:111111 6> 2:000000 2> 3:010001 4> 4:100010 4> 5:010111
4> 6:111010 3>
7:000010 2> 8:010000 4> 9:110111 2> 10:111011 4> 11:000111
6> 12:111000 2>
13:111101 2> 14:101111 4> 15:000100 2> 16:001000 2> 17:011001
6> 18:100110 3>
19:000011 4> 20:110000 3> 21:101001 2> 22:100101 2> 23:100000
2> 24:000001 3>
25:111001 4> 26:100111 2> 27:100001 6> 28:011110 2> 29:010010
6> 30:101101 3>
31:011100 2> 32:001110 3> 33:111100 4> 34:001111 4> 35:101000
4> 36:000101 2>
37:110101 4> 38:101011 6> 39:010100 4> 40:001010 3> 41:100011
2> 42:110001 4>
43:011111 2> 44:111110 3> 45:011000 4> 46:000110 3> 47:011010
2> 48:010110 3>
49:011101 4> 50:101110 4> 51:001001 4> 52:100100 1> 53:110100
6> 54:001011 2>
55:001101 2> 56:101100 3> 57:110110 4> 58:011011 3> 59:110010
2> 60:010011 1>
61:110011 6> 62:001100 3> 63:010101 6> 64:101010 3>
closure = 3, closure_offset = 1, overlap = 3, closure_sum = 9
This should be read by row rather than by column. Each entry represents a hexagram and is of the form n:abcdef m>, where n is the hexagram number (the number of the hexagram's position in the King Wen Sequence), abcdef represents the polarity of the lines of the hexagram (where f is the bottom line and a the top, and 0 = yin and 1 = yang) and m is the transition number (a.k.a. first order of difference).
Figure 18B in The Invisible Landscape (p.131 of the 1975 edition) is:
The WEN_GRPH program displays the same pair of graphs, with a report on whether closure occurs. If you run the program with no command line parameters, i.e. WEN_GRPH only (so that the King Wen Sequence is used for input), the screen display should be as below:
There are two graphs and two sets of axes. The upper graph (mauve) is relative to the upper horizontal axis and the right vertical axis. The lower graph (green) is relative to the lower horizontal axis and the left vertical axis.
Both graphs represent the transitions associated with the hexagram sequence. The values on the horizontal axes are hexagram numbers and the values on the vertical axes are transition numbers. For example, in the King Wen Sequence six lines change from Hexagram #63 to Hexagram #64, three lines change from #64 to #1, six lines change from #1 to #2, two lines change from #2 to #3, and so on. The upper and the lower graphs convey the same information. In fact the lower graph (as initially displayed when WEN_GRPH.EXE acts on the data in KING_WEN.SEQ) may be obtained by rotating the upper graph through 180 degrees and rotating it right by one position.
The lower graph may be moved vertically by means of the up- and down-arrow keys, and it may be rotated left or right by means of the left- and right-arrow keys. As it says on the screen, closure occurs when the end-points of the two graphs coincide and there are one or more overlapping segments at the left or the right.
I can now explain the meaning of the terms degree of closure, etc., that were used in the previous section in the definitions of Wen sequences of type 2 and of type 3.[4]
The degree of closure is the number of segments which overlap at the left plus the number which overlap at the right, which is 2 + 1 = 3 in the case of the King Wen Sequence. (As you move the lower graph up or down, or rotate it left or right, the degree of closure is automatically calculated.)
The closure offset is the number of positions that the upper graph must be rotated to the right, after having been rotated through 180 degrees, in order to coincide with the lower graph (1 for the King Wen Sequence).
Closure sum is the sum of the left-most value of the lower graph and the right-most value of the upper graph. In the case of the King Wen Sequence this is 6 + 3 = 9.
Internal segment overlap occurs if there are segments of the two graphs which overlap but which are not among any overlapping segments occurring at the left or at the right.
There remains to be explained (for screens of this type) only the two rows of numbers displayed below the closure report.
The numbers in the upper row are the differences between the upper and lower graphs. Negative differences are shown with the minus sign above the number. If you move the lower graph up and down you will find that the numbers in the upper row change (but the numbers in the lower row do not). For closure to occur the first and last number in this row must be 0 and there must be at least one additional 0 adjacent to at least one of them. The degree of closure is the number of such zeros minus two.
Each number in the upper row is the difference between a transition number and the transition number which occurs in the same position when the graph of transition numbers is rotated and shifted so as to produce closure.
The numbers in the lower row are analogous to the numbers in the upper row but whereas the numbers in the upper row are differences between first order differences, the numbers in the lower row are differences between second order differences, where the second order differences are the differences between adjacent first order differences.[5]
Both rows of numbers play an essential role in the derivation of the set of 384 numbers. If the screen you are viewing shows that closure is present then pressing the Enter key will cause the software to calculate these numbers and to write them to a file.
