PAIR MATRICES OF THE I CHING

(Copyright © 2003 C. J. Lofting)

[last version 19th August 2003]

(Supportive pages:

Masking and the I Ching

I Ching Matrices)

I have here grouped what we call the vertically and horizontally derived sequences of hexagrams applicable to a specific pair of hexagram 'opposites'. The ROOT horizontal matrix reflects the derivation of a sequence of hexagrams from the basic 'yin/yang' pair concept. This root sequence is derived from recursion of the notions of yin/yang. This root pattern is manifest in the horizontal matrix for 01/02 given below:

HORIZONTAL 01/02

01 43 14 34 09 05 26 11
10 58 38 54 61 60 41 19
13 49 30 55 37 63 22 36
25 17 21 51 42 03 27 24
44 28 50 32 57 48 18 46
06 47 64 40 59 29 04 07
33 31 56 62 53 39 52 15
12 45 35 16 20 08 23 02

Thus the above unfolds into the 64-hexagram sequence:

This sequence is the binary sequence of the traditional I Ching where it reflects the order of hexagrams derived from recursion. The sequence reflects the 'shades of grey' or the 'colour spectrum' between the pure expression of yangness (hexagram 01) and the pure expression of yinness (hexagram 02) (see more comments below) [A mathematics aside]

The question now is how do we derive the other horizontal sequences we seem to find in the I Ching? If I take the above sequence of hexagrams and rotate each hexagram and write down the number of this 'new' hexagram I will get the following sequence:

If we apply recursion to a hexagram, working from the bottom up, we derive what is called the vertical sequence of a hexagram. This sequence reflects the hexagram from applying line changes but also relationships of a line 'exaggeration'. The above sequence is the vertical sequence for the hexagram pair of 01/02, depending on which 'end' of the sequence you start, start at the 01 end and the sequence reflects changing lines for hexagram 01, from no lines (01) to all lines (and so 02) remembering that in any hexagram there are 64 possible line change patterns. Start at the 02 ends and the sequence reflects changing lines from no lines (02) to all lines (01).

We thus see a pattern, from the recursion of yin/yang I derive the horizontal sequence. Rotate the hexagrams in that sequence gives me a vertical sequence. I can thus work backwards, derive the vertical sequence through a hexagram and rotate the resulting hexagrams to give me a horizontal sequence for some pair of hexagram opposites. These two sequences are keys to uncovering some details about the qualities of hexagrams. Note that I have grouped the sequences that share the same hexagram pair as poles of both sequences but this does not necessarily mean that the horizontal sequence is derived from the vertical sequence for that pair. Thus for hexagram pair 01/02 the vertical is derived from the horizontal for the same pair. On the other hand, the horizontal of hexagram pair 23/43 is derived from rotation of the vertical pair of 24/44 etc.

From additional work on the I Ching ICPlus shows how the concepts of five-phase are manifest in these sequences. I have given just one presentation below of the five-phase phases associated with the sequences in that the same pattern applies for ALL of these sequences:

VERTICAL 01/02

All of these sequences reflect TWO sequences enfolded into ONE. As such these sequences are read top-to-bottom and bottom-to-top. Thus in the above vertical sequence the top-to-bottom order gives you the results of applying recursion to hexagram 01. The bottom-to-top order gives you the results of applying recursion to hexagram 01's structural opposite, hexagram 02. This matrix expressed as a single sequence, the sequence previously shown, as:

01 44 13 33 10 06 25 12 09 57 37 53 61 59 42 20 14 50 30 56 38 64 21 35 26 18 22 52 41 04 27 23 43 28 49 31 58 47 17 45 05 48 63 39 60 29 03 08 34 32 55 62 54 40 51 16 11 46 36 15 19 07 24 02

The five-phase labels given above for ROWS of the matrix also apply to COLUMNS of the matrix thus, when reading the matrix top-down the order of five-phase columns, read left-to-right, is as given in column 1 of the table (headed "matrix read top to bottom"). Thus for hexagram 01 the first two hexagrams are 01 and 44 and these reflect five-phase filtration phase (aka EARTH). For hexagram 02, reading bottom-up, the first two hexagrams are 02 and 24 and these too reflect the five-phase filtration path (aka EARTH).

The vertical focus appears to be five-phase with emphasis on raw/refined. Thus raw qualities that feed hexagram 02 are reflected in the first two hexagrams of the WOOD - raw PRODUCTION row for 02 where we find the pair 23 & 27, thus 23+27=02. This pair reflects the EARTH phase WITHIN the WOOD phase of raw production. Working downwards for hexagram 01 we find the corresponding pair for WOOD - raw PRODUCTION as 43 and 28, thus 43+28=01 and so on. Thus for hexagram 02, we can identify the basic characteristics as being reflected in MIXING the characteristics of 23 with those of 27. I have listed this relationship for all vertical sequences given below for hexagram pairs. (There is a LOT more details here but that is up to the user to flesh-out and you may like to check out the Five-phase I Ching pages.)

