The basic symmetry of I Ching
B. Svarog
We suggest that the
original symbolic system of I ching
was designed on the same principles that are assumed to be a basis of the kalavada
system‚ which elements had reached us through the teachings of Pythagoreans and
Plato, as well as through esoteric cosmogonies of archaic epoch (fixed‚ in
particular‚ in the first chapters of Genesis).
First of all let us define
an uncomplicated operation, the result of which is a third gua (an I chin drawing) made
from two others. So‚ if a strong line Yang,
enclosed in a given position, plays a role of a switch on a relay principle ("there
is a signal") and changes the value of the latter into the opposite one, than
a weak line Yin expresses an absence
of a change in a given position ("no signal").
Such interaction of the
lines answers the boolean law of logical adding:
0
+ 0 = 0,
0
+ 1 = 1,
1
+ 0 = 1,
1
+ 1 = 0.
The difference from the usual
arithmetical adding here is concluded in the fact that two units together give
a zero‚ i.e. the following rank is not taken into account (in the binary scale
of notation 1+1=10).
For the simplicity we
shall say‚ that A applies to B‚ and herewith we get C:
A
(B) = C.
The
law of transposition A (B) = B (A) = C, as well as inverse relations: À (Ñ) = Â è Â (Ñ) = À are equitable for
the operation given.
If the sequence of
trigrams in binomial "order of generating" (Table 1) is taken as a source, the operation Kun (
), applied to
these sequences, leaves all trigrams without changing ("a unity
element"), but the operation Kian (
) transforms all guas into
their opposites, reversing the source order of trigrams ("an inverse
element")*:
Table
1.
The rest six trigrams,
changing one lines and leaving without changing the other, form from the source
order (1) the following derived
sequences (Table 2):
Table 2.
From brought above it becomes
clear, that if opposites (A-O) are linked by Kian (eighth position), the
parallel (A-P) - by 
(third position), the coordinated Ê1 and Ê2 (À - Ê1 è À - Ê2) are linked with the source trigram A by Jen
and Dui 
accordingly (fifth and seventh positions).
If two trigrams are opposites, their co-ordinates are in opposition too, but if
they are parallel, their co-ordinates coincide and herewith are parallel themselves.
The way of location of hexagrams
in families or "houses" by Khan-dynasty scientist Ching Fan (77- 33 B.C.) was based
on trigrams ÀÎÏÊ1Ê2 relations of. Ching
Phang was also known for his work of the musical
theory lui (the quint system).
When marking a source
trigram by the letter A the family of eight hexagrams about the given trigram is
formed up as follows**:
It is easy to see‚ that
eight relations between five forming trigrams ÀÎÏÊ1Ê2 (corresponding to
number of five elements) in each of
Ching Fan’s families in turn are no other than eight trigrams - Table 3.
We are sure‚ that similarly to this order‚ shown in families‚ a certain
end-to-end relation must define the sequence of all the 64 I ching hexagrams.
Table
3.
Taken for the base a
square 8x8 trigrams in binomial "order of generating" (1)‚ corresponding to Former sky‚ and "having multiplied"
each trigram in a column by each trigram in a line in accordance with the above
mentioned logical procedure (Tables 1
and 2)‚ we receive the pattern of matrix
8õ8, characterizing these
secondary relations between trigrams.
For painting the chequers we will use colours of five elements‚ and
different tones of a colour in the case, when one and the same element
corresponds to two trigrams - Fig.
1.
In Fig. 1.
trigrams Kian and Dui answer an element of Metal (blue and white
colours of squares), trigrams Kun and
Gen - an element of Earth (yellow and orange colour), trigrams Sun and Jen - an element of Tree (green and blue colour), trigram Li - an element of Fire (red)‚ and
trigram Kan - of Water (black).
These correspondences answer the accepted order of elements in ba gua of the Former and Following sky
and Lo shu - the "9 fields" square.
In the Chinese cosmology Earth
and Metal mark the two diametrically-opposite principles - passive and
active‚ or lower and upper limits of the world’s Octave. Besides, the element Metal
can be identified with Ether ("Heaven")
- quintessence of
neo-Platonists‚ as well as the Spirit
of the near-East and Christian traditions; the passive origin in the latter
corresponds to the deep and the waters:
"In the
beginning God created the heavens and the earth.
Now the earth was formless and empty.
Darkness was on the surface of the deep.
God’s
Spirit was hovering over the surface of the waters"
(Gen. 1:1, 2).
In 
) - otherwise "Creation"‚
the element of Metal (Ether) and the sign of the odd (“strong line” Yang and triad), "waters" of Genesis correspond to Earth = Kun (
) - "Accomplishment"‚
and the sign of the even (“weak line”
Yin and duad).