The program actually creates two text files. The first is a file called KING_WEN.TBZ, which is identical to the table of "intermediate and final values" in the DOS software manual, and practically the same as as that contained in Appendix II of the original edition of The Invisible Landscape.
The second is the file KING_WEN.TWZ, the same as the DATA.TWZ file provided with the Timewave Zero software. This is the "standard" set of 384 numbers, which when read by the Timewave Zero software produced the "standard" timewave (now considered obsolete in orthodox Novelty Theory).
If we now run WEN_GRPH with the command line parameter X (i.e. WEN_GRPH X) the program will perform the exactly the same constuction except that the half-twist is omitted, and the two files KING_WEN.TBX and KING_WEN.TWX are produced.
The Sheliak numbers are given in SHELIAK.TWX. (These were supplied by John Sheliak, and are not generated by the WEN_GRPH program.) The Sheliak numbers are quite different from either the standard numbers (KING_WEN.TWZ) or the alternate numbers (KING_WEN.TWX). In fact, they are so different that one wonders whether the Sheliak construction is actually quite different from the one which McKenna describes.
There is a fourth set of 384 numbers, which generates another timewave: HUANG_TI.TWZ. They are discussed below in The Huang Ti Numbers and in The Zero Date (where it is shown that they generate a timewave which seems to have a better fit with the historical record).
Derivation of the Original 384 Numbers
In this part I shall consider what happens when you press Enter, when viewing the graphs described in the previous part, in order to obtain a file containing the data points which can be used in the construction of a fractal timewave, i.e., what the computational steps are which start from a hexagram sequence (a permutation of the numbers 1 through 64) and lead to a set of 384 numbers such as in the KING_WEN.TWZ file.This explanation follows the computational process which is specified in the C source code files. It describes the same process as is described in Derivation of the Timewave from the King Wen Sequence of Hexagrams, but in more mathematical terms.
Suppose we have a hexagram sequence such as the King Wen Sequence. Let diff1[] be an array[6] of 64 positive integers where diff1[i], for 1<=i<=64, is the number of lines which change from hexagram i to hexagram i+1 (or, if i = 64, from hexagram 64 to hexagram 1). From this we seek to generate an array total[i] of non-negative integers, where 0<=i<=383. These will be the 384 numbers which are our goal.
First we must calculate the difference of differences. Let diff2[i] be an array of 64 integers where diff2[i], for 1<=i<=64, is defined as follows:
For 1<=i<=63,
and
diff1[] is the array of first-order differences (FOD) and diff2[] is the array of second-order differences.
Now we define an array of numbers distance1[i], for 0<=i<=64, containing the vertical screen distances (i.e. the number of lines of the screen) between the points in the two graphs that are displayed by the WEN_GRPH program for the particular hexagram sequence under consideration. These numbers are shown in the first row of numbers on the screen, beneath the closure report.
The C source code in the labels_2() function which calculates the values in distance1[] is as follows (the variables start, row and row1 are concerned with the positioning of the graphs on the screen):
for ( i=0; i<=64; i++ )
{
if ( ( j = start - i ) < 1 )
j += 64;
distance1[i] = ( row+6-diff1[j] ) - ( row1+1+diff1[i?i:64] );
}
Now we define an array of numbers distance2[i], for 0<=i<=64, containing the differences in the slopes of the segments in the two graphs that are displayed by the WEN_GRPH program for the particular hexagram sequence under consideration. Consider, for example, the above graphs for the King Wen Sequence. The first three segments coincide, and so the difference in slopes is 0 for these segments. In the next segment of the upper graph there is a change of 5, whereas in the corresponding segment in the lower graph there is a change of 2, so the difference in slope (or the difference in the differences) is 3. And so on for the other segments of the two graphs, as given in the second row of numbers on the screen.
The code in the labels_2() function which calculates the values in distance2[] is as follows:
for ( i=0; i<=64; i++ )
{
if ( ( j = start - i - 1 ) < 1 )
j += 64;
if ( j < 1 )
j += 64;
distance2[i] = diff2[i?i:64] - diff2[j];
}Now we need to define nine arrays, each of 384 integers. These arrays are named dist_lin[], angle_lin[], dist_tri[], angle_tri[], dist_hex[], angle_hex[], dist_sum[], angle_sum[] and total[].
Firstly the dist_lin[] and angle_lin[] arrays are defined as follows:
For 0<=j<=63, dist_lin[j] = distance1[64-j] and
angle_lin[j] = distance2[64-j],
where distance1[] and distance2[] were defined above. Now we
must change the sign of half of the 64 numbers in angle_lin[], as
follows:[7]
For 1<=j<=32, angle_lin[j] = -angle_lin[j]
Now we define the remaining 320 numbers in these two arrays in terms of the first 64, as follows:
For 64<=j<=383, dist_lin[j] = dist_lin[j%64] and
angle_lin[j] = angle_lin[j%64],
where j%64 is the remainder when j is divided by
64.