HORIZONTAL 01/02

Using the horizontal format it is easier to cover the range of relationships all reflecting 01/02 characteristics and so pure yin/yang relationships. The roots of this are covered in the Masking page. Thus the above unfolds into the 64-hexagram sequence:

This sequence is the binary sequence of the traditional I Ching where it reflects the order of hexagrams derived from applying yin/yang recursively. The sequence reflects the 'shades of grey' or the 'colour spectrum' between the pure expression of yangness (hexagram 01) and the pure expression of yinness (hexagram 02). The overall relationship is thus of 01 to 02 and the aspects of that relationship are reflected in the relationship of the 32 pairs of hexagrams that make-up this sequence. Thus, reading from the left:

01 is to 43 as 01 is to 02

14 is to 34 as 01 is to 02

09 is to 05 as 01 is to 02

You can also hold 01 as a constant where the relationships to each hexagram reflect increasing degrees of difference ultimately reflected in the 01 to 02 relationship, thus 01 is to 43 as 01 is to 02

01 is to 14 as 01 is to 02

01 is to 34 as 01 is to 02

etc., where the relationship gets increasingly 'different'.

43 28 49 31 58 47 17 45
05 48 63 39 60 29 03 08
34 32 55 62 54 40 51 16
11 46 36 15 19 07 24 02
01 44 13 33 10 06 25 12
09 57 37 53 61 59 42 20
14 50 30 56 38 64 21 35
26 18 22 52 41 04 27 23

02+24 = 23, where pruning is made-up of the devotion of 02 combined with the sense of 'returning' in 24.

44+01 = 43, where the seeding, the 'spreading of the word', of 43 is made-up of the singlemindedness, the percevering of 01 combined with the persuading, seducing nature of 44.

HORIZONTAL - 23/43 pruning/seeding

23 02 20 08 35 16 12 45
52 15 53 39 56 62 33 31
04 07 59 29 64 40 06 47
18 46 57 48 50 32 44 28
27 24 42 03 21 51 25 17
22 36 37 63 30 55 13 49
41 19 61 60 38 54 10 58
26 11 09 05 14 34 01 43

Unfolded the sequence is:

This sequence covers the 'shades of grey' or the 'colour spectrum' across the pair 23/43. As such:

23 is to 02 as 23 is to 43

20 is to 08 as 23 is to 43

etc.,

or hold 23 constant and map its relationship to all other hexagrams in the sequence as we move from 'minute' difference (02) to total difference (43).

VERTICAL 14/08 directing/uniting

14 50 30 56 38 64 21 35
26 18 22 52 41 04 27 23
01 44 13 33 10 06 25 12
09 57 37 53 61 59 42 20
34 32 55 62 54 40 51 16
11 46 36 15 19 07 24 02
43 28 49 31 58 47 17 45
05 48 63 39 60 29 03 08

20+42 = 08, where the augmenting of 42 is combined with the admiring of 20.

34+32 = 14, where the invigorating of 34 is combined with the comitting of 32.

HORIZONTAL 14/08 directing/uniting

14 34 01 43 26 11 09 05
38 54 10 58 41 19 61 60
30 55 13 49 22 36 37 63
21 51 25 17 27 24 42 03
50 32 44 28 16 46 57 48
64 40 06 47 04 07 59 29
56 62 33 31 52 15 53 39
35 16 12 45 23 02 20 08

14 is to 34 as 14 is to 08

01 is to 43 as 14 is to 08

etc.,

VERTICAL 34/20 invigorating/admiring

34 32 55 62 54 40 51 16
11 46 36 15 19 07 24 02
43 28 49 31 58 47 17 45
05 48 63 39 60 29 03 08
14 50 30 56 38 64 21 35
26 18 22 52 41 04 27 23
01 44 13 33 10 06 25 12
09 57 37 53 61 59 42 20

08 + 03 = 20, admiring, where the uniting of 08 is combined with the sprouting of 03.

14+50 = 34, invigorating, where the directing of 14 is combined with the transforming of 50.

HORIZONTAL 20/34

20 08 23 02 12 45 35 16
53 39 52 15 33 31 56 62
59 29 04 07 06 47 64 40
57 48 18 46 44 28 59 32
42 03 27 24 25 17 23 51
37 63 22 36 13 49 30 55
61 60 41 19 10 58 38 54
09 05 26 11 01 43 14 34

20 is to 08 as 20 is to 34

23 is to 02 as 20 is to 34

etc

VERTICAL 09/16 small gaining/foreseeing

09 57 37 53 61 59 42 20
01 44 13 33 10 06 25 12
26 18 22 52 41 04 27 23
14 50 30 56 38 64 21 35
05 48 63 39 60 29 03 08
43 28 49 31 58 47 17 45
11 46 36 15 19 07 24 02
34 32 55 62 54 40 51 16

35+21 = 16, foreseeing, where the progressing of 35 (bring something into the light) is combined with the problem solving of 21.

05+48 = 09, small gaining, where the waiting (networking, serving one's dues) of 05 is combined with the foundation setting of 48.

HORIZONTAL 09/16

09 05 26 11 01 43 14 34
61 60 41 19 10 58 38 54
37 63 22 36 13 49 30 55
42 03 27 24 25 17 23 51
57 48 18 46 44 28 59 32
59 29 04 07 06 47 64 40
53 39 52 15 33 31 56 62
20 08 23 02 12 45 35 16

09 is to 05 as 09 is to 16

26 is to 11 as 09 is to 16

etc

VERTICAL 05/35 waiting/progressing

05 48 63 39 60 29 03 08
43 28 49 31 58 47 17 45
11 46 36 15 19 07 24 02
34 32 55 62 54 40 51 16
09 57 37 53 61 59 42 20
01 44 13 33 10 06 25 12
26 18 22 52 41 04 27 23
14 50 30 56 38 64 21 35

16+51 = 35, progressing, where the forseeing of 16 is combined with the 'enlightening' of 51.

09+57 = 05, waiting, where the small gaining of 09 is combined with the cultivating of 57.

