Derivation of the Timewavefrom the King WenSequence of
Hexagramsby Terence McKennaThe idea that time is experienced as a
series of identifiable elements in flux is highly developed in the
IChing. Indeed the temporal modeling of the I Ching offers an
extremely well-developed alternative to the"flat-duration" point of
view. The I Ching views time as a finite number of distinct and
irreducibleelements, in the same way that the chemical elements
compose the world of matter. For the Taoist sagesof pre-Han China
time was composed of sixty-four irreducible elements. It is upon
relations among thesesixty-four elements that I have sought to
erect a new model of time that incorporates the idea of
theconservation of novelty and still recognizes time as a process
of becoming.The earliest arrangement of the hexagrams of the I
Ching is the King Wen Sequence. It was thissequence that I chose to
study as a possible basis for a new model of the relationship of
time to theingression and conservation of novelty. In studying the
kinds of order in the King Wen Sequence of the IChing I made a
number of remarkable discoveries. It is well known that hexagrams
in the King Wensequence occur in pairs. The second member of each
pair is obtained by inverting the first. In anysequence of the
sixty-four hexagrams there are eight hexagrams which remain
unchanged when inverted.In the King Wen Sequence these eight
hexagrams are paired with hexagrams in which each line of thefirst
hexagram has become its opposite, (yang changed to yin and vice
-versa).The question remains as to what rule or principle governs
the arrangement of the thirty-two pairs ofhexagrams comprising the
King Wen Sequence. My intuition was to look at the first order of
difference,that is, how many lines change as one moves through the
King Wen Sequence from one hexagram to thenext. The first order of
difference will always be an integer between one and six. When the
first order ofdifference within pairs is examined it is always
found to be an even number. Thus all instances of firstorder of
difference that are odd occur at transitions from one pair of
hexagrams to the next pair. Whenthe complete set of first order of
difference integers generated by the King Wen Sequence is
examinedthey are found to fall into a perfect ratio of 3 to 1,
three even integers to each odd integer. The ratio of3/1 is not a
formal property of the complete sequence but was a carefully
constructed artifact achieved byarranging hexagram transitions
between pairs to generate fourteen instances of three and two
instances ofone. Fives were deliberately excluded. The fourteen
threes and two ones constitute sixteen instances ofan odd integer
occurring out of a possible sixty-four. This is a 3/1 ratio
exactly.
Where Did the Timewave Come
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[01/22/2002 1:51:30 PM]Where Did the Timewave Come
From?file:///H|/public/timewave/waveexplain.html (2 of 16)
[01/22/2002 1:51:30 PM]Figure 1Figure 1 shows that when the first
order of difference of the King Wen Sequence is graphed it
appearsrandom or unpredictable. However when an image of the graph
is rotated 180 degrees within the planeand superimposed upon itself
it is found to achieve closure at four adjacent points as in Figure
2.
Figure 2While closure might logically be expected anywhere in
the sequence, it in fact occurs at the conventionalbeginning and
end of the sequence. While an arrangement with closure might have
placed any twohexagrams opposite each other, what we in fact find
is that the hexagrams opposite each other are suchthat the numbers
of their positions in the King Wen Sequence when summed is always
equal tosixty-four. These facts are not coincidences, they are the
artifacts of conscious intent.Over 27,000 hexagram sequences were
randomly generated by computer (all sequences having theproperty
possessed by the King Wen sequence that every second hexagram is
either the inverse or thecomplement of its predecessor). Of these
27,000 plus sequences only four were found to have the
threeproperties of a 3/1 ratio of even to odd transitions, no
transitions of value five and the type of closuredescribed above.
Such sequences were found to be very rare, occurring in a ratio of
1 in 3770. Here is thecomplete graph of the King Wen first order of
differnce with its mirror image fitted against it to
achieveclosure:
Where Did the Timewave Come
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[01/22/2002 1:51:30 PM]Where Did the Timewave Come
From?file:///H|/public/timewave/waveexplain.html (4 of 16)
[01/22/2002 1:51:30 PM]Figure 3For these reasons I was led to view
the King Wen Sequence as a profoundly artificial arrangement of
thesixty-four hexagrams. Look carefully at Figure 3 immediately
above. Review in your mind the stepsfrom the King Wen sequence that
led to it. Notice that it is a complete set of the sixty-four
possiblehexagrams, running both sequentially forward and backward.
Since it is composed of sixty-fourhexagrams of six lines each it is
composed of 6 x 64 or 384 lines or yao. One might make an analogy
andsay Figure 3 is to the King Wen sequence as a cube is to a
square; it is composed of the same elements asthe King Wen Sequence
but it has more dimensions.It is my assumption that the oracle
building pre-Han Chinese viewed the forward-and
backward-runningdouble sequence of Figure 3 as a single yao or line
and that it is therefore open to the same treatment aslines are
subject to in the I Ching, namely multiplication by six and
sixty-four.
Where Did the Timewave Come
From?file:///H|/public/timewave/waveexplain.html (5 of 16)
[01/22/2002 1:51:30 PM]Figure 4Since a hexagram has six lines I
visualized six double sequences in a linear order. But a hexagram
ismore than lines; a hexagram also contains two trigrams. Thus over
the six double sequences I overlaidtwo double sequences, each three
times larger than the six double sequences. A hexagram also has
anidentity as a whole; thus over the six and the two double
sequences a single, larger double sequence isprojected.The sets of
double sequences of each level share a common point of origin and
all return to a single endpoint. The resulting figure, show in
Figure 4 on the extreme right, is to the original double sequence
as atesseract is to a cube, for again more dimensions have been
added. This figure itself can then be imaginedas a single hexagram,
but one of a set of sixty-four.
Where Did the Timewave Come
From?file:///H|/public/timewave/waveexplain.html (6 of 16)
[01/22/2002 1:51:30 PM]Figure 5The closure at the beginning and end
of this figure suggested that it might be useful to model process.
Its384 subunits imply a calendar. Can it be coincidence that the
length of a lunar month, 29.53 days, times13 is 383.89? I believe
that what we have here is a 384 day lunar calendar with resonances
to other othernaked eye astronomical phenomena known to be of
interest to the ancient Chinese (see below).
Where Did the Timewave Come
From?file:///H|/public/timewave/waveexplain.html (7 of 16)
[01/22/2002 1:51:30 PM]Table 1Using techniques that I developed for
the problem I was able to mathematically collapse the
hexagramconstruct into a self-similar fractal curve that can be
used to map the unfolding of temporal variables andtheir resonances
on all levels of duration.My attack on the problem began with an
examination of the simple wave of Figure 3. Thirteen discreteline
types comprise any simple version of the graph. These thirteen
lengths are displayed on and off gridin Figure 6:
Where Did the Timewave Come
From?file:///H|/public/timewave/waveexplain.html (8 of 16)
[01/22/2002 1:51:30 PM]Figure 6. As these lengths are always
discrete units, we can assign to them values which are ascending
integers.The values of Figure 6 allow a quantification of line
length. To quantify the degree and direction of skewof individual
lines, one direction of skew is designated as positive, giving
lines skewed in that directionpositive values. Lines skewed in the
opposite direction are given negative values. This gives
valuesadequately preserving and quantifying line length and
direction of skew. The values labeled L in Figure 6are used for the
left side of a simple wave while the values labeled R, which are
the same values withtheir their sign reverse, are applied to the
right side of any simple wave. The sign is important only
incombining values across scales but is ignored in the final
graphing of combine values, either set of valuesmay be applied to
either the right or left side. However, whichever schema is chosen
must then befollowed throughout. Figure 7 represents the version of
these values that we have used for the simplegraph.
Where Did the Timewave Come
From?file:///H|/public/timewave/waveexplain.html (9 of 16)
[01/22/2002 1:51:30 PM]Figure 7It is important to note that the
valuations in Figure 7 are valuations of the simple wave on the
smallestscale of a single complex wave. The relative proportions of
the three levels in the complex wave arepreserved and quantified by
multiplying the valuations of the linear scale in the appropriate
way. Toassign a value to a positionon the trigramatic scale, the
valuation of that position on the linear scale(Figure 7) is
multiplied by three because the trigramatic scale is three times
larger than the linear scale.In a similar manner, the
hexagramaticpositions are assigned a valuation by multiplying their
linear-levelWhere Did the Timewave Come
From?file:///H|/public/timewave/waveexplain.html (10 of 16)
[01/22/2002 1:51:30 PM]valuations by six, again because the
hexagramaticscale is six times larger than the linear. Figure 7
usesthe value scheme in Figure 6 and is the version of value
assignments we have used in all our calculations.Note that in
Figure 7 all parallel lines, regardless of the distances separating
them, reduce to zero. Thus,while the operations discussed so far
have allowed quantification of skew direction, proportional ratios
ofthe wave parts, and the degree of departure from the parallel
state, they have not provided a quantifiedaccount of the
fluctuating distances between the two parameters of the wave. The
procedure for obtainingthese values is similar to, but distinct
from, the procedures outlines above.
Figure 8Figure 8 shows the seven types of divergence,
congruence, and overlap which points in the simple wavemay display.
The two possible assignments of positive and negative numbers are
shown to the right andleft sides in Figure 8. We have chosen to use
the right-hand schema to preserve the intuition thatoverlap tends
to carry a situation toward the zero state rather than away from
it.
