[What is presented here is a property of recursion that is reflected in interpretations of hexagrams pairs etc
in the I Ching].
A tool useful in computer programming is that of 'masking'. Basically one uses a pattern of 'bits' (0s and 1s)
to draw-out a pattern in a particular sequence of bits. We can detect this process at work in the I Ching through
our use of recursion where we have:
[0] <the undifferentiated whole>
[1] yin (0)/ yang (1)
[2] 00, 01 / 10, 11 [digrams]
[3] 000, 001, 010, 011 / 100, 101, 110, 111 [trigrams]
...
...
[6] 000000 ..... / ..... 111111 [hexagrams]
Thus each level gives us a finer set of distinctions that are the PARTS of the WHOLE at [0]. (also see
graphic )
The process of recursion ensures the whole is in all parts (each level is in fact the whole just re-presented with
finer details) and that includes the prime distinction at [1] above such that at level [3] we find PAIRS of trigrams
that reflect interpretations of the trigrams from the perspective of level [1], the basic yin/yang pair.
Thus at level [3] we have pairings of (000,001), (101,011), (100,101), (110,111) interpreted in the context of
the general qualities of [1]. This means that in the pair 000,001 the 000 element reflects qualities of 'yin-ness'
compared to the qualities of 001 that reflect qualities of 'yang-ness' where these differences fall within the
bounds of the sameness of both elements, namely their initial bits of 00 - the top line of a trigram/hexagram,
here expressed as the last bit in a three-bit sequence, is thus the 'difference'.
This interpretation is applied across all of the above pairs allowing for detailed analysis of the differences
in the elements of these pairs created by the process of recursion.
Given this 'pair perspective' we can also apply level [2] as a mask in that it draws out the qualities from sets
of fours, quartets, as does the application of level [3] that draws out qualities of sets of eights - octets.
When we move to level [6], the level of the 64 hexagrams, so we have five previous levels usable as filters for
interpreting differences (and samenesses) in hexagram relationships. Thus, for a basic analysis of 64-hexagram
sequences we have at least (a) the qualities of the individual hexagrams (and so each hexagram can be interpreted
as a whole, as T'ai chi, - level [0]) and (b) the shared qualities of pairs of hexagrams as determined by the nature
of the pair at [1]. IOW the pair at level [1] act as a mask to bring out relationships in the hexagram pairs at
level [6].
In the binary sequence of the I Ching the pair at level [1] reflects the 'pure' differences of, the competitive
nature of, yin vs yang as well as the cooperative nature of yin and yang. The differences are LOCAL at level [6]
focusing on pairing adjacent hexagrams as in hex 02 - 000000 compared to hex 23, 000001 where the difference is
in the top line and reflects subtle differences in concepts of integration (yin top line) vs differentiation (yang
top line). As we move 'outwards' from these pairs so the differences become increasingly global until we reach
the level of pairing hex 02 ( 000000 at one end of the sequence) with hexagram 01 (111111 at the other end of the
sequence) where the difference is globally 'opposite' (light vs dark) but also globally 'cooperative' (male + female).
What we notice here is that the poles of the 64-hexagram sequence, level [6], reflect the properties of the generic
pairing at level [1] - thus at [1] I have '0/1' whereas at [6] I have '000000/111111' - IOW the generic quality
remains constant.
Thus, for the eight hexagrams of the binary sequence of hexagrams starting at the hexagram 02 end (000000) we have:
02, 23, 08, 20, 16, 35, 45, 12......
Applying the level [1] filter, so in the pair of 02/23, 02 is to 23 as 02 is to 01, 23 is to 02 as 01 is to 02.
Or 08 is to 20 as 02 is to 01 etc etc to the other end of the sequence where we have 43 is to 01 as 02 is to 01
(but limited to a top line difference). With the association of 'integrating' to yin and 'differentiating' to yang
so hexagrams 02 and 23 are BOTH reflective of devotion to others/another (determined by the base trigram) but differ
at the top line where the yin line of 02 reflects a focus on devotion expressed through integration whereas the
yang line of 23 reflects a focus on devotion expressed through differentiation through the act of 'pruning', of
cutting back the chaff to reveal the wheat ('stripping the bed' where the bed symbolises what one 'rests' upon
- one's belief system)
This particular order of hexagrams, derived from recursion of yin/yang gives us fundamental relationships of yin/yang
described at level [1] as 'yin/yang' and at level [6] as whole hexagrams but also as pairs reflecting the fundamental
yin/yang relationship. As such we can order the sequence of hexagrams into relational pairs that move from a tight
association to a distant opposite/complementary association e.g.