HORIZONTAL 05/35

35 16 12 45 23 02 20 08
56 62 33 31 52 15 53 39
64 40 06 47 04 07 59 29
50 32 44 28 16 46 57 48
21 51 25 17 27 24 42 03
30 55 13 49 22 36 37 63
38 54 10 58 41 19 61 60
14 34 01 43 26 11 09 05

35 is to 16 as 35 is to 05

12 is to 45 as 35 is to 05

etc

VERTICAL 26/45 holding firm/congregating

26 18 22 52 41 04 27 23
14 50 30 56 38 64 21 35
09 57 37 53 61 59 42 20
01 44 13 33 10 06 25 12
11 46 36 15 19 07 24 02
34 32 55 62 54 40 51 16
05 48 63 39 60 29 03 08
43 28 49 31 58 47 17 45

12+25 = 45, where congregating is made up of the neutralising of 12 (neutralise the attacks of others on one's faith) combined with the asserting of 25 (stand up to assert one's opinion, disentangle from others)

11+46 = 26, where holding firm is made up of the harmonising of 11 combined with the increased entanglement of 46.

HORIZONTAL 26/45

26 11 09 05 14 34 01 43
41 19 61 60 38 54 10 58
22 36 37 63 30 55 13 49
27 24 42 03 21 51 25 17
18 46 57 48 50 32 44 28
04 07 59 29 64 40 06 47
52 15 53 39 56 62 33 31
23 02 20 08 35 16 12 45

26 is to 11 as 26 is to 45

09 is to 05 as 26 is to 45

etc

VERTICAL 11/12 harmonising/neutralising

11 46 36 15 19 07 24 02
34 32 55 62 54 40 51 16
05 48 63 39 60 29 03 08
43 28 49 31 58 47 17 45
26 18 22 52 41 04 27 23
14 50 30 56 38 64 21 35
09 57 37 53 61 59 42 20
01 44 13 33 10 06 25 12

45+17 = 12, neutralising being made up of the sense of congregating to celebrate (45) combined with the choice of a particular following (17).

26+18 = 11, harmonising being made up of the sense of holding firm to traditions etc (26) combined with a sense of correcting corruptions (18).

HORIZONTAL 11/12

12 45 35 16 20 08 23 02
33 31 56 62 53 41 52 15
06 47 64 40 59 29 04 07
44 28 50 32 57 48 18 46
25 17 21 51 42 03 27 24
13 49 30 55 37 63 22 36
10 58 38 54 61 60 41 19
01 43 14 32 09 05 26 11

12 is to 45 as 12 is to 11

35 is to 16 as 12 is to 11

etc

VERTICAL 10/15 treading/levelling

10 06 25 12 01 44 13 33
61 59 42 20 09 57 37 53
38 64 21 35 14 50 30 56
41 04 27 23 26 18 22 52
58 47 17 45 43 28 49 31
60 29 03 08 05 48 63 39
54 40 51 16 34 32 55 62
19 07 24 02 11 46 36 15

52+22 = 15, levelling is made up of the self-restraining (quality-control) of 52 combined with the sense of covering-up, gloss-over, facade, beautyfy of 22.

58+47 = 10, treading is made up of the intense self-reflection of 58 combined with the sense of enforced enclosure of 47.

HORIZONTAL 10/15

10 58 38 54 61 60 41 19
01 43 14 32 09 05 26 11
25 17 21 51 42 03 27 24
13 49 30 55 37 63 22 36
06 47 64 40 59 29 04 07
44 28 50 32 57 48 18 46
12 45 35 16 20 08 23 02
33 31 56 62 53 41 52 15

10 is to 58 as 10 is to 15

38 is to 54 as 10 is to 15

etc

VERTICAL 58/52 self-reflecting/self-restraining

58 47 17 45 43 28 49 31
60 29 03 08 05 48 63 39
54 40 51 16 34 32 55 62
19 07 24 02 11 46 36 15
10 06 25 12 01 44 13 33
61 59 42 20 09 57 37 53
38 64 21 35 14 50 30 56
41 04 27 23 26 18 22 52

15+36 = 52, self-restraing/discerning is made up of the levelling of 15 combined with the uncompromising (hide the light) of 36.

10+06=58, self-reflecting/intensity is made up of the treading of 10 with the compromising of 06.

HORIZONTAL 58/52

52 15 53 39 56 62 33 31
23 02 20 08 35 16 12 45
18 46 57 48 50 32 44 28
04 07 59 29 64 40 06 47
22 36 37 63 30 55 13 49
27 24 42 03 21 51 25 17
26 11 09 05 14 34 01 43
41 19 61 60 38 54 10 58

52 is to 15 as 52 is to 58

53 is to 39 as 52 is to 58

etc.,

VERTICAL 38/39 opposing/obstructing

38 64 21 35 14 50 30 56
41 04 27 23 26 18 22 52
10 06 25 12 01 44 13 33
61 59 42 20 09 57 37 53
54 40 51 16 34 32 55 62
19 07 24 02 11 46 36 15
58 47 17 45 43 28 49 31
60 29 03 08 05 48 63 39

53+37 = 39, obstructing is made up of maturing of 53 combined with the rigid structuring of 37.

54+40 = 38, opposing is made up of immaturing of 54 combined with the loose structuring of 40.