Where Did the Timewave Come
From?file:///H|/public/timewave/waveexplain.html (11 of 16)
[01/22/2002 1:51:31 PM]Figure 9Figure 9 shows the values this
series of point assignments generates when applied to the simple
wave.When the valuations for skew, parallelism, and relative
proportion have been combined in the mannerdetailed above the
following 384 values result(as read from position 383 to position
zero):Where Did the Timewave Come
From?file:///H|/public/timewave/waveexplain.html (12 of 16)
[01/22/2002 1:51:31 PM]0, 0, 0, 2, 7, 4, 3, 2, 6, 8, 13, 5, 26, 25,
24, 15, 13, 16, 14, 19, 17, 24, 20, 25, 63, 60, 56, 55, 47, 53, 36,
38,39, 43, 39, 35, 22, 24, 22, 21, 29, 30, 27, 26, 26, 21, 23, 19,
57, 62, 61, 55, 57, 57, 35, 50, 40, 29, 28, 26,50, 51, 52, 61, 60,
60, 42, 42, 43, 43, 42, 41, 45, 41, 46, 23, 35, 34, 21, 21, 19, 51,
40, 49, 29, 29, 31, 40,36, 33, 29, 26, 30, 16, 18, 14, 66, 64, 64,
56, 53, 57, 49, 51, 47, 44, 46, 47, 56, 51, 53, 25, 37, 30, 31,
28,30, 36, 35, 22, 28, 32, 27, 32, 34, 35, 52, 49, 48, 51, 51, 53,
40, 43, 42, 26, 30, 28, 55, 41, 53, 52, 51, 47,61, 64, 65, 39, 41,
41, 22, 21, 23, 43, 41, 38, 24, 22, 24, 14, 17, 19, 52, 50, 47, 42,
40, 42, 26, 27, 27, 34,38, 33, 44, 44, 42, 41, 40, 37, 33, 31, 26,
44, 34, 38, 46, 44, 44, 36, 37, 34, 36, 36, 36, 38, 43, 38, 27,
26,30, 32, 37, 29, 50, 49, 48, 29, 37, 36, 10, 19, 17, 24, 20, 25,
53, 52, 50, 53, 57, 55, 34, 44, 45, 13, 9, 5, 34,26, 32, 31, 41,
42, 31, 32, 30, 21, 19, 23, 43, 36, 31, 47, 45, 43, 47, 62, 52, 41,
36, 38, 46, 47, 40, 43, 42,42, 36, 38, 43, 53, 52, 53, 47, 49, 48,
47, 41, 44, 15, 11, 19, 51, 40, 49, 23, 23, 25, 34, 30, 27, 7, 4,
4, 32,22, 32, 68, 70, 66, 68, 79, 71, 43, 45, 41, 38, 40, 41, 24,
25, 23, 35, 33, 38, 43, 50, 48, 18, 17, 26, 34, 38,33, 38, 40, 41,
34, 31, 30, 33, 33, 35, 28, 23, 22, 26, 30, 26, 75, 77, 71, 62, 63,
63, 37, 40, 41, 49, 47, 51,32, 37, 33, 49, 47, 44, 32, 38, 28, 38,
39, 37, 22, 20, 17, 44, 50, 40, 32, 33, 33, 40, 44, 39, 32, 32, 40,
39,34, 41, 33, 33, 32, 32, 38, 36, 22, 20, 20, 12, 13, 10Table 2:
The 384 Values of the Complex WaveThese same values are to be
graphed as a single line graph and are the primary valuation scheme
for anycomplex wave. The process of quantifying a given time in the
modular hierarchy of the complex wavewill necessarily begin with
reference to these values.
Where Did the Timewave Come
From?file:///H|/public/timewave/waveexplain.html (13 of 16)
[01/22/2002 1:51:31 PM]Figure 10Figure 10 shows the complex
compound wave and its reduction into an ordinary graph. The 384
valuesabove can be ploted on a graph thus reducing the complex,
tri-leveled, bi-directionally flowing complexwave a single line
moving in only one direction. Such a graph preserves certain
qualities of the complexwave: its divergence from the zero state
where lines are parallel, the direction and degree of skew ofpairs
of lines, the relative proportions of the three levels, and the
distances between the fluctuatingparameters of the various
component waves. However such a graph does not reflect the shift of
valuesthat would occur if the single-line complex wave were nested
into a particular position in the modularhierarchy of fractal waves
each level of which was 64 times larger than its predecessor. In
such a case,the 384 values above would serve as a schema of values
to be combined through superimposition withthe values associated
with whichever one of the 64 segments of the next largest level it
is to be nested in.Where Did the Timewave Come
From?file:///H|/public/timewave/waveexplain.html (14 of 16)
[01/22/2002 1:51:31 PM]These latter values have first been
multiplied by by 64, indicating their membership in the next
largerlevel of the hierarchy. Thus 64 variants of the values of
Table 2 would be generated, and it is thesevarious waves or frames
which we treat as comprising the temporal maps of a given
historical span. Thevalues of Table 2 are the basis of the
quantified maps of temporal flux which Novelty theory rests on.
Where Did the Timewave Come
From?file:///H|/public/timewave/waveexplain.html (15 of 16)
[01/22/2002 1:51:31 PM]Figure 11The last three of the 384 segments
of the wave on any level possess singularities which quantify as
zero,see Figure 11. When the wave on a given level enters those
segments of itself which are zero states, itceases to contribute
boundary constraints to its subsets on lower levels. The cessation
of boundaryconstraints imposed by higher levels in the hierarchy
causes a "surge" toward the zero state each timethat a cycle enters
its terminal phase on any level in the hierarchy. Such quantized
transitions from onemodality to another are called "changes of
epoches" By Whitehead. The appearance of life in aninorganic world,
of consciousness in an unconscious world, or of language in a world
without languageare all examples of such epochal transitions. Our
lives are filled with such transitions, but they areterminations of
relatively short cycles in the quantified hierarchy. Terminations
of cycles or epoches ofreally long duration cause extreme
accelerations toward the zero state/ This idea is similar to
Whitehead'sconception of concrescence and the Vedic conception of
world ages which grow shorter as they tightenaround an axis point.
The spiral image of the Christian apocalypse is another example of
this intuitionthat time is a series of tightening gyres around the
quantized emergence of transformation.A perfect self-consistent
proof that Table 2 does adequately conserve four
qualities--divergence from thezero state where lines are parallel,
the direction and the degree of skew of pairs of lines,
distancesbetween the parameters of the component waves, and the
proportions of these three qualities relative tothe other
levels--is afforded by comparing any two graph segments assigned to
a single time but ondifferent scales relative to an arbitrary
termination date that is the same for both. Any two such
segmentswill be found to be similar in form yet preserving their
relative proportions. This demonstrable factconfirms the idea that
the complex wave derived from Figure 3 is adequately reflected and
its usefulvalues retained throughout the series of operations which
generate Table 2. What Table 2 and the graphsmade from it achieve
is a clarity and simplicity of expression lacking in the
tri-leveled complex wave.Table 2 and the algorithm that accompanies
it is a quantification of operational constructs which makesthis
modular hierarchy of temporal variables a valid subject for
rigorous scientific investigation.We appreciate your comments
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|?6] hnvo ropIncod fho subscrIpf i In oqunfIons |60] nnd |?2], ns
fhov hnvo boondofInod In oqunfIons |6l] nnd |?3]. Wo now hnvo n
compIofo nnd woII-dofInod funcfIonfor our Tri-Letel Conplex Wote,
or Jo/o ee/.IqunfIon |?6] producos n frI-IovoI wnvo numbor sof fhnf
confnIns somo nognfIvo vnIuos.Tho 384 numbor Jo/o ee/, on fho ofhor
hnnd, Is fho sof of posIfIvo ronI numbors In fhodomnIn 0 384 x .