Now we define the arrays dist_tri[], angle_tri[], dist_hex[] and angle_hex[] in terms of the arrays dist_lin[] and angle_lin[], thus:
For 0<=j<=191, dist_tri[j] = 3*dist_lin[j/3] and angle_tri[j] = 3*angle_lin[j/3]
For 192<=j<=383, dist_tri[j] = dist_tri[j-192] and angle_tri[j] = angle_tri[j-192]
For 0<=j<=383, dist_hex[j] = 6*dist_lin[j/6] and angle_hex[j] = 6*angle_lin[j/6]
Now we can define dist_sum[] in terms of the arrays dist_lin[], dist_tri[] and dist_hex[], and we can define angle_sum[] in terms of the arrays angle_lin[], angle_tri[], angle_hex[], as follows:
For 0 <=j<=384, angle_sum[j] = angle_lin[j] + angle_tri[j] + angle_hex[j] and dist_sum[j] = dist_lin[j] + dist_tri[j] + dist_hex[j]
Finally we obtain the array total[] in terms of angle_sum[] and dist_sum[] as follows:
For 0<=j<=383, total[j] = abs(angle_sum[j]) + abs(dist_sum[j]), where abs() is the absolute value of its argument.
The table in KING_WEN.TBZ contains the values of some of the arrays in the case of the derivation of the standard 384 numbers from the King Wen Sequence. The columns in that table correspond to the arrays mentioned above as follows:
Distances
Lin Tri Hex Sum
dist_lin[pos] dist_tri[pos] dist_hex[pos] dist_sum[pos]
Angles
Lin Tri Hex Sum
angle_lin[pos] angle_tri[pos] angle_hex[pos] angle_sum[pos]
Total corresponds to total[pos].Note that the lines in the table are listed "backwards", with the last line containing angle_lin[0], etc., and the first line contianing angle_lin[383], etc. This is for consistency with the table in Appendix II of the original edition of The Invisible Landscape.
The table in KING_WEN.TBX contains the values of some of the arrays in the case of the derivation of the alternate 384 numbers from the King Wen Sequence, i.e., the numbers generated without the half-twist.
The Huang Ti Numbers
Originally there was only one set of 384 numbers, namely, those given in Appendix II of the 1975 edition of Dennis and Terence McKenna's The Invisible Landscape. This set was not queried until 1997, and so was the orthodox set of 384 numbers for a period of over twenty years. Since this set of numbers wa produced by Leon Taylor and Royce Kelley it is appropriate to call it the Kelley numbers.The classical construction of the 384 numbers is made on the basis of the King Wen Sequence. In 1994 (as mentioned above) I discovered a component of this construction, called the "half-twist", which was necessary for constructing the 384 numbers, but which was not mentioned in the description that McKenna had given — see Derivation of the Timewave from the King Wen Sequence of Hexagrams). Matthew Watkins later wrote an evaluation of the premises of the Timewave theory in which he drew attention to the lack of justification for this half-twist (see "Autopsy for a Mathematical Hallucination?"). Thus arose the possibility that the "true" set of 384 numbers was the one obtained without the half-twist, and that the classical set of 384 numbers was "erroneous". This second set of 384 numbers may be called the Watkins numbers (although Watkins did not discover this set).
That the set of the original 384 numbers was "erroneous" was the position taken by McKenna and John Sheliak in November 1997. They claimed that when the construction (proceding from the King Wen Sequence) was formalized in terms of vector algebra, no justification for the half-twist could be found. It was thus omitted and John Sheliak produced a set of 384 numbers said to be produced by the equations given in Sheliak's paper. These may be called the Sheliak numbers. [Since these numbers are not the same as the numbers which result from McKenna's construction with the half-twist omitted, it follows that Sheliak's construction is erroneous. — PM, 2008]
The three preceding sets of 384 numbers have one thing in common — they are all generated in some manner from the King Wen Sequence of I Ching hexagrams.
It has emerged [insofar as I invented this fiction. — PM, 2008] that there was an older tradition concerning the arrangement of the I Ching hexagrams, dating back to the Yellow Emperor, Huang Ti. This venerable being (considered by some to have been an extraterrestrial) taught humans many skills. To Huang Ti is attributed [by me, tongue-in-cheek. — PM, 2008] an arrangement of the I Ching hexagrams which precedes that of King Wen. The Huang Ti Sequence is as follows, where each number n represents the hexagram at the nth position in the King Wen Sequence):
1 2 33 34 4 3 26 25 40 39 31 32 36 35 28 27 24 23 45 46 60 59 18 17 43 44 57 58 8 7 54 53 6 5 47 48 37 38 12 11 42 41 30 29 63 64 56 55 15 16 19 20 10 9 14 13 21 22 49 50 51 52 62 61 |
The Huang Ti Sequence is thus: 1, 2, 33, 34, ..., 62, 61.