HORIZONTAL 38/39

38 54 10 58 41 19 61 60
14 34 01 43 26 11 09 05
21 51 25 17 27 24 42 03
30 55 13 49 22 36 37 63
64 40 06 47 04 07 59 29
50 32 44 28 16 46 57 48
35 16 12 45 23 02 20 08
56 62 33 31 52 15 53 39

38 is to 54 as 38 is to 39

10 is to 58 as 38 is to 39

etc.,

VERTICAL 54/53 immaturing/maturing

54 40 51 16 34 32 55 62
19 07 24 02 11 46 36 15
58 47 17 45 43 28 49 31
60 29 03 08 05 48 63 39
38 64 21 35 14 50 30 56
41 04 27 23 26 18 22 52
10 06 25 12 01 44 13 33
61 59 42 20 09 57 37 53

39+63 = 53, maturing is made up of the sense of obstructing (against the 'mindless' flow - 39) combined with the sense of completing, closure (63).

38+64 = 54, immaturing, is made up of the sense of opposition (38) combined with the sense of incompleting, remain open (64)

HORIZONTAL 53/54

53 39 52 15 33 31 56 62
20 08 23 02 12 45 35 16
57 48 18 46 44 28 59 32
59 29 04 07 06 47 64 40
37 63 22 36 13 49 30 55
42 03 27 24 25 17 23 51
09 05 26 11 01 43 14 34
61 60 41 19 10 58 38 54

53 is to 39 as 53 is to 54

52 is to 15 as 53 is to 54

etc.,

VERTICAL 61/62 yielding-softening / portraying

61 59 42 20 09 57 37 53
10 06 25 12 01 44 13 33
41 04 27 23 26 18 22 52
38 64 21 35 14 50 30 56
60 29 03 08 05 48 63 39
58 47 17 45 43 28 49 31
19 07 24 02 11 46 36 15
54 40 51 16 34 32 55 62

56+30 = 62, portraying (exaggerating traditions to ensure cultural integration) made up of loyalty (56) combined with a sense of direction, an ideology (30)

60+29 = 61, softening (empathy etc) made up of standardising (60 [sameness]) combined with containing/controlling (29)

HORIZONTAL 61/62

61 60 41 19 10 58 38 54
09 05 26 11 01 43 14 34
42 03 27 24 25 17 23 51
37 63 22 36 13 49 30 55
59 29 04 07 06 47 64 40
57 48 18 46 44 28 59 32
20 08 23 02 12 45 35 16
53 39 52 15 33 31 56 62

61 is to 60 as 61 is to 62

41 is to 19 as 61 is to 62

etc.,

VERTICAL 60/56 standardising/loyalty

60 29 03 08 05 48 63 39
58 47 17 45 43 28 49 31
19 07 24 02 11 46 36 15
54 40 51 16 34 32 55 62
61 59 42 20 09 57 37 53
10 06 25 12 01 44 13 33
41 04 27 23 26 18 22 52
38 64 21 35 14 50 30 56

62+55 = 56, loyalty made up of exaggerating traditions (62) combined with diversity (abundance) (55)

61+59 = 60, standardising made up of hard exterior (and so rigid covering) combined with dispelling of illusions (59)

HORIZONTAL 60/56

56 62 33 31 52 15 53 39
35 16 12 45 23 02 20 08
50 32 44 28 16 46 57 48
64 40 06 47 04 07 59 29
30 55 13 49 22 36 37 63
21 51 25 17 27 24 42 03
14 34 01 43 26 11 09 05
38 54 10 58 41 19 61 60

56 is to 62 as 56 is to 60

33 is to 31 as 56 is to 60

etc.,

VERTICAL 41/31 concentrating-distilling/wooing, intense stimulating

41 04 27 23 26 18 22 52
38 64 21 35 14 50 30 56
61 59 42 20 09 57 37 53
10 06 25 12 01 44 13 33
19 07 24 02 11 46 36 15
54 40 51 16 34 32 55 62
60 29 03 08 05 48 63 39
58 47 17 45 43 28 49 31

33+13 = 31, wooing/intense stimulating made up of trickary (draw in - 33) and likemindedness, associating (13)

19+07 = 41, concentrating/distilling made up of approaching (bring together high and low - 19) and uniformity (07)

HORIZONTAL 41/31

41 19 61 60 38 54 10 58
26 11 09 05 14 34 01 43
27 24 42 03 21 51 25 17
22 36 37 63 30 55 13 49
04 07 59 29 64 40 06 47
18 46 57 48 50 32 44 28
23 02 20 08 35 16 12 45
52 15 53 39 56 62 33 31

41 is to 19 as 41 is to 31

61 is to 60 as 41 is to 31

etc.,

VERTICAL 19/33 approaching/retreating-tricking

19 07 24 02 11 46 36 15
54 40 51 16 34 32 55 62
60 29 03 08 05 48 63 39
58 47 17 45 43 28 49 31
41 04 27 23 26 18 22 52
38 64 21 35 14 50 30 56
61 59 42 20 09 57 37 53
10 06 25 12 01 44 13 33

31+49 = 33, tricking/retreating made up of intense stimulation/wooing (31) combined with revolting, revealing (49)

41+04 = 19, approaching made up of concentrating (41) and masking (04)

HORIZONTAL 33/19

33 31 56 62 53 41 52 15
12 45 35 16 20 08 23 02
44 28 50 32 57 48 18 46
06 47 64 40 59 29 04 07
13 49 30 55 37 63 22 36
25 17 21 51 42 03 27 24
01 43 14 32 09 05 26 11
10 58 38 54 61 60 41 19

33 is to 31 as 33 is to 19

56 is to 62 as 33 is to 19

etc.,

VERTICAL 13/07 associating/uniforming

13 33 01 44 25 12 10 06
37 53 09 57 42 20 61 59
30 56 14 50 21 35 38 64
22 52 26 18 27 23 41 04
49 31 43 28 17 45 58 47
63 39 05 48 03 08 60 29
55 62 34 32 51 16 54 40
36 15 11 46 24 02 19 07

04+41 = 07, uniforming is made up of masking (04) and concentrating (41).