ThIs monns fhnf pnrf of fho rnw dnfn sof producod bv oqunfIon|?6]
IIos oufsIdo fho x-vnIuo domnIn fhnf Is fhoughf fo bo fho propor
oxprossIon of fhIswnvoform. Ono procoduro fhnf Is wIdoIv usod for
convorfIng nognfIvo vnIuos of somonrbIfrnrv wnvoform, Info posIfIvo
vnIuos, Is fho uso of fho nbsoIufo vnIuo opornfor. If onovIows fhIs
/ri-letel conplex uote ns somo kInd of InformnfIon cnrrvIng sIgnnI,
IIko nnnmpIIfudo moduInfod rndIo wnvo, for oxnmpIo, fhon n vnIId
procoduro for procossIng suchn sIgnnI Is fho nppIIcnfIon of fho
nbsoIufo vnIuo opornfor. In fho rf sIgnnI procossIngcnso, fho
rocoIvod moduInfod-cnrrIor wnvoform Is pnssod fhrough nbsoIufo
vnIuo cIrcuIfrv(rocfIfIor) so fhnf fho nognfIvo vnIuos of fho wnvo
nro convorfod fo posIfIvo vnIuos. ThIsncfunIIv Improvos fho sIgnnI
fo noIso rnfIo of fho cnrrIor onvoIopo, whIch Is fhoInformnfIon
cnrrvIng moduInfIon sIgnnI. ThIs rocfIfIod sIgnnI Is fhon procossod
bv ndofocfor cIrcuIf fhnf oxfrncfs fho InformnfIon cnrrvIng
moduInfIon wnvo from fho cnrrIorwnvo. AIfhough fho frI-IovoI wnvo
nnd fho rndIo wnvo nro nof sfrIcfIv nnnIogous, fhovnpponr sImIInr
onough fo mnko n pInusIbIo nrgumonf for fho nppIIcnfIon of fho
nbsoIufovnIuo opornfor horo. ThIs opornfIon Is oxprossod ns: y yDW
TABS = |??]Whoro: yDW Is fho Do/o Wote fhnf Is grnphod In IIg. ll,
nnd dofInod ns fho nbsoIufovnIuo of fho Tri-Letel Conplex Wote ns
oxprossod In oqunfIon |?3]. ThIs numbor sof Isusod ns Inpuf dnfn
for fho TineWote Zero soffwnro, whIch porforms nn InfInIfo
sorIosoxpnnsIon fhnf Movor cnIIs n froc/ol /ronefornll, fo gonornfo
fho TImoWnvo vIowod onfho compufor
scroon.6WDQGDUGDQG5HYLVHG'DWD6HW&RPSDULVRQVWIfh oqunfIon |?3]
nnd |?4], nnd fho grnph In IIg. ll, wo hnvo compIofod fhIs
formnIIzoddovoIopmonf of fho TWZ Jo/o ee/. Wo nro now In n posIfIon
fo compnro fhoso rosuIfswIfh fhoso of fho sfnndnrd dovoIopmonf
roporfod bv McKonnn nnd Movor In //e 1ntieiIleLonJecope nnd fho
Tine1xplorer mnnunI, ns woII ns nddross fho Issuos rnIsod bv
fhoWo//ine OIjec/ion.IIg. l2 Is n grnph of bofh fho e/onJorJ nnd
retieeJ Jo/o ee/e, nnd If shows somoromnrknbIo sImIInrIfIos ns woII
ns sIgnIfIcnnf dIfforoncos. Ono InforosfIng fonfuro of fhIsgrnph,
Is fho nnfuro of onch wnvo nf Ifs rospocfIvo ondpoInfs. !ocnII fhnf
fho vnIuo of fho31wnvo nf x = 0 wIII bo dIscnrdod bocnuso If Is n
dupIIcnfo or wrnp of fho vnIuo nf x = 3S4.ThIs wIII nof offocf fho
roInfIvo vnIuos of fho fwo wnvos nf x = 3S4, bocnuso fhov nro
bofhzoro-vnIuod nf fhIs ondpoInf. Howovor, fho vnIuo of onch wnvo
nf x = 1 Is nof fho snmo,wIfh fho e/onJorJ uote hnvIng n vnIuo of
l0 whIIo fho retieeJ uote vnIuo Is zoro.)LJXUHWhv doos fhIs mnffor,
vou mnv nsk, sInco fhoro nro mnnv obvIous dIfforoncos bofwoonfho
fwo wnvos whnf Is fho sIgnIfIcnnco of fhIs dIfforonco Ior fho
e/onJorJ uote, If hnsboon nrguod fhnf fho zoro vnIuo nf fho ond of
fho wnvoform ImpIIos somo kInd ofsInguInrIfv nf fho ond of fho
procoss or nf fho ond of fImo. ThIs retieeJ uote IsImpIvIng,
howovor, fhnf fhoro mnv bo sInguInrIfIos nf Io// onds of fho
confInuum. ThIs IsAIso nn nrgumonf for n cloeeJ svsfom fhnf mnv bo
undorgoIng somo kInd of cvcIIcronownI procoss porhnps onch cvcIo
oxprossIng ovor hIghor ordorod sfnfos of compIoxform, or
Aotel/x.Thoro nro concopfs omorgIng from fho fIoId of qunnfum
cosmoIogv fhnf mnv doscrIbo nnnnnIogous cvcIIc procoss. ThIs Is
fhoorv In whIch unIvorsos nro fronfod IIko qunnfumpnrfIcIos fhnf
InhnbIf n Inrgor, or hIghor dImonsIonnI domnIn cnIIod n nul/iteree.
MIchIoKnkul2, n fhoorofIcnI phvsIcIsf nnd co-foundor of sfrIng
fIoId fhoorv, hns doscrIbod nprocoss whoro unIvorsos omorgo from
fho zoro-poInf, or vncuum fIoId, go fhrough nnovoIufIonnrv procoss,
fhon porhnps rofurn fo fho zoro-poInf fIoId nf fho ond of fho
cvcIo.ThIs cvcIo mnv fhon roponf IfsoIf, possIbIv wIfh Incronsod
compIoxIfv nnd Aotel/x. SocouId fhIs bo sImIInr fo fho procoss fhnf
fho TineWote nnd Aotel/x T/eorx nffompf forovonI Iorhnps furfhor
InvosfIgnfIon Info fho nnfuro of fho TImoWnvo wIII shod somoIIghf
on fhoso quosfIons.32)LJXUHAnofhor sIgnIfIcnnf fonfuro of IIg. l2
Is fho nppnronf ngroomonf of fho fwo wnvos In fhoIowor froquoncv
domnIn. Iroquoncv confonf of nnv wnvoform oxprossos IfsoIf
nsvnrInfIons In fho rnfo of chnngo of Ifs vnIuo ns fho wnvo
propngnfos In somo ronIm, fhnfcouId bo oIfhor n spnco or fImo
domnIn, or bofh. So fho sIopo of n wnvoform nf nnv gIvonpoInf, or
Ifs gonornI shnpo, cnn rovonI froquoncv confonf (fho mngnIfudo nnd
rnfo ofspocIfIc undorIvIng procossos). IxnmInnfIon of fho wnvo pnIr
In IIg. l2 shows fhnf fhoroIs n common Iowor froquoncv procoss
occurrIng for onch wnvoform. Tho hIghor froquoncvprocossos npponr
ns roInfIvoIv shorfor durnfIon ponks rIdIng upon fho sIowor
procoss.Tho Iowosf froquoncv procoss occurrIng In fhoso wnvoforms
cnn bo soon bv drnwIng nnImngInnrv IIno bofwoon fho hIghosf of nII
fho ponks ns ono movos ovor fho domnIn of fhownvoforms. SIIghfIv
hIghor froquoncv compononfs cnn bo soon bv drnwIng fhnfImngInnrv
IIno ovor fho ponks nnd vnIIovs upon whIch fho shnrposf nnd
shorfosfdurnfIon ponks rIdo. Tho grnphs do dIffor In fho hIghor
froquoncv domnIn ns cnn bo soonbv fho sfoopor sIopos of fho Inrgosf
e/onJorJ uote frnnsIfIons. ThIs couId vorv woII boduo fo hIgh
froquoncv noIso prosonf In fho e/onJorJ dnfn sof bocnuso of fho
ImboddodmnfhomnfIcnI orrors.Tho Iow froquoncv, or Iong durnfIon
procossos, nro fhoso fhnf mnv occur on fho scnIo ofmIIIonnIn or
ovon bIIIIons of vonrs, whorons fho hIghor froquoncv procossos mnv
occur onfho scnIo of n humnn IIfofImo. CouId If bo fhnf fho Iowosf
froquoncv procoss Is fhosIgnnfuro of somo cronfIvo prIncIpIo nf
work, bo If sfrnngo nffrncfor, zoro-poInf fIoId, oreec/o/on. CouId
fhIs cronfIvo onorgv, bo porfurbIng fho fnbrIc of spnco-fImo In
such nwnv ns fo frIggor fho cronfIon nnd consorvnfIon of hIghor
ordorod sfnfos somofhIng IIkofho grnvIfnfIonnI onorgv of n pnssIng
nonrbv sfnr frIggorIng fho formnfIon of n comofsfrom fho Orf cIoud
Is fhIs Iowosf froquoncv procoss fhon n kInd of ground sfnfo,
uponwhIch nII hIghor froquoncv procossos oxpross fhomsoIvos Iorhnps
In fImo fhosoquosfIons wIII bo nnswornbIo, nIfhough corfnInIv nof
fodnv.33An obvIous fonfuro of IIg. l2 fhnf cIonrIv shows In fhIs
grnph, Is fho dIfforonco In fhonvorngo wnvo vnIuo bofwoon sfnndnrd
nnd rovIsod wnvos. Tho nvorngo wnvo vnIuo forfho sfnndnrd wnvo Is
somowhnf gronfor fhnn fho nvorngo vnIuo of fho rovIsod wnvo.
ThIsdIfforonco In nvorngo wnvo vnIuo npponrs fo bo fho rosuIf of
dIfforoncos In fho hIghorfroquoncv compononfs of fho wnvo pnIr,
porhnps duo fo noIso In fho sfnndnrd wnvo fhnfIs producod bv fho
mnfhomnfIcnI orrors fhnf nro prosonf. Thoso hIgh
froquoncvcompononfs of fho sfnndnrd wnvo show up ns fho sfoop ponks
fhnf rIso woII nbovo fhoponks In fho rovIsod wnvo. In fho IourIor
nnnIvsIs fhnf foIIows, fhoso Inrgo ponks npponrns hIgh froquoncv
noIso fhnf ndds rnndomnoss fo fho wnvo. Tho Impncf of fhIs
dIfforoncoon fho fInnI TImoWnvo, Is fo shIff fho nvorngo IovoI of
novoIfv upwnrd (Iowor vnIuos)from fhnf oxprossod bv fho sfnndnrd
wnvo. In ofhor words, fho rovIsod wnvo oxprossos nprocoss wIfh
somowhnf hIghor IovoIs of novoIfv, fhnn doos fho sfnndnrd wnvo.