This is a Wen sequence of type 2. The King Wen Sequence has the following characteristics:
The Huang Ti Sequence has the following:
It thus has a much higher closure value than does the King Wen Sequence. Hexagram sequences with such a high closure value are vary rare.
We can obtain these values for the Huang Ti Sequence by running the WEN_GRPH program on the hexagram sequence file HUANG_TI.SEQ, just as previously we ran it on the file KING_WEN.SEQ. The DOS command line WEN_GRPH Z HUANG_TI.SEQ produces the following:
WEN_GRPH.EXE, Version 2.1, Copyright 1994,1998 Peter Meyer Using half_twist and hexagram sequence file "HUANG_TI.SEQ". Hexagrams and transition values: 1:111111 6> 2:000000 4> 33:111100 4> 34:001111 4> 4:100010 4> 3:010001 4> 26:100111 4> 25:111001 4> 40:001010 4> 39:010100 1> 31:011100 2> 32:001110 3> 36:000101 4> 35:101000 4> 28:011110 6> 27:100001 1> 24:000001 2> 23:100000 3> 45:011000 4> 46:000110 3> 60:010011 2> 59:110010 2> 18:100110 6> 17:011001 2> 43:011111 2> 44:111110 1> 57:110110 4> 58:011011 3> 8:010000 2> 7:000010 2> 54:001011 6> 53:110100 3> 6:111010 4> 5:010111 3> 47:011010 2> 48:010110 3> 37:110101 4> 38:101011 3> 12:111000 6> 11:000111 4> 42:110001 2> 41:100011 3> 30:101101 6> 29:010010 3> 63:010101 6> 64:101010 2> 56:101100 2> 55:001101 2> 15:000100 2> 16:001000 3> 19:000011 4> 20:110000 3> 10:111011 2> 9:110111 2> 14:101111 2> 13:111101 2> 21:101001 2> 22:100101 3> 49:011101 4> 50:101110 4> 51:001001 4> 52:100100 2> 62:001100 6> 61:110011 2> closure = 9, closure_offset = 1, overlap = 9, closure_sum = 8 Press a key for graphical display ...
Then we obtain the following:
The 384 numbers which result from this construction are in HUANG_TI.TWZ. These are called the Huang Ti numbers. See The Zero Date for a discussion of the Timewave which is generated by these numbers.
There are thus four sets of numbers, with four associated timewaves. [A fifth set, based on an 8x8 magic square discovered by Benjamin Franklin, has in recent years been introduced — PM, 2008.] Which timewave has the best fit with the historical record is a matter for further study.
Obtaining the WEN_GRPH software
The WEN_GRPH.EXE software, and the C source code files, are provided as a free extra with the Fractal Time software and its user manual when a user license for the latter software is purchased. All the WEN_GRPH files are in a ZIP file called WEN_GRPH.ZIP, which is itself contained in the downloadable ZIP file.Purchasers of a user license prior to 2008-05-01 may download the new ZIP file containing the WEN_GRPH software (in addition to the Fractal Time software), free of charge, in the same way as they downloaded the old ZIP file.
Footnotes
[1] References in this section are to the 1994 edition of The Invisible Landscape (published by HarperSanFrancisco) unless otherwise noted.[2] The Invisible Landscape, p.140.
[3] The concept of paired hexagrams was defined above.
[4] This is an intuitive explanation. Precise definitions of these terms are required for the calculations performed by the program, but an explanation of the C source code for the WEN_GRPH program would involve us in technical matters that most readers would probably prefer to avoid. However, for those who are interested, the source code for WEN_GRPH is provided on this web site and also in the WEN_GRPH.ZIP file.
[5] This is a little difficult to explain in English, though further elucidation is contained in the next section. Those interested in knowing the exact meaning of these numbers should study the source code for the function labels_2() in the module GRAPH.C.
[6] An array of numbers is a sequence a0, a1, ..., an of numbers.
[7] This is the mysterious "half twist". As I said in the 1994 documentation for the Timewave Zero software, "The reason for this is not well understood at present, and is a question which awaits further research." The research was never performed, for in November 1997 Terence McKenna announced that the half-twist had been an error all along, and that a new approach to the construction of the 384 numbers had removed such alleged errors, resulting in the "correct" 384 numbers (the Sheliak numbers). It is doubtful, however, that the Sheliak construction correctly captures the construction of the 384 numbers as described in the original edition of The Invisible Landscape. The latter was justified by "revelation". If the Sheliak construction differs, is it to be justified by a different "revelation"? If so, we have not been told.
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