49+31 = 13, associating is made up of unmasking ( revolting/reveling - 49) and intense stimulating (wooing - 31)

HORIZONTAL 13/07

13 49 30 55 37 63 22 36
25 17 21 51 42 03 27 24
01 43 14 32 09 05 26 11
10 58 38 54 61 60 41 19
33 31 56 62 53 41 52 15
12 45 35 16 20 08 23 02
44 28 50 32 57 48 18 46
06 47 64 40 59 29 04 07

13 is to 49 as 13 is to 07

30 is to 55 as 13 is to 07

etc.,

VERTICAL 49/04 masking/unmasking

49 31 43 28 17 45 58 47
63 39 05 48 03 08 60 29
55 62 34 32 51 16 54 40
36 15 11 46 24 02 19 07
13 33 01 44 25 12 10 06
37 53 09 57 42 20 61 59
30 56 14 50 21 35 38 64
22 52 26 18 27 23 41 04

07+19 = 04, masking is made up of uniforming (07) and approaching (19)

13+33 = 49, umasking is made up of associating (like-mindedness 13) and trickary/draw-in (33)

HORIZONTAL 04/49

04 07 59 29 64 40 06 47
18 46 57 48 50 32 44 28
23 02 20 08 35 16 12 45
52 15 53 39 56 62 33 31
41 19 61 60 38 54 10 58
26 11 09 05 14 34 01 43
27 24 42 03 21 51 25 17
22 36 37 63 30 55 13 49

04 is to 07 as 04 is to 49

59 is to 29 as 04 is to 49

etc.,

VERTICAL 30/29 guiding-direction-setting/containing-controlling

30 56 14 50 21 35 38 64
22 52 26 18 27 23 41 04
13 33 01 44 25 12 10 06
37 53 09 57 42 20 61 59
55 62 34 32 51 16 54 40
36 15 11 46 24 02 19 07
49 31 43 28 17 45 58 47
63 39 05 48 03 08 60 29

59+61 = 29, containing/controlling is made up of dispelling illusions (59) and hard exterior, soft interior - yielding, empathising (61)

55+62 = 30, guiding-direction-setting is made up of abounding/diversity (55) and portraying, exaggerating a tradition/ideology etc (62)

HORIZONTAL 30/29

30 55 13 49 22 36 37 63
21 51 25 17 27 24 42 03
14 34 01 43 26 11 09 05
38 54 10 58 41 19 61 60
56 62 33 31 52 15 53 39
35 16 12 45 23 02 20 08
50 32 44 28 16 46 57 48
64 40 06 47 04 07 59 29

30 is to 55 as 30 is to 29

13 is to 49 as 30 is to 29

etc.,

VERTICAL 55/59 abounding/dispelling

55 62 34 32 51 16 54 40
36 15 11 46 24 02 19 07
49 31 43 28 17 45 58 47
63 39 05 48 03 08 60 29
30 56 14 50 21 35 38 64
22 52 26 18 27 23 41 04
13 33 01 44 25 12 10 06
37 53 09 57 42 20 61 59

29+60 = 59, dispelling illusions is mad up of containment/control (29) combined with standardising (60)

30+56 = 55, abounding, is made up of direction-setting/guidance (30) combined with loyalty (56)

HORIZONTAL 55/59

59 29 04 07 06 47 64 40
57 48 18 46 44 28 59 32
20 08 23 02 12 45 35 16
53 39 52 15 33 31 56 62
61 60 41 19 10 58 38 54
09 05 26 11 01 43 14 34
42 03 27 24 25 17 23 51
37 63 22 36 13 49 30 55

59 is to 29 as 59 is to 55

04 is to 07 as 59 is to 55

etc.,

VERTICAL 37/40 rigid structuring/relaxed structuring

37 53 09 57 42 20 61 59
13 33 01 44 25 12 10 06
22 52 26 18 27 23 41 04
30 56 14 50 21 35 38 64
63 39 05 48 03 08 60 29
49 31 43 28 17 45 58 47
36 15 11 46 24 02 19 07
55 62 34 32 51 16 54 40

64+38 = 40, relaxed structuring is made up of incompleting/remain open (64) combined with opposing (38)

63+39 = 37, rigid structuring is made up of closure (63) combined with obsticles (39)

HORIZONTAL 37/40

37 63 22 36 13 49 30 55
42 03 27 24 25 17 23 51
09 05 26 11 01 43 14 34
61 60 41 19 10 58 38 54
53 39 52 15 33 31 56 62
20 08 23 02 12 45 35 16
57 48 18 46 44 28 59 32
59 29 04 07 06 47 64 40

37 is to 63 as 37 is to 40

22 is to 36 as 37 is to 40

etc.,

VERTICAL 63/64 incompleting/completing

63 39 05 48 03 08 60 29
49 31 43 28 17 45 58 47
36 15 11 46 24 02 19 07
55 62 34 32 51 16 54 40
37 53 09 57 42 20 61 59
13 33 01 44 25 12 10 06
22 52 26 18 27 23 41 04
30 56 14 50 21 35 38 64

40+54 = 64, incompleting is made up of loose structuring (40) combined with immaturing (initial 'burst' of enery (54))