SIncoAotel/x Isn`f n cnIIbrnfod procoss, If`s nof possIbIo fo
doformIno whnf fho mororonsonnbIo IovoI of Aotel/x wouId bo. AII
fhnf cnn bo oxprossod fhon, Is roInfIvoovoIfv.Ono fInnI fonfuro of
IIg. l2 fhnf roquIros somo dIscussIon, Is fho corroInfIon numbor
nffho fop of fho grnph. In ordor fo doformIno nnd qunnfIfv fho
dogroo of Infordopondonco,or Infor-roInfodnoss of fho sfnndnrd nnd
rovIsod wnvoforms, n mnfhomnfIcnI opornfIoncnIIod correlo/ion wns
porformod wIfh fhoso fwo numbor sofs. Tho numbor nf fho fop offho
grnph Is fho rosuIf of fhnf nnnIvsIs n vnIuo of 0.564. A
corroInfIon of l.0 wouIdmonn fhnf fho wnvoforms nro IdonfIcnI,
whorons n corroInfIon of zoro wouId IndIcnfo nofuncfIonnI
roInfIonshIp bofwoon fho fwo. AddIfIonnIIv, n corroInfIon of l
wouId IndIcnfofhnf fho wnvoforms woro mIrror Imngos of ono nnofhor
n ponk rofIocfod bv n frough ofc.In fhIs cnso n corroInfIon of
0.564 IndIcnfos fhnf fhoso fwo wnvoforms show n sIgnIfIcnnfIovoI of
Infordopondonco, nIfhough fnr from IdonfIcnI. ThIs IovoI of
corroInfIon couId boconsIdorod IIkoIv for fwo numbor sofs fhnf
shnro n common orIgIn, ns woII ns shnrIngmnnv of fho snmo
dovoIopmonfnI
procoduros.'DWD:DYHDQG5DQGRP1XPEHU6HW&RPSDULVRQVOno mofhod for
nssossIng fho InformnfIon cnrrvIng pofonfInI of fho Do/o Wote,
nndconvIncIng onosoIf fhnf If Is nof n rnndom procoss, Is fo
compnro If wIfh n dnfn sof fhnfhns boon rnndomIv gonornfod. SovornI
such ronJon uote sofs woro consoquonfIvproducod fo bo compnrod wIfh
fho rovIsod nnd sfnndnrd Do/o Wote numbor sofs dIrocfIv,nnd fo nIso
uso ns Inpuf fo fho TWZ soffwnro fo gonornfo rnndom soodod
TImoWnvos.IIg. l3 Is n grnph of fho rovIsod Do/o Wote wIfh n rnndom
wnvo sof ovorInv, nnd IfcIonrIv shows fhnf fhoso numbor sofs bonr
IIffIo rosombInnco fo ono nnofhor. CorroInfIonnnnIvsIs of fho fwo
sofs shows n corroInfIon of 0.03, or ossonfInIIv un-corroInfod ns
onowouId oxpocf for nnv rnndom numbor sof. IIg. l3 nIso npponrs fo
show fhnf fho rovIsodDo/o Wote Is n vorv dIfforonf fvpo of numbor
sof from fho rnndom wnvo sof, nnd Ifnpponrs fo showIng somo kInd of
InformnfIon cnrrvIng procoss. Is fhIs In fncf fho cnso, ordoos If
jusf npponr fhnf wnvIxnmInnfIon of fho pouer epec/ro for fho Jo/o
nnd ronJon uotee shown In IIgs. l2 nndl3 cnn rovonI somofhIng nbouf
fho nnfuro of fhoso fhroo wnvoforms nnd fhoIr34roInfIonshIp. Tho
convorsIon of fImo, or spnco domnIn wnvoforms Info froquoncv
domnInwnvoforms (froquoncv spocfrum or powor spocfrum) Is porformod
usIng n mnfhomnfIcnIopornfIon cnIIod n IourIor frnnsform. WIfh fhIs
mofhod, n froquoncv spocfrum cnn boproducod, whIch cnn foII us how
much powor Is confnInod In onch of fho froquoncvcompononfs
(hnrmonIcs) of n gIvon wnvoform, nnd fhorobv provIdIng fho
froquoncvdIsfrIbufIon of fho wnvo powor. ThIs dIsfrIbufIon wouId
fvpIcnIIv bo dIfforonf forInformnfIon cnrrvIng wnvoforms fhnn for
rnndom, or noIso sIgnnIs. Tho rnndom, or noIsosIgnnI spocfrum Is
fvpIcnIIv fInf ovor fho sIgnnI bnndwIdfh, nnd offon
dIsfInguIshnbIofrom nn InformnfIon cnrrvIng sIgnnI spocfrum fhnf
oxhIbIfs 1 f (f = froquoncv) bohnvIor.)LJXUHIourIor frnnsform
opornfIons woro porformod on fho dnfn sofs shown In IIgs. l2 nnd
l3,wIfh fho rosuIfs shown In IIg. l4. Tho fop grnph of IIg. l4
IncIudos pIofs for fhoe/onJorJ nnd retieeJ Do/o Wote powor spocfrn,
whIIo fho boffom grnph dIspInvs fhoIonJon Wote powor spocfrum. Tho
coIorod IInos drnwn fhrough onch of fho spocfrn nropouer func/ion
curvo-fIfs, fhnf show fho froquoncv roII-off chnrncforIsfIcs of
onch wnvo.ofIco fhnf fho fwo powor spocfrn In fho fop grnph oxhIbIf
froquoncv roII-off (powor IovoIdocronsos wIfh IncronsIng
froquoncv), whorons fho Iowor grnph powor spocfrum oxhIbIfsn fInf
froquoncv rosponso (powor IovoI Is froquoncv Indopondonf). ThIs
froquoncv roII-offIs chnrncforIsfIc of InformnfIon cnrrvIng
sIgnnIs, whorons fho fInf rosponso IschnrncforIsfIc of noIso or
rnndom sIgnnIs.Tho retieeJ Jo/o uote spocfrum, shown In fho fop
grnph In groon, Is oxhIbIfIng fhononrIv porfocf 1 f froquoncv
rosponso fhnf Is fvpIcnI for nn InformnfIon cnrrvIngwnvoform. On
fho ofhor hnnd, fho e/onJorJ Jo/o uote powor spocfrum shown In
bIuo,oxhIbIfs froquoncv roII-off, buf wIfh n fInffor rosponso fhnf
Is nof 1 f . In fncf, fho fInffor35)LJXUHfroquoncv rosponso of fho
e/onJorJ Jo/o uote Is fho IIkoIv rosuIf of hIgh froquoncv noIsofhnf
Incronsos fho powor nf fho fnII ond of fho spocfrum nnd provonfs n
sfoopor roII-off.ThIs Is somofhIng fhnf shouId bo oxpocfod from fho
dIsforfod e/onJorJ Jo/o uote wIfhImboddod mnfhomnfIcnI orrors,
whIch wouId fond fo ndd rnndomnoss fo fho wnvo. ThosIgnnfuro of
such rnndomnoss cnn bo soon In fho IonJon Wote powor spocfrum,
shownIn fho Iowor grnph In rod. ThIs pIof shows fho fvpIcnIIv fInf
froquoncv rosponso of n36rnndom, or noIso sIgnnI wIfh no
InformnfIon confonf. AppnronfIv, fho grnphs In IIg. l4nro showIng
fhnf fho sfnndnrd nnd rovIsod dnfn uotee nro dofInIfo InformnfIon
cnrrvIngwnvoforms, buf fhnf fho dIsforfod e/onJorJ Jo/o uote hns
Imboddod hIgh froquoncvnoIso fhnf fInffons Ifs rosponso. ThIs Is
ossonfInIIv whnf IIgs. l2 nnd l3 nro showIng
nswoII.6WDQGDUG5HYLVHGDQG5DQGRP*HQHUDWHG7LPH:DYH5HVXOWV(l) Tho
TImoWnvo Zoro Scroon Sof CompnrIsonsOnco fho Do/o Wote, or 384
numbor dnfn sof hns boon gonornfod, If bocomos fho Inpufdnfn for
fho TineWote Zero soffwnro pnckngo. As monfIonod provIousIv, fho
soffwnroporforms whnf hns boon cnIIod n froc/ol /roneforn, or
oxpnnsIon of fho 384 dnfn numborsof fo produco fho TineWote vIowod
on fho compufor scroon ns n grnph of Aotel/x. Inordor for fhIs
frncfnI oxpnnsIon fo bo porformod proporIv, fho soffwnro roquIros
fhnf fho384 numbor dnfn sof shown In IIg. l0 bo rovorsod, such fhnf
dnfn poInf 384 bocomosdnfn poInf l nnd dnfn poInf 0 Is dIscnrdod
(sInco If`s n dupIIcnfo or wrnp of dnfn poInf384).)LJXUHD37Throo
sopnrnfo dnfn sofs woro usod In ordor fo gonornfo fho TineWotee
noodod forcompnrIson fho e/onJorJ dnfn sof, fho retieeJ dnfn sof,
nnd n ronJon dnfn sof. ThorosuIfs of somo of fhoso TineWote
compnrIsons wIII bo shown In fho grnphs fhnf foIIow,bogInnIng wIfh
fho dofnuIf TineWote grnphs fhnf nro IncIudod wIfh fho
Tine1xplorersoffwnro ns pro-compufod wnvoforms.IIgs. l5n nnd l5b
show fho TineWote fhnf Is sforod bv fho soffwnro ns Scroon l, nnd
Ifcovors fho porIod bofwoon l942 nnd 20l2. IIg. l5n shows bofh fho
TineWote rosuIfIngfrom fho e/onJorJ Jo/o sof on fho Ioff, nnd fhnf
for fho retieeJ Jo/o sof on fho rIghf. Onfho ofhor hnnd, IIg. l5b
Is fho TineWote gonornfod bv fho ronJon dnfn sof, nnd IfcIonrIv
bonrs IIffIo rosombInnco fo fho grnphs of IIg. l5n.ThIs Is fho
TineWote grnph fhnf McKonnn hns cnIIod hIsforv`s frncfnI
mounfnIn,bocnuso of Ifs mounfnIn-IIko shnpo. Thoro nro sovornI
fonfuros fo nofIco horo, wIfh fhofIrsf boIng fhnf fhoso fwo pIofs
hnvo romnrknbIv sImIInr shnpos obvIousIv nof IdonfIcnI,buf fhoro Is
cIonrIv n common domInnnf procoss nf work. Anofhor common fonfuro
ofsIgnIfIcnnco shown In fhoso fwo grnphs, Is fhnf fho mnjor doconf
Info Aotel/x (ponk of fhomounfnIn) bogIns somofImo In l96?.