It is obviously, that one counter
position (Heaven and the Earth, the odd and the even) is insufficient for the description of
"variety of all the things", and therefore other symmetry relations
would to be generated from the base of these principles.
But earlier determined by
us logical operation between two source trigrams brings once again to the same Kian and Êun:
The outlet from the dead
end ancient theorists saw in making a symmetry of axial type (a cross) as a
fetus of lengthiness - by the image of that
what the Pythagoreans named tetrax,
but the Chinese - Four images (Si sian):
The twin counter position Kian - Êun (Heaven - Earth) and Li - Kan (Fire -Water) is fixed by order of the Former and the Following
sky ba gua. Let’s take into account,
that trigrams Li and Êàn possess the same characteristic of internal symmetry
as Kian and Êun: they are not changed by turning over (reverse)‚ and presenting the opposites - pass over to their pair by changing lines (yin « yang). At the same time Li turns over (reverse)
any asymmetrical trigram‚ but Êàn transforms a
trigram into its parallel (reverse opposite) - Table
2. Relations in the first quadruplets are expressed as follows:
"Multiplying" Li and Êàn returns us once
again to a Kian (inverse element). However,
it is impossible to say that the principle Êun
does not play a role in the first quadruplet: though "multiplying" by
Êun gives the same trigram (that
corresponds to an identical element) ‚ the "inwardly symmetrical"
trigrams Êàn and Li could not be formed without an introduction of weak lines. Otherwise,
the opposition of Kian «Êun 
marks the first symmetry ration, on the basis
of which all the rest rations are formed.
It is inquisitively, that
if we take numeric (binary) values of these trigrams, they will give the
follows:
The
vertical axis forms Êun - do
(0‚ as well as 8 =1000 in the binary scale of notation‚ that is to say an octave)
- and the seventh step
si = 7: by them are determined
extreme limits of an îctave space ("Heaven - Earth")***. But on a
horizontal axis the “dynamic” 2 and 5 (in the amount giving
7) are located, i.e. the fifth step = quint (31) ‚ and the second step = whole tone (32) ‚ corresponding to the first
segments of octave interval and the base of development of all following harmonic
numbers (Pythagorean tetrax). We also
know, that the quint answers a harmonic ration 7 : 5 (3.5 tone to 2.5
tone). In the musical correspondences of ba gua Li (
) - Êàn(
) also give a quint ration.
The Li trigram is compared to element Fire and quality of the light
(red colour of squares in Fig.1),
the trigram Êàn - to element Water and the dark (black colour of squares). Genesis in the first chapter denominates the relation of a second
pair of trigrams by following verses:
" God said, "Let there be light," and there was light.
God saw the light, and saw that it was
good. God divided the light from the
darkness.
God called the light Day, and the
darkness he called Night. There was evening
and there was morning, one day"
(G.
1: 3-5).
In this first aeon (cycle) of Creation appear three quint
steps (fa, do, sol) and the first quadruple
(tetrax) of an Octave (îctave interval‚ a quint‚ a quart and whole tone). The following
"numbers of creation" or the rest of the notes of a diatonic gamma (“Tetrax
B”, by A.Losev) must comprise with necessity the beginnings
of the first tetrax.
In the genitive matrix of
64 hexagrams of Fig. 1‚ containing
colours of trigrams and corresponding them elements‚ eight of going on the perimeter
squares 2x2 contain red and black cells Li
and Êan (Fire and Water), while the central and angular - only the colours
of source elements Earth and Metal. These angular squares note the
cardinal perpendicular axes in the 12-step circle of the Octave: fa – si and rå – lab (sol#) - its quaternary pattern‚ while the squares,
including colours of Li and Êàn‚ correlate with both main diatonic and chromatic changed
notes. Only red and black cells from those going on periphery adjoin to angular
squares of Heaven and Earth.
But in order to form a
second quadruplex and to complete a seven step gamma of an octave one more
condition of symmetry is required, which the already received four trigrams do
not possess. Corresponding to them two main procedures - a formation of an îpposite trigram (change all the line into
opposites‚ or operation Kian), as
well as a reverse trigram (turning them
over‚ or operation Li) do not give a
way to create four staying trigrams. The
transformation is coming to rescue, known from the divinatory practice of I ching as building of the internal
hexagram (chu gua):
In this way of structure
realignment of hexagrams two upper lines
are taken from the lower and one lower line from the upper trigram, and then the
upper line from the lower and two lower lines from the upper trigram.