37+53 = 63, completing is made up of rigid structuring (37) combined with maturing (gradual development) (53)

HORIZONTAL 63/64

64 40 06 47 04 07 59 29
50 32 44 28 16 46 57 48
35 16 12 45 23 02 20 08
56 62 33 31 52 15 53 39
38 54 10 58 41 19 61 60
14 34 01 43 26 11 09 05
21 51 25 17 27 24 42 03
30 55 13 49 22 36 37 63

64 is to 40 as 64 is to 63

06 is to 47 as 64 is to 63

etc.,

VERTICAL 22/47 facading, gloss-overing/forced enclosing

22 52 26 18 27 23 41 04
30 56 14 50 21 35 38 64
37 53 09 57 42 20 61 59
13 33 01 44 25 12 10 06
36 15 11 46 24 02 19 07
55 62 34 32 51 16 54 40
63 39 05 48 03 08 60 29
49 31 43 28 17 45 58 47

06+10 = 47, forced enclosing, made up of compromising (06) and treading (conduct - 10)

36+15 = 22, facading, made up of uncompromising (36 - hide light) and levelling (modesty - 15)

HORIZONTAL 22/47

22 36 37 63 30 55 13 49
27 24 42 03 21 51 25 17
26 11 09 05 14 34 01 43
41 19 61 60 38 54 10 58
52 15 53 39 56 62 33 31
23 02 20 08 35 16 12 45
18 46 57 48 50 32 44 28
04 07 59 29 64 40 06 47

22 is to 36 as 22 is to 47

37 is to 63 as 22 is to 47

etc.,

VERTICAL 36/06 uncompromising/compromising

36 15 11 46 24 02 19 07
55 62 34 32 51 16 54 40
63 39 05 48 03 08 60 29
49 31 43 28 17 45 58 47
22 52 26 18 27 23 41 04
30 56 14 50 21 35 38 64
37 53 09 57 42 20 61 59
13 33 01 44 25 12 10 06

47+58 = 06, compromising made of forced enclosing (47) and self-reflecting/intensity (58)

22+52 = 36, uncompromising made of facading, gloss-over (22) and block/discernment (52)

HORIZONTAL 06/36

06 47 64 40 59 29 04 07
44 28 50 32 57 48 18 46
12 45 35 16 20 08 23 02
33 31 56 62 53 41 52 15
10 58 38 54 61 60 41 19
01 43 14 32 09 05 26 11
25 17 21 51 42 03 27 24
13 49 30 55 37 63 22 36

06 is to 47 as 06 is to 36

64 is to 40 as 06 is to 36

etc.,

VERTICAL 25/46 disentangling/entangling

25 12 10 06 13 33 01 44
42 20 61 59 37 53 09 57
21 35 38 64 30 56 14 50
27 23 41 04 22 52 26 18
17 45 58 47 49 31 43 28
03 08 60 29 63 39 05 48
51 16 54 40 55 62 34 32
24 02 19 07 36 15 11 46

18+26 = 46, entangling made up of correcting corruption (18) and holding firm to beliefs/traditions (26)

17+45 = 25, disentangling (stand up to assert) made up of acquiring a belief (17) and celebrating through congregating (45)

HORIZONTAL 25/46

25 17 21 51 42 03 27 24
13 49 30 55 37 63 22 36
10 58 38 54 61 60 41 19
01 43 14 32 09 05 26 11
12 45 35 16 20 08 23 02
33 31 56 62 53 41 52 15
06 47 64 40 59 29 04 07
44 28 50 32 57 48 18 46

25 is to 17 as 25 is to 46

21 is to 51 as 25 is to 46

etc.,

VERTICAL 17/18 following/correcting

17 45 58 47 49 31 43 28
03 08 60 29 63 39 05 48
51 16 54 40 55 62 34 32
24 02 19 07 36 15 11 46
25 12 10 06 13 33 01 44
42 20 61 59 37 53 09 57
21 35 38 64 30 56 14 50
27 23 41 04 22 52 26 18

46+11 = 18, correcting made up of entangling and harmonising

25+12 = 17, following made up of disentangling (25) and neutralisating (12)

HORIZONTAL 17/18

18 46 57 48 50 32 44 28
04 07 59 29 64 40 06 47
52 15 53 39 56 62 33 31
23 02 20 08 35 16 12 45
26 11 09 05 14 34 01 43
41 19 61 60 38 54 10 58
22 36 37 63 30 55 13 49
27 24 42 03 21 51 25 17

18 is to 46 as 18 is to 17

57 is to 48 as 18 is to 17

etc.,

VERTICAL 21/48 problem solving/founding

21 35 38 64 30 56 14 50
27 23 41 04 22 52 26 18
25 12 10 06 13 33 01 44
42 20 61 59 37 53 09 57
51 16 54 40 55 62 34 32
24 02 19 07 36 15 11 46
17 45 58 47 49 31 43 28
03 08 60 29 63 39 05 48

57+09 = 48, founding made up of cultivating (57) and small gaining (09)

51+16 = 21 problem solving made up of enlightening (51) and foreseeing/planning (16)

HORIZONTAL 21/48

21 51 25 17 27 24 42 03
30 55 13 49 22 36 37 63
38 54 10 58 41 19 61 60
14 34 01 43 26 11 09 05
35 16 12 45 23 02 20 08
56 62 33 31 52 15 53 39
64 40 06 47 04 07 59 29
50 32 44 28 16 46 57 48