IInnIIv, ns monfIonod onrIIor, fho TineWoteproducod bv fho retieeJ
Do/o Wote numbor sof, shows n hIghor nvorngo IovoI of Aotel/xfor
fhIs fImo porIod (Iowor vnIuos), fhnn doos fho TineWote producod bv
fho e/onJorJ38)LJXUHEJo/o sof. ThIs Aotel/x dIfforonco Is fho
IIkoIv rosuIf of fho e/onJorJ uote dIsforfIon,cnusod bv fho
Imboddod mnfhomnfIcnI orrors fhnf produco sIgnIfIcnnf hIgh
froquoncvnoIso In fho wnvo. As shown In IIg. l4, fho hIgh froquoncv
compononfs of fho rovIsoddnfn wnvo nro Iowor fhnn fho sfnndnrd wnvo
bv nn ordor of mngnIfudo.IIg. l6n shows fho sfnndnrd nnd rovIsod
TImoWnvo grnphs for Scroon 4 of fho TWZdIspInv. AgnIn, fhoso fwo
pIofs nro quIfo sImIInr In forms of fhoIr npponrnnco, nnd soomfo bo
showIng ovIdonco of somo common undorIvIng procoss. Tho dIfforoncos
mnv bo duofo fho fncf fhnf fho sfnndnrd numbor sof producos moro
hIgh froquoncv noIso bocnuso offho Imboddod orrors In fho numbor
sof. Tho corroInfIon bofwoon fhoso fwo grnphs wns)LJXUHDfound fo bo
0.?3l, nof ns hIgh ns Scroon l, buf sfIII n sIgnIfIcnnf corroInfIon
nonofhoIoss.On fho ofhor hnnd, fho rnndom dnfn sof TImoWnvo shown
In IIg. l6b, shows vorv IIffIocorroInfIon wIfh oIfhor of fho grnphs
In IIg. l6n. ThIs Is oxpocfod, sInco rnndom numborsofs nro bv
dofInIfIon, un-corroInfod wIfh nnv ofhor numbor sof.A compIofo sof
of compnrIsons IIko fhoso shown In IIgs. l5 nnd l6 woro porformod
on nIIfho TImoWnvo Zoro scroon sofs (Scroons l-l0) wIfh vorv
sImIInr rosuIfs. Tho corroInfIonrosuIfs for fho TWZ Scroon sof
compnrIsons rnngod from n Iow of 0.?3 fo n hIgh of 0.9839wIfh nn
nvorngo corroInfIon of 0.86, showIng fhnf fho sfnndnrd nnd rovIsod
TImoWnvosIn fhIs scroon sof woro romnrknbIv sImIInr. ThIs wns nof
fho cnso for ofhor TImoWnvosfhnf woro oxnmInod, whIch wIII bo shown
Infor. In ofhor cnsos of TineWote compnrIson,fho dIfforoncos
bofwoon fho sfnndnrd nnd rovIsod wnvos, npponrs fo show fhnf
fhoretieeJ TineWote oxprossos n Aotel/x procoss hnvIng boffor
nIIgnmonf wIfh knownhIsforIcnI procoss somofhIng ono wouId oxpocf
from n moro procIso formnIIznfIonprocoss. Moro nnnIvsIs Is
corfnInIv In ordor, buf fho dnfn fhus fnr sooms fo mnko
fhnfcnso.)LJXUHE(2) CompnrIsons for Ofhor SIgnIfIcnnf HIsforIcnI
IorIodsSovornI ofhor TImoWnvo porIods hnvIng hIsforIcnI
sIgnIfIcnnco woro oxnmInod forcompnrIson, buf fho fwo roporfod horo
nro fho porIods from l895-l925, nnd from l935-l955. Tho fIrsf
porIod IncIudos mnjor ndvnncos In phvsIcs nnd fochnoIogv, ns woII
ns nworId wnr; nnd fho socond porIod IncIudos fho dovoIopmonf nnd
uso of nucIonr wonpons,ns woII ns fwo mnjor wnrs. IIg. l? Is n
grnph of fho TImoWnvo compnrIson for fho l895-40l925 porIod, nnd
ngnIn fhoso pIofs nro romnrknbIv sImIInr In form. SovornI
sIgnIfIcnnfdnfos nro mnrkod wIfh groon nnd rod nrrows fo sIgnIfv
Aotel nnd HoIi/uol phonomonn.Tho fIrsf poworod fIIghf hnppons nf
KIffvhnwk on ocombor l?, l903; foIIowod bvIInsfoIn`s SpocInI Thoorv
of !oInfIvIfv (ST!) on Juno 30, l905; ConornI !oInfIvIfv Inl9l5,
nnd fho WorId Wnr I porIod of l9l4-l9l8. Tho ovonfs fhnf wouId bo
consIdorodnovoI (mnnnod fIIghf nnd bronkfhroughs In phvsIcs) nII
occur nf ovoIfv froughs orovoIfv dosconfs. Tho HnbIfunI phonomonon
(wnr), on fho ofhor hnnd, npponrs fo drIvowhnf sooms fo bo n vorv
novoI porIod, bnck Info hnbIf. Whon bofh novoI nnd
hnbIfunIphonomonon nro occurrIng sImuIfnnoousIv, fhov bofh
InfIuonco fho shnpo of fhoTImoWnvo. WWI mnv hnvo drIvon fho wnvo
furfhor Info hnbIf fhnn If dId, If If woron`ffor fho sImuIfnnoous
occurronco of vorv novoI phonomonn. Ior oxnmpIo, fho work on
fhoConornI Thoorv of !oInfIvIfv occurs In fho mIdsf of WorId Wnr I
wIfh Ifs snmo 'O!IhnbIfunI nnfuro. Tho moro novoI procoss of n
sIgnIfIcnnf ndvnncomonf In scIonfIfIcknowIodgo, ncfunIIv npponrs fo
suppross whnf wouId bo n mnjor nsconf Info hnbIf, nndncfunIIv
drIvIng fho wnvo Info novoIfv froughs.)LJXUHofIco fhnf fho sfnndnrd
TImoWnvo on fho Ioff doosn`f show fho rogrossIon Info hnbIfdurIng
fho IIrsf WorId Wnr fho rovIsod TImoWnvo cIonrIv doos. ThIs Is ono
cnso InwhIch fho rovIsod TImoWnvo npponrs fo provIdo n boffor
doscrIpfIon of fho ovoIfvprocoss fhnn doos fho sfnndnrd TImoWnvo.
Howovor, fhIs Is somofhIng fhnf shouId booxpocfod for n procoss
wIfh n moro procIso nnd consIsfonf mnfhomnfIcnI modoI.41IIg. l8
shows fho l9l5 fImo porIod, for whIch fho fwo wnvos oxhIbIf n
subsfnnfInIdIsngroomonf. WIfh fho oxcopfIon of n brIof fwo-monfh
porIod, fho sfnndnrd TImoWnvoshows n sfondv dosconf Info ovoIfv.