Only in this way four trigrams of the second tetrax are to be received, which possess one more, additional to the
first two, parameter of asymmetry: their reversing brings to another
trigram. Three differences (or base kind
of asymmetry)‚ applied together‚ are capable now to give five types of main
relations between trigrams (Table 3):
(A) - "the unity
element", (O) - complementary
opposition or "the inverse element", a parallel (P) and two "co-coordinated"
(Ê1 and Ê2):
Three types of internal symmetry
generate five relations ÀÎPÊ1Ê2 - corresponding to a
number of five elements - which‚ therefore‚
generate about eight trigrams in each of Ching Fan’s family of hexagrams. As
far as the generating matrix of I ching
itself comprises three, five and eight beginnings (accordingly to the first three
levels of Octave numbers), these principles enter then as the patterns into all
processes, resulting in empirical time gaps, according to the ideology of founders
of the doctrine about Changes (I).
It is possible to make sure
that similar "abstract algebra" is far from being trivial; however,
its explications were already in use a millennium B.C.
Four trigrams of the second
tetrax are bound by relations of four trigrams of the first tetrax:
In the esoterism of Generation the stage of forming of the second
tetrax (building the seven-step gamma
of first Cosmos) is denominated by following words:
"God said, "Let there be an expanse in the midst of the waters, and
let it divide
the waters from the waters.
God made the expanse, and divided the
waters which were under the expanse
from the waters which were above the expanse, and it was so.
God called the expanse sky. There was
evening and there was morning, a
second day"
(G. 1: 6-8).
«An expanse in the midst of the waters» in this instance
corresponds to a trigram Gen 
- “Mountain”
- otherwise a Cave
- and to the Earth element. In Fig. 1 squares of Gen are
painted orange, and they attendant to walking diagonally from the left lower
corner yellow squares of Êun‚ also designating Earth.
The opposite Gen is to be Dui 
- "
The opposition Kian ─ Êun - "Heaven─Earth"
is parallel to the opposition Li─Êàn - "Fire─Water" or "Light - Dark". In Fig. 1. red squares of Li (Light) just adjoins
to white squares of Dui and blue squares of Kian (element Metal) in
upper-left and lower-right sectors, while the black squares of Êàn (Dark) are
connected with orange squares of Gen and yellow squares of Êun (element Earth)
in the lower-left and upper-right sectors of diagram.
It is necessary to say
about the last two trigrams Sun
- "Wind" or Motion (Energy)‚ and Jen 
- "Thunder" or Message‚ both marked by the
element Tree (green and blue squares
of diagram). Their position in the matrix of 64 hexagrams (Fig.1) is distinguished from earlier considered Dui and Gen, to which they are turned over (reversed). If the latter accompany
the elements of Heaven and Earth,
then the Sun is connected with
squares of Water, and Jen - with squares of Fire. Here we can trace the same
correspondence that has been marked by us for proceeding pairs of opposites.
It will not to be excessive
in such a case to quote a passage from the Bible:
«The earth brought forth grass, herbs yielding seed after their kind, and
trees bearing
fruit, with its seed in it, after their kind: and God saw that it was good.
There was evening and there was morning, a
third day"
(G. 1: 12, 13).
According to the plan of
the Generation Book‚ "the first day" of Creation is denoted in the fifth verse of the first chapters,
"the second day" in the eighth
verse, and "the third day" - in the thirteenth verse. After that what has
been said above it does not seem a strained interpretation if conclusion, that
"formally-logical side of Creation" in Moses times was considered in
the full conformity with the stated above about the Octave in its relation to time.
In the thirteenth verse a building of eight trigrams and relation pattern between them is completed,
expressed by 64 hexagrams of I ching,
and by this the base of unfolding spiral of time is laid, making its first
cycle in the thirteenth quint step of the Octave.
«God said, "Let there be lights in the expanse of sky...
God made the two great lights:
the greater light to rule the day, and the lesser light to rule the night. He
also made the stars...
God saw that it was good.
There was evening and there was morning, a fourth day"
(G.
"Absurdity" of Generation Book’s composition seemed to rationalists
to be an acknowledgement of their atheistic arguments: after all why it could happen‚
that God created light and "trees bearing
fruit" earlier‚ than luminaries celestial? Six verses of a fourth
day‚ describing creation of luminaries‚
just note a transition from the abstract order to concrete values of manifested
time - the rotations "of signs, and
of seasons, and of days and years" in the full consent with the presentation
sequence of Thimeus (38 c-d). We have defined this stage of building as a
fifth level of Octave numbers: as the module of its micro chromatic cycles acts
the number eighteen.
Not less that impresses,
that in the Bible language of realties an initial stage of Creation to a certain
extent strictly follows the names of the abstract first principles, assumed as
a basis on the other edge of the land - in ancient
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