21 is to 51 as 21 is to 48

25 is to 17 as 21 is to 48

etc.,

VERTICAL 51/57 enlightening/cultivating

51 16 54 40 55 62 34 32
24 02 19 07 36 15 11 46
17 45 58 47 49 31 43 28
03 08 60 29 63 39 05 48
21 35 38 64 30 56 14 50
27 23 41 04 22 52 26 18
25 12 10 06 13 33 01 44
42 20 61 59 37 53 09 57

48+05 = 57, cultivating (become influencial) made up of foundation-setting (48) and waiting (bidding one's time, serving dues - 05)

21+35 = 51, enlightening made up of problem solving (21) and progressing (bring something into the 'light' - 35)

HORIZONTAL 51/57

57 48 18 46 44 28 59 32
59 29 04 07 06 47 64 40
53 39 52 15 33 31 56 62
20 08 23 02 12 45 35 16
09 05 26 11 01 43 14 34
61 60 41 19 10 58 38 54
37 63 22 36 13 49 30 55
42 03 27 24 25 17 23 51

57 is to 48 as 57 is to 51

18 is to 46 as 57 is to 51

etc.,

VERTICAL 42/32 augmenting/committing

42 20 61 59 37 53 09 57
25 12 10 06 13 33 01 44
27 23 41 04 22 52 26 18
21 35 38 64 30 56 14 50
03 08 60 29 63 39 05 48
17 45 58 47 49 31 43 28
24 02 19 07 36 15 11 46
51 16 54 40 55 62 34 32

50+14 = 32, committing made up of transforming (50) and directing (14)

03+08 = 42, augmenting made up of sprouting (03) and uniting (08)

HORIZONTAL 42/32

42 03 27 24 25 17 23 51
37 63 22 36 13 49 30 55
61 60 41 19 10 58 38 54
09 05 26 11 01 43 14 34
20 08 23 02 12 45 35 16
53 39 52 15 33 31 56 62
59 29 04 07 06 47 64 40
57 48 18 46 44 28 59 32

42 is to 03 as 42 is to 32

27 is to 24 as 42 is to 32

etc.,

VERTICAL 03/50 sprouting/transforming

03 08 60 29 63 39 05 48
17 45 58 47 49 31 43 28
24 02 19 07 36 15 11 46
51 16 54 40 55 62 34 32
42 20 61 59 37 53 09 57
25 12 10 06 13 33 01 44
27 23 41 04 22 52 26 18
21 35 38 64 30 56 14 50

32+34 = 50, transforming made up of committing (32) and invigorating (34)

42+20 = 03, sprouting made up of augmenting (42) and admiring (20)

HORIZONTAL 03/50

50 32 44 28 16 46 57 48
64 40 06 47 04 07 59 29
56 62 33 31 52 15 53 39
35 16 12 45 23 02 20 08
14 34 01 43 26 11 09 05
38 54 10 58 41 19 61 60
30 55 13 49 22 36 37 63
21 51 25 17 27 24 42 03

50 is to 32 as 50 is to 03

44 is to 28 as 50 is to 03

etc.,

VERTICAL 27/28 hungering/exceeding

27 23 41 04 22 52 26 18
21 35 38 64 30 56 14 50
42 20 61 59 37 53 09 57
25 12 10 06 13 33 01 44
24 02 19 07 36 15 11 46
51 16 54 40 55 62 34 32
03 08 60 29 63 39 05 48
17 45 58 47 49 31 43 28

44+01 = 28, exceeding made up of persuading (44) and persevering (singlemindedness - 01)

24+02 = 27, hungering made up of returning (24) and devoting (02)

HORIZONTAL 27/28

27 24 42 03 21 51 25 17
22 36 37 63 30 55 13 49
41 19 61 60 38 54 10 58
26 11 09 05 14 34 01 43
23 02 20 08 35 16 12 45
52 15 53 39 56 62 33 31
04 07 59 29 64 40 06 47
18 46 57 48 50 32 44 28

27 is to 24 as 27 is to 28

42 is to 03 as 27 is to 28

etc.,

VERTICAL 24/44 returning/seducing

24 02 19 07 36 15 11 46
51 16 54 40 55 62 34 32
03 08 60 29 63 39 05 48
17 45 58 47 49 31 43 28
27 23 41 04 22 52 26 18
21 35 38 64 30 56 14 50
42 20 61 59 37 53 09 57
25 12 10 06 13 33 01 44

28+43 = 44, seducing/persuading made up of exceeding (28) and seeding (43)

27+23 = 24, returning made up of hungering (27) and pruning (23)

HORIZONTAL 24/44

44 28 50 32 57 48 18 46
06 47 64 40 59 29 04 07
33 31 56 62 53 41 52 15
12 45 35 16 20 08 23 02
01 43 14 32 09 05 26 11
10 58 38 54 61 60 41 19
13 49 30 55 37 63 22 36
25 17 21 51 42 03 27 24

44 is to 28 as 44 is to 24

50 is to 32 as 44 is to 24

etc.,

An Aside: What is the difference between a changing line vs an exaggerating line? Here we mimic something we see in DNA strands in that, when in a particular context, those genes in the DNA that are applicable will seem to 'swell-up', reflecting their activity. In a hexagram, that hexagram can be intepreted as a unique form, a 'cell' type and as such that type can 'shape-shift' given a context, in other words the general type will change its facade to fit-in or reflect a context and that facade is reflected in the observed line-change activity.