Tho rovIsod TImoWnvo, howovor, shows moro ofwhnf ono mIghf oxpocf
for n pInnof ombroIIod In gIobnI confIIcf. AddIfIonnIIv, fho
rovIsodTImoWnvo shows sovornI Insfnncos whoro fho doformInod mnrch
Info hnbIf Is oIfhorsIowod or fompornrIIv rovorsod; nnd wIfh fho
pubIIcnfIon of fho gonornI fhoorv In onrIvl9l6, fho IovoI of ovoIfv
bocomos foo gronf for fho forcos of hnbIf, nnd fho wnvo
pIungos.ThIs fIguro provIdos n good oxnmpIo of how fho sfnndnrd nnd
rovIsod TImoWnvos cnnoxhIbIf bohnvIornI dIvorgonco, nnd how fhIs
dIvorgonco fonds fo nffIrm fho Improvodnccurncv of fho rovIsod
wnvoform. !of us now fnko n Iook nf nnofhor porIod fhnf mosf ofus
nro fnmIIInr wIfh fho porIod fhnf IncIudos WorId Wnr II, nucIonr
onorgvdovoIopmonf, nnd fho Koronn Wnr.)LJXUHIIguro l9 shows fho
sfnndnrd nnd rovIsod TImoWnvo compnrIson grnphs for fho
porIodl935-l955, nnd fhoro nro obvIous sImIInrIfIos nnd cIonr
dIfforoncos bofwoon fho fwownvos. Iofh grnphs show fhnf WWII bogIns
nnd onds durIng sfoop nsconfs Info hnbIf,buf fhov doscrIbo somowhnf
dIvorgIng procossos, for much of fho mIddIo porIod of fhownr. Tho
rovIsod TImoWnvo shows fhnf n vorv novoI procoss Is nppnronfIv nf
work formuch of fho porIod of fho wnr. Tho sfnndnrd TImoWnvo doos
show novoI InfIuoncos, bufIf Is noIfhor ns consIsfonf nor drnmnfIc
ns for fho rovIsod TImoWnvo. Somo vorv pofonfnovoI procoss sooms fo
bo occurrIng durIng much of fho wnr porIod, nnd fhnf procoss mnvbo
supprossIng n mnjor nsconf Info hnbIf fhnf mIghf ofhorwIso bo
hnpponIng. CouId fhIsnovoI procoss bo fho dovoIopmonf of nucIonr
scIonco nnd fochnoIogv, ovonfunIIv IondIng42fo fho producfIon nnd
uso of nucIonr wonpons Thnf mnv bo nn offonsIvo nofIon, buf
Iof`sfnko n cIosor Iook nf If.Tho dovoIopmonf of nucIonr scIonco Is
ronIIv nbouf bocomIng moro nwnro nndknowIodgonbIo of n procoss fhnf
powors fho sun nnd fho sfnrs moro nwnro of jusf how nvorv poworfuI
nspocf of nnfuro works. Whnf ono fhon doos wIfh such knowIodgo Is
ndIfforonf procoss onfIroIv nnd InrgoIv n mnffor of conscIousnoss
nnd mnfurIfv. As wocnn soo from fho rovIsod TImoWnvo grnph, fho
momonf fhnf fhIs knowIodgo Is convorfodfo wonpons fochnoIogv fho
nucIonr oxpIosIon nf TrInIfv SIfo In ow MoxIco fho wnvobogIns n
sfoop nsconf Info hnbIf.Tho uso of fhIs nwosomo powor ngnInsf ofhor
humnn boIngs In HIroshImn nnd ngnsnkIoccurs shorfIv nffor fho fosf
nf TrInIfv SIfo, nnd occurs on n vorv sfoop nscondIng sIopo
ofhnbIf. Iorhnps fho procoss of bocomIng moro nwnro of nnfuro, nnd
oursoIvos Is vorvnovoI Indood. If Is fho sncrod knowIodgo of fho
shnmnn, who rofurns from nn ImmorsIonInfo nn nspocf of nnfuro, wIfh
guIdnnco or honIIng for hor or hIs poopIo. Wo soom fo hnvoIosf fho
sonso of sncrod knowIodgo wIfh Ifs nccompnnvIng rosponsIbIIIfv,
somowhoronIong fho wnv. Iorhnps If Is fImo fo rognIn fhnf sonso,
nnd rocInIm rosponsIbIIIfv for ourknowIng.)LJXUHTho rovIsod
TImoWnvo of IIg. l9 nIso shows fho porIod of fho Koronn wnr ns n
vorv sfoopnsconf Info hnbIf, nIfhough somofhIng occurrIng onrIv In
l952 dId momonfnrIIv rovorsofho hnbIfunI
frond.43&RUUHODWLRQ'DWDDQG7LPH:DYH&RPSDULVRQVCorroInfIon
nnnIvsIs wns porformod for nII fho dnfn sofs compnrod In fhIs
roporf, ns woIIns fho romnInIng oIghf TWZ scroon sofs nof shown
horo, nnd soIocfod fImo porIods. ThIsfvpo of nnnIvsIs nIIows us fo
oxnmIno fho roInfIonshIp bofwoon dnfn sofs, nnd osfImnfofhoIr
dogroo of Infordopondonco I.o. how sImIInr fhoIr InformnfIon
confonf Is. ThorosuIfs of fhoso nnnIvsos nro shown grnphIcnIIv In
IIg. 20, nnd fhov IncIudo fho fonTineWote scroons IncIudod wIfh fho
TWZ soffwnro, nIno soIocfod hIsforIcnI wIndows,nnd fho 384 numbor
dnfn sofs. In nII cnsos shown, fho rovIsod nnd rnndom dnfn sofs
nroboIng corroInfod (compnrod) wIfh fho sfnndnrd dnfn sof. SInco
nnv numbor sof corroInfodwIfh IfsoIf, hns n corroInfIon cooffIcIonf
of ono, fho bIuo IIno nf fho fop of fho grnphroprosonfs fho
sfnndnrd dnfn soIf-corroInfIon.!ocnII fhnf n corroInfIon of l
sIgnIfIos numbor sofs fhnf hnvo IdonfIcnI InformnfIonconfonf, n
corroInfIon of zoro sIgnIfIos no common InformnfIon confonf, nnd n
corroInfIonof l monns fhnf fho numbor sofs InformnfIon confonf
oxhIbIf mIrror Imngo bohnvIor wnvo ponks fo wnvo vnIIovs ofc. Tho
groon IIno In fho grnph shows fho dogroo ofcorroInfIon bofwoon fho
rovIsod wnvoform nnd fho sfnndnrd wnvoform, for onch of fhosopnrnfo
TImoWnvos fhnf woro oxnmInod. Tho rod IIno shows fho corroInfIon
IovoIbofwoon wnvos gonornfod bv fho rnndom soodod dnfn sofs, nnd
fhoso gonornfod bv fhosfnndnrd dnfn sof. Tho fIrsf poInf of onch
IIno, Is fho corroInfIon cooffIcIonf for onch of fho384 numbor dnfn
sofs oxnmInod ronJon, retieeJ, nnd e/onJorJ dnfn sofs.)LJXUH44Tho
fIrsf fonfuro fo nofIco nbouf fho retieeJ nnd e/onJorJ dnfn sof
corroInfIons shown InIIg. 20, Is fho fncf fhnf fho rovIsod 384
numbor dnfn sof shows n corroInfIon wIfh fhosfnndnrd numbor sof of
nbouf 60 - n compnrIson fhnf Is shown In IIg. l2. ThIs Is
nsIgnIfIcnnf cross-IInkIng of InformnfIon confonf, buf somofhIng
fhnf ono mIghf oxpocf fornumbor sofs wIfh n common bnso nnd vorv
sImIInr dovoIopmonfnI procoduros. Tho noxffonfuro of sIgnIfIcnnco
Is fho fncf fhnf fho corroInfIon bofwoon fho retieeJ nnd
e/onJorJTineWotee, for nII fon TWZ scroon sofs, Is boffor fhnn ?0
nnd ns hIgh ns 98, showIngn vorv hIgh IovoI of Infordopondonco. Tho
fImo porIods roprosonfod bv fhoso fonTImoWnvo scroons, rnngos from
4 vonrs fo 36,000 vonrs, whIch Is InboIod on fho grnph.Tho durnfIon
of fhoso TImoWnvo porIods mnv hnvo n bonrIng on fho IovoI of
corroInfIon,ns wo shnII soo In n momonf.IogInnIng wIfh fho porIod
l895-l925, fho grnph shows moro scnffor In fho corroInfIonbofwoon
sfnndnrd nnd rovIsod dnfn sofs, whIch rnngos from nbouf 98 down fo
8, wIfhono nnfI-corroInfIon of 95. ofIco fhnf fho corroInfIon
npponrs worso for vorv shorffImo porIods, ono fo fwo monfhs or so.
Ono possIbIo oxpInnnfIon Is fhnf fho vorv shorffImo porIod
TImoWnvos nro gonornfod bv n vorv fow dnfn poInfs In ofhor words n
Iowwnvo snmpIIng froquoncv or rnfo. A smnII, nnd undor-snmpIod
Inpuf dnfn sof wouId nddn hIghor IovoI of noIso fo fho wnvo sIgnnI,
nnd consoquonfIv produco fho hIghor dnfnscnffor obsorvod. Tho
snmpIIng fhoorom, from InformnfIon fhoorv, sfnfos fhnf
nIInsIng(noIso gonornfIon) bogIns fo occur whon fho sIgnnI snmpIIng
rnfo bocomos Ioss fhnn fwIcofho hIghosf froquoncv compononf of fho
snmpIod sIgnnI. ThIs Is corfnInIv somofhIng fhnfmnv bo occurrIng In
fho mnfhomnfIcs of TImoWnvo gonornfIon.AddIfIonnIIv, ns monfIonod
provIousIv, fhIs dIfforonco couId bo fho consoquonco of hnvIngnn
Improvod modoI of fho procoss. If Is Imporfnnf fo romombor fhrough
nII of fhIscompnrIson nnnIvsIs, fhnf fho sfnndnrd dnfn sof Is
gonornfod bv n procoss wIfh ImboddodfInws - nof onough fo dosfrov
fho InformnfIon confonf of fho wnvo sIgnnIs, buf onough focnuso
somo dIsforfIon of fhnf InformnfIon confonf. ThIs corroInfIon
nnnIvsIs IsInforosfIng, prImnrIIv bocnuso If Ionvos fho sfnndnrd
TineWote Infncf, moro or Ioss buffho Imporfnnf poInf fo romombor Is
fhnf ovon wIfh Iow corroInfIon fho retieeJ Jo/o ee/npponrs fo
produco n boffor TineWote.If Is probnbIo fhnf fho vnrInfIons wo
obsorvo In IIg. 20 nro fho rosuIf of Io// fhodIsforfIon of fho
InformnfIon confonf of fho 384 numbor Jo/o ee/, ns n rosuIf
ofmnfhomnfIcnI orrors, onJ fho Iow dnfn wnvo snmpIIng rnfo fhnf
occurs for shorf durnfIonTineWotee (nn unoxnmInod buf pInusIbIo
fhosIs). If Is nIso Imporfnnf fo poInf ouf horo,fhnf whon wo do soo
sIgnIfIcnnf dIfforoncos In fho TImoWnvos gonornfod bv fho
e/onJorJnnd retieeJ dnfn sofs, fhoso dIfforoncos hnvo rovonIod n
retieeJ TImoWnvo of gronfornccurncv. Howovor, If Is Imporfnnf fhnf
wo oxnmIno n sIgnIfIcnnf vnrIofv of nddIfIonnITImoWnvo porIods, fo
gnfhor moro sfnfIsfIcs on fho funcfIonIng of fho rovIsod wnvo;
buffho dnfn In hnnd so fnr, soom fo bo suggosfIng fhnf fho
mnfhomnfIcnI formnIIznfIon of fhodnfn sof gonornfIng procoss, doos
Improvo fho TineWote nccurncv.Anofhor sIgnIfIcnnf fonfuro of fho
rovIsod dnfn corroInfIon pIof In IIg. 20 fhnf shouId bomonfIonod
horo, Is fho fncf fhnf fho corroInfIon cooffIcIonf for fho l9l5
porIod Is nonrIv -l, sIgnIfvIng nn nnfI-corroInfIon or mIrror Imngo
roInfIonshIp bofwoon fho wnvos. ThIs Is45fho TImoWnvo compnrIson
fhnf Is shown If IIg. l8. If ono woro fo pInco nn
ImngInnrvfwo-sIdod mIrror bofwoon fho sfnndnrd nnd rovIsod TImoWnvo
grnphs, fhon fhorofIocfIon on oIfhor sIdo of fho mIrror wouId
cIosoIv rosombIo fho wnvo fhnf Is on fhoofhor sIdo honco fho
doscrIpfIon of nnfI-corroInfIon ns n mIrror Imngo roInfIonshIp.AIso
nofIco, fhnf n groon doffod IIno mnrks fho nvorngo of nII fho
sfnndnrd/rovIsod wnvocorroInfIons nf nbouf ?0.Tho rod IIno of IIg.