02, 23 <minimal difference>
02, 08
02, 20
02, 16
...
...
02, 01 <extreme difference>
What is of interest with the 'naturally' derived sequence of 64 hexagrams, where we have applied recursion to yin/yang,
is that if I rotate all of the hexagrams in the sequence I will derive a new sequence which is in fact the sequence
of changing line combinations possible through the hexagrams of 01 or 02 (depending on which direction you read
the sequence - see the example for hexagram 01).
Thus, for the eight hexagrams of the binary sequence of hexagrams starting at the hexagram 02 end we have:
02, 23, 08, 20, 16, 35, 45, 12......
Rotate these and we have:
02, 24, 07, 19, 15, 36, 46, 11 where this sequence reflects the initial eight hexagrams derived through applying
recursion to hexagram 02, where the order of change is:
02 - no change
24 - change line 1 of 02
07 - change line 2 of 02
19 - change lines 1 and 2
15 - change line 3
36 - change lines 1 and 3
46 - change lines 2 and 3
11 - change lines 1, 2, and 3
..
..
..
..
01 - change all lines of 02
(these two types of sequences are shown for all hexagrams here)
What is of interest here is that, from the derivation of the binary sequence of hexagrams I can use level [1] of
that derivation to interpret hexagram pairs at level [6]. Rotate that level [6] sequence and I get a sequence for
changing line patterns through hexagrams 01/02, a sequence also explicitly derivable from applying recursion to
either hexagram 01 or hexagram 02. Implicit here is that by applying recursion to any hexagram and
rotating the resulting sequence I derive another sequence that in some way identifies pairs of hexagrams that share
a theme determined by the hexagrams at the poles of the sequence - in other words this derived sequence is equivalent
of the 'natural' binary sequence and as such reflects the same properties.
IOW - for hexagram 23, the first eight hexagrams in the changing line sequence (reflecting recursion of 23) are:
23, 27, 04, 41, 52, 22, 18, 26... to 43
Rotate these gives us:
24, 27, 03, 42, 51, 21, 17, 25.... to 44
What is this sequence? Functionally it is the equivalent of the 'naturally derived' binary sequence of the I Ching
that focuses on hexagrams 01/02 but is here applied to the pairing of 24/44 - IOW the pairs in this sequence reflect
a 24/44 masking that brings out the qualities of 24/44 across all of the hexagrams. For example, the 03,42 relationship
is 'about' the newness of 24, the 'beginning' (or 'born again', return to), interacting with the element of transition
(seduction, persuation to the 'other' side etc) of 44 expressed in the 'augmentation' of 42.
In the pairing of 17, 25 we can identify the same process where 17 focuses on a 'new' faith, a new belief, and
25 focuses on the expression and so an act of persuation (or attempted). Overall the particular sequence reflects
the rule:
X is to Y as 24 is to 44, Y is to X as 44 is to 24.
Thus in the 'true' binary sequence, derived from recursion of yin/yang, the pairs at level [6] reflect X is to
Y as 01 is to 02, 02 is to 01. Thus 02,23 is a small difference of devotion issues - integrating (02) vs differentiating
(23), whereas 02, 01 is a huge difference in integration overall (02) vs differentiation over all (01).
In the traditional sequence, where the sequence is given as 01 to 64 and reflects a derivation
of the sequence through recursion, the focus is of X is to Y as 01 is to 64, 64 is to 01. (and so 01 is to
02 as 01 is to 64, 03 is to 04 as 01 is to 64, 05 is to 06 as 01 is to 64 etc etc etc)
Each possible recursively-derived sequence of hexagrams serves as a tool for analysis of relationships of hexagrams
through analogy at the particular level as well as at the general level. This capability stems from the method
used by the species with which to derive information - recursion (see
the IDM pages)
Note the above is not complete - there are issues re the properties of the hexagram sequences of changing-line
patterns and their link to five-phase theory - for
a set of recursively-derived sequences for each hexagram see the pairs matrix page.