Thus we have states where lines dont change, or more so dont complete their change, they just become 'unstable' in their being active. As such a 'changing line 1' in hexagram 02 does not actually change but reflects qualities associated with that line as being 'expressed' by the hexagram overall. Thus, if we extend the changing line patterns of the I Ching to their fullest manifestation then we dont have 6 changing lines, we have symbols of 12 fixed lines that include line states that are interpretable as at the transition phase from yin-to-yang or yang-to-yin but NEVER change into either yin or yang when viewed from the 6-line perspectives.

What this means is we have three states of (1) no change, (2) unstable, (3) change applicable to each line, and combinations of. No change keeps the hexagram 'as is', an unstable line keeps the hexagram 'as is' BUT also adds some 'colour' to the expression of the hexagram. A changed line reflects a 'mutation', a transcendence, of some form and as such a line reflects a POTENTIAL to move from being considered 'unstable' to actually changing.

For example, in hexagram 01, a changing base line reflects an expression of values associatable with where the line is 'going', as in to hexagram 44. BUT it has not got to 44 (if it had our hexagram would BE 44, not 01 apparently changing into 44). We thus see the traditional line comment in the form of a warning about not being too hasty in expressing one's yangness - IOW dont be seduced by the yangness - where seduction is a characteristic of hexagram 44. As such we see here a manifestion of an aspect of hexagram 01, a manifestation describable by analogy to the qualities of the associated hexagram if the change DID occur.

In other words, given hexagram 01 with changing line 1 does NOT necessarily mean hex 01 changing into 44 but hex 01 reflecting a state of hex 01 behaving LIKE hexagram 44. Thus the 'archetype' that is hex 01 never changes, it is 'eternal' but it can 'shape-shift' and so reflect aspects of all other hexagrams through line activity.

As outlined above, since the vertical sequence of hexagrams 01/02 reflects the rotation of the horizontal sequence of 01/02 it is possible to derive a 'binary sequence' that is horizontal for all hexagrams by (a) deriving their vertical sequence and then (b) rotating all hexagrams in that sequence to derive a horizontal sequence.

A Mathematics Aside: In the conversion of the traditional binary sequence of the I Ching to a matrix we move into the realm of isomorphisms. In the matrix :

we find the standard properties of matrices but in particular note a pattern here that reflects properties of what is called a Hermitian matrix. We note that (a) the lead diagonal (highlighted above) is made up of 'pure' hexagram forms where the base trigram is repeated. (b) for each row m the corresponding column n contains the trigrams of the row element swapped (NOT rotated) E.g. in row 1 column 2 we find hexagram 43 that reflects the swapping of the trigrams of hexagram 10, mapped to column 1 row 2; in row 1 column 3 we have 14 that is the trigram swap of row 3 column 1, hexagram 13 etc. As such we see a sort of reflection about the diagonal - which is what we see in a Hermitian matrix of complex numbers where (a) the diagonal is 'pure' in the form of real numbers only, reflecting a complex number and its conjugate sharing the same space (only possible where a + bi = a - bi with b = 0) (b) all of the other positions are complex numbers where the conjugate of column n row m (as in a + bi) is mapped to column m row n (as in a - bi) - as we see in the form of swapping of trigrams above.

In other words, for any given matrix, we can derive the 'conjugate' of a hexagram. The 'pure' forms in the diagonal reflect their own conjugate, eleswhere we have 10 as the conjugate of 43, 13 of 14, 49 of 38, 25 of 34, 23 of 15 etc etc etc - what these patterns indicate is that the focus is at the level of trigrams rather than hexagrams in that the states reflect different orderings of trigrams. Since these matrices are 8 x 8 (2⁶) so the basic unit of measure is at the level of the 8 trigrams. We need to move to a matrix of 64 x 64 to deal with hexagram relationships but expressed in the form of dodecagrams (2¹²).

There is a basic structure at work here, (using matrices we are delving into the area of operator theory) as reflected in studies on the concept of numbers and the derivation of meaning. (Also note that any focus on Hermitian matrices moves us into methodologies applied to quantum mechanics and the use of eigenvalues as well as issues re the Riemann Hypothesis and the zeroes of the zeta-function - the overall focus being on the properties of the lead diagonal in this type of matrix. From there we move into prime number theory etc so I think one can see how people can see so much in the I Ching!)

Note that in the above matrix that the given numbers encode hexagram STRUCTURE. We can convert these numbers to 'true' numbers reflecting the binary ordering of the hexagrams and so derive such concepts as the characteristic polynomial of the matrix, the zeroes of that polynomial (the eigenvalues) and the trace of the matrix.

The main point to note here is the 'common ground' of the matrix of the I Ching hexagrams derived from recursion and the generic properties of matrices and in particular that of the Hermitian matrix a matrix format fundamental to research areas in both Mathematics and Physics.

In conclusion note that when we rotate the hexagrams in the horizontal sequence to derive the vertical sequence so (a) the order of the diagonal changes but we keep the same 'pure' hexagrams and (b) the swapping pattern is retained:

...and so 09/44 reflects trigram swapping (and the reflection across the diagonal) as does 13/14, 33/26, 37/50 etc etc. If we focus on another matrix with the same hexagrams making up the lead diagonal, just different ordering:

...we find that the same 'rules' apply in that the reflections across the diagonal are ones focused on swapping trigrams - as in 21/55, 13/14, 39/38 etc etc where the swapping for the 'pure' forms encoded into the lead diagonal reflects invarience.

Since we are dealing with operator theory, where these matrices reflect properties and method of an operator, we are here dealing with some 'deep' fundamentals of the I Ching, and our neurology in general.