20 shows fho corroInfIon of fho rnndom numbor gonornfod wnvos,wIfh
fho sfnndnrd dnfn sof. Iv dofInIfIon, fho rnndom dnfn sofs shouId
show IIffIo or nocorroInfIon wIfh oIfhor fho sfnndnrd or rovIsod
dnfn sofs, nor wIfh nnv ofhor rnndomnumbor sof. In sovornI cnsos In
IIg. 20, fhIs furns ouf fo bo fruo, buf fhoro nro nIsosovornI cnsos
In whIch fho rnndom sof corroInfIon Is nof nonr zoro, ns ono wouId
oxpocf.In gonornI, fho rod IIno pIof of IIg. 20 shows n much Iowor
IovoI of corroInfIon wIfh fhoe/onJorJ numbor sof fhnn doos fho
retieeJ sof ns oxpocfod. Inch dnfn poInf on fho rodIIno, howovor,
Is ncfunIIv nn nvorngo of oIfhor fwo, or sovon rnndom numbor
sofcorroInfIons. In ofhor words, oIfhor fwo or sovon rnndom numbor
corroInfIons woronvorngod fo produco onch poInf on fho rod IIno
grnph. If furns ouf fhnf mosf of fhosIxfoon corroInfIon poInfs
producod bv nvorngIng onIv fwo rnndom sofs, hnvo much moroscnffor
fhnn do fho four poInfs producod bv nvorngIng sovon rnndom sof
corroInfIons.Tho 384 numbor ronJon Jo/o ee/, nnd fho porIods
l895-l925, l905, nnd l9l5, woro nIIproducod bv nvorngIng sovon
rnndom sof corroInfIons. Tho vIoIof doffod IIno runnIngfhrough fho
rnndom numbor sof corroInfIons, Is fho nvorngo corroInfIon IovoI
for nII fhornndom sofs shown, nnd If shows n vorv Iow nvorngo
corroInfIon of nbouf 5.If Is nIso possIbIo fhnf fho snmo procoss
proposod for producIng fho Inrgor corroInfIonscnffor of fho rovIsod
dnfn sof, couId bo nf work for fho rnndom dnfn sofs I.o.
shorfdurnfIon fImo porIods wIfh Iow snmpIIng froquoncIos, couId bo
cnusIng dnfn scnffor duofo noIso. If n smnII numbor of fho 384 Jo/o
file poInfs nro usod fo gonornfo n e/or/ perioJTineWote, fhon fhoro
Is n much hIghor probnbIIIfv of corroInfIon bofwoon fho rnndomsofs
nnd fho TImoWnvo numbor sofs. WIfhouf furfhor InvosfIgnfIon,
howovor, fhIs Is nspocuInfIvo, If pInusIbIo fhosIs.Tho grnphs of
IIg. 20 do show fhnf fho e/onJorJ nnd retieeJ dnfn sofs nnd
fhoIrdorIvnfIvo TineWotee nro romnrknbIv woII corroInfod. In fho
rogIons whoro fhocorroInfIon wonkons, or bronks down onfIroIv, fho
rovIsod TineWote npponrs fo show novoIfv procoss fhnf Is In cIosor
ngroomonf wIfh known hIsforIcnI procoss. In nddIfIon,fho pIofs In
IIg. 20 mnv bo rovonIIng n procoss whorobv shorf porIod TineWotee
producosnmpIIng noIso fhnf wonkons fho corroInfIon. ThIs dnfn
supporfs fho vIow, fhnf fhoInformnfIon confonf of fho e/onJorJ
TineWote Is somowhnf dIsforfod, buf nof dosfrovod;nnd suggosfs fhnf
fho retieeJ TineWote nnd Ifs pieceuiee lineor func/ion Is nbIo
focorrocf fhIs dIsforfIon, nnd provIdos nn Improvod oxprossIon of
fho ovoIfv procoss.&RQFOXGLQJ5HPDUNVTho dovoIopmonf of fho 384
numbor Jo/o ee/ from fho sof of Iire/ OrJer of Difference(IO)
Infogors hns boon oxprossod ns n procoss fhnf Is pieceuiee lineor
In nnfuro. ThIsprocoss InvoIvos fho combInnfIon nnd oxpnnsIon of
sfrnIghf-IIno sogmonfs, whIch cnn bo46oxprossod mnfhomnfIcnIIv ns n
pieceuiee lineor func/ion. Tho e/onJorJ Jetelopnen/ hnsboon
doscrIbod bv McKonnn nnd Movor In fho TineWote Zero documonfnfIon
nnd Inofhor roporfs. Iuf fhIs procoss IncIudos n procodurnI sfop
cnIIod fho hnIf fwIsf, fhnf Isnof consIsfonf wIfh fho sfrucfuro of
pIocowIso IInonr mnfhomnfIcs; nnd consoquonfIvproducos n dIsforfIon
of fho IO InformnfIon confonf. WnfkIns oInbornfod on fhIs Insomo
dofnII, In hIs woII-documonfod oxposo on fho nnfuro of fho /olf
/uie/, In whIch hodoscrIbod fho dIsforfIons nnd InconsIsfoncIos
InvoIvod. Ho fhon concIudod fhnf fhIsdIsforfIon wouId rondor fho
TineWote monnIngIoss, ns n ronIIsfIc grnphIcnI dopIcfIon offho
Aotel/x procoss ns hnd boon doscrIbod bv McKonnn. I mnInfnIn fhnf
fhIs concIusIonwns promnfuro, nnd nppnronfIv Incorrocf.Tho retieeJ
Jetelopnen/ of fho 384 numbor Jo/o ee/ IncIudos fho uso of
mnfhomnfIcs fhnfcorrocfIv oxprossos fho pieceuiee lineor
dovoIopmonf procoss, nnd fhoroforo producos nnundIsforfod oxpnnsIon
of fho IO numbor sof. Tho TineWote fhnf rosuIfs from fhIsoxpnnsIon
procoss, Is fhon nn nccurnfo rofIocfIon of fho IO numbor sof,
provIdod fho sofcnn bo doscrIbod or modoIod bv n pIocowIso IInonr
funcfIon. Tho pieceuiee lineor func/iondoscrIbod horo, mnv onIv bo
nn npproxImnfIon fo somo moro conplex func/ion fhnf hnsvof fo bo
found. In fncf, I wouId nrguo fhnf fhIs Is quIfo IIkoIv for n
phonomonon orprocoss of fhIs nnfuro, whIch furfhor sfudv mnv shod
somo IIghf on. If wo nssumo fhnffho retieeJ TineWote Is n
ronsonnbIv nccurnfo rofIocfIon of fho InformnfIon confonf offho IO
numbor sof, fhon fho e/onJorJ TineWote shouId hnvo n dogroo of
nccurncvproporfIonnI fo Ifs dogroo of corroInfIon wIfh fho retieeJ
TineWote. As wo hnvo soonfhus fnr, fhoso fwo TineWotee show nn
oteroge correlo/ion of nbouf ?0, so fhnf fhoe/onJorJ uote hns nn
nvorngo nccurncv of nbouf ?0 whon compnrod wIfh fho retieeJuote.
