King Wen sequence

The King Wen sequence (Chinese: 文王卦序) is an arrangement of the sixty-four divination figures in 易經 Yì Jīng, the I Ching or Book of Changes. They are called hexagrams in English because each figure is composed of six 爻 yáo—broken or unbroken lines, that represent 陰 yin or 陽 yang respectively.

The King Wen sequence is also known as the received or classical sequence because it is the oldest surviving arrangement of the hexagrams. Its true age and authorship are unknown. Traditionally, it is said that 周文王 Zhōu Wén Wáng (King Wen) arranged the hexagrams in this sequence while imprisoned by 商紂王 Shāng Zhòu Wáng in the 12th century BC. A different arrangement, the binary sequence named in honor of the mythic culture hero 伏羲 Fú Xī, originated in the Song Dynasty. It is believed to be the work of scholar 邵雍 Shào Yōng (1011–1077 AD). As mirrored by the 先天 Earlier Heaven and 後天 Later Heaven arrangements of the eight trigrams, or 八卦 bā guà, it was customary to attribute authorship to these legendary figures. Of the two hexagram arrangements, the King Wen sequence is, however, of much greater antiquity than the Fu Xi sequence.^[1]

Structure of the sequence

The 64 hexagrams are grouped into 32 pairs. For 28 of the pairs, the second hexagram is created by turning the first upside down (i.e. 180° rotation). The exception to this rule is for the 8 symmetrical hexagrams that are the same after rotation (1 & 2, 27 & 28, 29 & 30, 61 & 62). Partners for these are given by inverting each line: solid becomes broken and broken becomes solid. These are indicated with icons in the table below.

Given the mathematical constraints of these simple rules, the number of lines that change within pair partners will always be even (either 2, 4, or 6). Whereas the number of lines that change between pairs depends on how the pairs are arranged, and the King Wen Sequence has notable characteristics in this regard. Of the 64 transitions, exactly 48 of them are even changes (32 within-pairs plus 16 between-pairs) and 16 are odd changes (all between-pairs). This is a precise 3 to 1 ratio of even to odd transitions. Of the odd transitions, 14 are changes of three lines and 2 are changes of one line. Changes of five are absent.

1  ↕  2	3  ~  4	5  ~  6	7  ~  8	9  ~ 10	11 ~ 12	13 ~ 14	15 ~ 16
17 ~ 18	19 ~ 20	21 ~ 22	23 ~ 24	25 ~ 26	27 ↕ 28	29 ↕ 30	31 ~ 32
33 ~ 34	35 ~ 36	37 ~ 38	39 ~ 40	41 ~ 42	43 ~ 44	45 ~ 46	47 ~ 48
49 ~ 50	51 ~ 52	53 ~ 54	55 ~ 56	57 ~ 58	59 ~ 60	61 ↕ 62	63 ~ 64

Dual hexagrams

The I Ching book was traditionally split up in two parts with the first part covering the first 30 hexagrams of the King Wen sequence and the second part with the remaining 34.^[2][3] The reason for this was not mentioned in the classic commentaries but was explained in later Yuan dynasty commentaries: 8 hexagrams are the same when turned upside down and the other 56 present a different hexagram if inverted. This allows the hexagrams to be displayed succinctly in two equal columns or rows of 18 unique hexagrams each; half of the 56 invertible hexagrams plus the 8 non-invertible.^[4][5]

Explanation

Over the centuries there were many attempts to explain this sequence. Some basic elements are obvious: each symbol is paired with an "upside-down" neighbor, except for 1, 27, 29, and 61 which are "vertically" symmetrical and paired with "inversed" neighbors.

Its combinatorial mathematical basis was explained for the first time in 2006.^[4]

Other hexagram sequences

• Binary sequence, also known as Fu Xi sequence or Shao Yong sequence
• Mawangdui sequence^[1]
• Eight Palaces sequence (attributed to Jing Fang).

See also

• Timewave zero
• Wen Wang Gua

References

1. 1 2 Marshall, Steve Yijing hexagram sequences
2. ↑ Hacker, Edward A.; Moore, Steve (6 May 2003). "A brief note on the two–part division of the received order of the hexagrams in the Zhouyi" (PDF). Journal of Chinese Philosophy. 30 (2): 219–221. doi:10.1111/1540-6253.00115. Retrieved 31 May 2010. 
3. ↑ Bent Nielsen (2003). A companion to Yi jing numerology and cosmology: Chinese studies of images and numbers from Han (202 BCE-220 CE) to Song (960-1279 CE). Routledge. p. 83. ISBN 978-0-7007-1608-1. Retrieved 31 May 2010. 
4. 1 2 Cook, Richard S. (2006). "《周易》卦序詮解 (Zhou yi guaxu quanjie)" (JPEG Image, 1024x793). Retrieved 22 May 2010.  STEDT Monograph 5: Classical Chinese Combinatorics: Derivation of the Book of Changes Hexagram Sequence. 660 pages. ISBN 0-944613-44-6. OCLC 77009740.
5. ↑ "Yijing Dao - Archive of Yijing-related scans from Chinese and other sources". 20 February 2010. Retrieved 19 May 2010. If you look at the scan of ztd601 you will see two rows of 18 hexagrams, and notice that the hexagrams that are different when inverted have the hexagram name written upside down above them (the table reads from right to left, with hexagram 1, Qian, upper right). This is the secret of it, a single hexagram is made to represent two hexagrams when its inverse differs. There are eight hexagrams the same both ways up, occurring in the following pairs: 1/2, 27/28, 29/30, and 61/62. If you look now at the diagram you can see that six of these hexagrams occur in the top row of 18 hexagrams while only two appear in the bottom row of 18. This means that the top row represents 30 individual hexagrams while the bottom row accounts for 34 hexagrams. This very clever and yet simple arrangement appears to be the reasoning behind the unequal division, which is actually an equal division when 'dual hexagrams' are used in this way. The same principle is also shown in ztd762. 

External links

• "Sloane's A102241 : Hexagrams of the Yi Jing interpreted in base 10". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. 
• 'Derivation of the Timewave from the King Wen Sequence of Hexagrams' -(Archive link)