Howovor, wo hnvo nIso soon fhIs corroInfIon ns hIgh ns 98, or ns
Iow ns 6, wIfhono cnso of n mIrror Imngo or nnfI-corroInfIon of
0.94.ThIs work hns sorvod fo cInrIfv nnd formnIIzo fho procoss bv
whIch fho 384 numborTImoWnvo Jo/o ee/ Is gonornfod. ThIs hns boon
dono bv showIng fhnf fho procoss IsdoscrIbnbIo wIfhIn fho frnmowork
of pIocowIso IInonr mnfhomnfIcs In gonornI, nnd vocformnfhomnfIcs
In pnrfIcuInr. Inch sfop hns boon doIInonfod nnd
formnIIzodmnfhomnfIcnIIv, fo gIvo fho procoss cInrIfv nnd
confInuIfv. Tho formnIIzod nnd rovIsoddnfn sof sorvos ns fho
foundnfIon of fho TineWote gonornfod bv fho TineWote Zerosoffwnro,
whIch Is vIowod ns n grnphIcnI dopIcfIon of n procoss doscrIbod bv
fho obb nndfIow of n phonomonon cnIIod Aotel/x. Aotel/x Is fhoughf
fo bo fho bnsIs for fho cronfIonnnd consorvnfIon of hIghor ordorod
sfnfos of compIox form In nnfuro nnd fho unIvorso.Tho rosuIfs
roporfod horo mnko no fInnI cInIms ns fo fho vnIIdIfv of fho
TineWote ns If Isoxprossod bv Aotel/x T/eorx, nor doos If cInIm
fhnf fho curronf TineWote Is fho bosfdoscrIpfIon of fhIs Aotel/x
procoss. If doos show fhnf fho propor mnfhomnfIcnI fronfmonfof fho
IO numbor sof, producos n TineWote fhnf npponrs fo bo moro
consIsfonf wIfhknown hIsforIcnI procoss. ThIs consIsfoncv Is
gonornI, howovor, nnd moro work noods fobo dono fo oxnmIno fho
spocIfIc rofIocfIons or projocfIons fhnf fho TImoWnvo mnv
borovonIIng. If Aotel/x T/eorx Is n vnIId hvpofhosIs, rofIocfIng n
ronI phonomonon Innnfuro, fhon ono wouId oxpocf fhnf If Is
vorIfInbIo In spocIfIc wnvs.If hns nIso soomod npproprInfo fo
oxnmIno somo of fho sfops In fhIs wnvo dovoIopmonfprocoss In forms
of fhoIr corrospondonco wIfh oIomonfs of phIIosophv nnd scIonco.
Tho47fIow of YIn nnd Ynng onorgv rofIocfod In fho oxprossIon of fho
forwnrd nnd rovorso bI-dIrocfIonnI wnvos, for oxnmpIo, fInds
phIIosophIcnI corrospondonco In n nnfurnI cvcIo
ofIIfo-donfh-robIrfh, or In fho procoss of fho shnmnnIc journov
ImmorsIon, ongngomonf,nnd rofurn. Corrospondonco cnn nIso bo found
In scIonco, In fho fIoIds of cosmoIogv,nsfronomv, nsfrophvsIcs, nnd
qunnfum phvsIcs fho IIfo cvcIos nnd mofIon of honvonIvbodIos,
qunrks, nnd unIvorsos; fho hnrmonIc nnd hoIogrnphIc nnfuro of IIghf
nnd wnvomochnnIcs; nnd fho cvcIIc frnnsformnfIon of mnffor fo
onorgv, nnd onorgv fo mnffor. ThorofIocfIon of nII nnfurnI
phonomonn nnd procossos ovor fho confInuum of oxIsfonco, fromfho
smnIIosf scnIos up fo fho Inrgosf scnIos, musf suroIv IncIudo
whnfovor procoss IsoccurrIng In fho I-ChIng ns woII. Tho quosfIon
Is, nro wo nro cIovor nnd conscIous onoughfo docIphor nnd oxpross
If corrocfIv nnd npproprInfoIv$FNQRZOHGJHPHQWVI wouId IIko fo fhnnk
Terence McKenna, for brIngIng fhIs InfrIguIng nnd provocnfIvo
nofIonInfo fho coIIocfIvo, nnd for fho courngo nnd forosIghf shown,
bv hIs wIIIIngnoss fo oponhImsoIf nnd hIs Idons fo scrufInv nnd
boundnrv dIssoIufIon. If fhoro Is nnv roIovnnco ormonnIng fo bo
found In fho TineWote or Aotel/x T/eorx, fhon If Is suroIv
somofhIng fhnfIs Inrgor fhnn ho, or nnv of us; nnd If Is nIso
somofhIng fhnf Is proporIv In fho domnIn ofnII humnn oxporIonco,
wIfh onch of us n wIfnoss, pnrfIcIpnnf, nnd confrIbufor.I wouId
nIso IIko fo oxpross mv fhnnks nnd npprocInfIon fo Mathew Watkins
for hIs work InoxposIng fho mnfhomnfIcnI InconsIsfoncIos, vngnrIos,
nnd procodurnI orrors of fhosfnndnrd TineWote Jo/o ee/ dovoIopmonf,
nnd chnIIongIng n fhoorv fhnf mnv hnvobocomo fnr foo sodonfnrv nnd
Inbrod for Ifs own good. Whnfovor fho fInnI oufcomo of fhIsondonvor
of Aotel/x T/eorx, ho hns sof fho onforprIso on Ifs propor courso
of opon nndcrIfIcnI InquIrv.I nm nIso gronfIv Indobfod fo Peter
Meyer for hIs skIII nnd forosIghf In cronfIng n TWZsoffwnro pnckngo
fhnf Is fIoxIbIo, nccossIbIo, nnd frIondIv fo fho sorIous
InvosfIgnfor.WIfhouf hIs OS vorsIon of TImoWnvo Zoro soffwnro, fhIs
work wouId hnvo boon muchmoro dIffIcuIf If nof ImpossIbIo. Ho hns
cronfod n soffwnro pnckngo fhnf mnkos fhosonofIons ronIIsfIcnIIv
fosfnbIo, In n roInfIvoIv sfrnIghfforwnrd mnnnor. ThIs mndo
IfpossIbIo for mo fo oxnmIno fho offocfs of fho rovIsod dnfn sof on
fho TImoWnvo IfsoIf, nswoII ns fncIIIfnfIng fho oxnmInnfIon of fho
dofnIIod sfrucfuro of fho wnvo In work fofoIIow.Mv fhnnks nIso fo
Dan Levy for hIs offor fo pubIIsh fhIs work on hIs !ovIfv sIfo, ns
woII nshosfIng nn upcomIng TineWote mnfhomnfIcnI nnnox fo Aotel/x
T/eorx. I wnnf nIso foncknowIodgo Brian Crissey nf IIuo Wnfor
IubIIshIng for hIs hoIp In InfogrnfIng fho nowprocoss Info fho
TImoWnvo Zoro soffwnro pnckngos nnd documonfnfIon. 1 Computer
Software program written by Meyer and others, based on a
mathematical relationship exhibited by the I-Ching, formulated and
reported by T. McKenna and D. McKenna, the Invisible Landscape,
Harper San Francisco, 1993,p. 1212 T. McKenna, the Invisible
Landscape, p. 1403 M. Watkins, Autopsy for a Mathematical
Hallucination, Terence McKennas Hyperborea at www.levity.com48 4 T.
McKenna, Time Explorer Manual, p60, the Invisible Landscape, pp.
140-1425 P. Meyer, http://www.magnet.ch/serendipity/twz/kws.html6
DeltaPoint, Inc., 22 Lower Ragsdale Dr., Monterey, CA 93940, (408)
648-40007 Microsoft Corp., One Microsoft Way, Redmond, WA 980528
McKenna, TimeExplorer Manual, PP. 60-63,
http://www.levity.com/eschaton/waveexplain.html9 H.B. Anderson,
Analytic Geometry with Vectors, p71, McCutchan Publishing Corp.,
Berkeley, Ca. 196610 T. McKenna, TimeExplorer software manual, pp.
62-6311 P. Meyer, TimeExplorer software manual, pp. 85-9112 M.
Kaku, What Happened BEFORE the Big Bang?, Astronomy, May 1996, pp.
34-41[John Sheliak] [email protected][Terence McKenna]
[email protected][return to Levity]
http://www.levity.com/eschaton/Filename: MathWave PaperIIDirectory:
D:\DG4\TWZ FilesTemplate: C:\Program
Files\MicrosoftOffice\Templates\NORMAL.DOTTitle: Delineation,
Specification, and Formalization of the TWZData Set Generation
Process - Philosophical, Procedural, and
MathematicalSubject:Author: John SheliakKeywords:Comments:Creation
Date: 11/11/97 9:50 PMChange Number: 2Last Saved On: 11/11/97 9:50
PMLast Saved By: John SheliakTotal Editing Time: 1 MinuteLast
Printed On: 11/11/97 9:51 PMAs of Last Complete PrintingNumber of
Pages: 48Number of Words: 13,802 (approx.)Number of Characters:
78,672 (approx.)
McKenna's TimewaveConventional King Wen Tabulation000 110 101
011 111 001 010 100000 1 34 5 26 11 9 14 43110 25 51 3 27 24 42 21
17101 6 40 29 4 7 59 64 47011 33 62 39 52 15 53 56 31111 12 16 8 23
2 20 35 45001 44 32 48 18 46 57 50 28010 13 55 63 22 36 37 30 49100
10 54 60 41 19 61 38 58The I Ching is an ancient chinese oracular
system whereinsix coins (or similar) ar consulted to obtain an
allegedlymystically-relevant maybe-not-random number in the range0
to 63 inclusive, known as a hexagram.The (binary) bits of this
number (or hexagram) areconventionally represented as either broken
or unbrokenhoriontal lines stacked vertically. Hexagrams are
oftenconsidered as the combination of two three-bit trigrams.The
traditional ordering of the sixty four "hexagrams" isusually
attributed to King Wen circa 1150 BC.This ordering, essentially one
of 64! > 1089 permutations ofthe set Z64 = {0,1,2,....,63} is
the numerical starting point ofTerrance McKenna's TimeWave theory.
I will write W(i) forthe i'th element of the cyclic King Wen
Ordering, startingwith i=0 and with the understanding that W(i) =
W(i Mod64) for i > 63 and for i