King Wen sequence

King Wen sequence

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The King Wen sequence is an arrangement of the sixty-four divination figures in 易經 Yì Jīng
I Ching
The I Ching or "Yì Jīng" , also known as the Classic of Changes, Book of Changes and Zhouyi, is one of the oldest of the Chinese classic texts...

, the I Ching or Book of Changes. They are called hexagrams
Hexagram (I Ching)
The I Ching book consists of 64 hexagrams.A hexagram is a figure composed of six stacked horizontal lines , where each line is either Yang , or Yin . The hexagram lines are traditionally counted from the bottom up, so the lowest line is considered line 1 while the top line is line 6...

in English because each figure is composed of six 爻 yáo—broken or unbroken lines, that represent 陰 yin or 陽 yang
Yin and yang
In Asian philosophy, the concept of yin yang , which is often referred to in the West as "yin and yang", is used to describe how polar opposites or seemingly contrary forces are interconnected and interdependent in the natural world, and how they give rise to each other in turn. Opposites thus only...

 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 of Zhou
King Wen of Zhou family name : Ji , Clan name : Zhou Personal name: Chang, known as Zhou Chang or Xibo Chang was the founder of the Zhou Dynasty and the first epic hero of Chinese history....

 (King Wen) arranged the hexagrams in this sequence while imprisoned by 商紂王 Shāng Zhòu Wáng
King Zhou of Shang
Emperor Xin of Shang was the last king of the Shang Dynasty. He was later given the pejorative posthumous name Zhòu . He is also called Zhou Xin or King Zhou . He may also be referred to by adding "Shang" in front of any of his names...

 in the 12th century BC. A different arrangement, the binary sequence named in honor of the mythic culture hero
Culture hero
A culture hero is a mythological hero specific to some group who changes the world through invention or discovery...

 伏羲 Fú Xī, originated in the Song Dynasty
Song Dynasty
The Song Dynasty was a ruling dynasty in China between 960 and 1279; it succeeded the Five Dynasties and Ten Kingdoms Period, and was followed by the Yuan Dynasty. It was the first government in world history to issue banknotes or paper money, and the first Chinese government to establish a...

. It is believed to be the work of scholar 邵雍 Shào Yōng
Shao Yong
Shao Yong , courtesy name Yaofu , named Shào Kāngjié after death, was a Song Dynasty Chinese philosopher, cosmologist, poet and historian who greatly influenced the development of Neo-Confucianism in China....

 (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.

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. The reason for this was not mentioned in the classic commentaries but was explained in later Yuan dynasty
Yuan Dynasty
The Yuan Dynasty , or Great Yuan Empire was a ruling dynasty founded by the Mongol leader Kublai Khan, who ruled most of present-day China, all of modern Mongolia and its surrounding areas, lasting officially from 1271 to 1368. It is considered both as a division of the Mongol Empire and as an...

 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.
The Creative, Heaven
 1 → 
||||||




䷁  坤 kūn
The Receptive, Earth
 2 → 
¦¦¦¦¦¦


䷂   屯 chún
Difficulty At The Beginning
 3 → 
|¦¦¦|¦
  ← 4 
䷃  蒙 méng
Youthful Folly


䷄  需
Waiting
 5 → 
|||¦|¦
  ← 6 
䷅  訟 sòng
Conflict


䷆  師 shī
The Army
 7 → 
¦|¦¦¦¦
  ← 8 
䷇  比
Holding Together


䷈  小畜 xiǎo chù
The Taming Power of the Small
 9 → 
|||¦||
 ← 10 
䷉  履
Treading (Conduct)


䷊  泰 tài
Peace
11 → 
|||¦¦¦
 ← 12 
䷋  否
Standstill (Stagnation)


䷌  同人 tóng rén
Fellowship with Men
13 → 
|¦||||
 ← 14 
䷍  大有 dà yǒu
Possession in Great Measure


䷎  謙 qiān
Modesty
15 → 
¦¦|¦¦¦
 ← 16 
䷏  豫
Enthusiasm

䷐  隨 suí
Following
17 → 
|¦¦||¦
 ← 18 
䷑  蠱
Work On What Has Been Spoiled (Decay)


䷒  臨 lín
Approach
19 → 
||¦¦¦¦
 ← 20 
䷓  觀 guān
Contemplation (View)


䷔  噬嗑 shì kè
Biting Through
21 → 
|¦¦|¦|
 ← 22 
䷕  賁
Grace

䷖  剝
Splitting Apart
23 → 
¦¦¦¦¦|
 ← 24 
䷗  復
Return (The Turning Point)


䷘  無妄 wú wàng
Innocence (The Unexpected)
25 → 
|¦¦|||
 ← 26 
䷙  大畜 dà chù
The Taming Power of the Great


䷚  頤
The Corners of the Mouth (Providing Nourishment)
27 → 
|¦¦¦¦|




䷛  大過 dà guò
Preponderance Of The Great
28 → 
¦||||¦


䷜  坎 kǎn
The Abysmal (Water)
29 → 
¦|¦¦|¦




䷝  離
The Clinging, Fire
30 → 
|¦||¦|



Influence (Wooing)
31 →
¦¦|||¦
← 32
䷟  恆 héng
Duration

䷠  遯 dùn
Retreat
33 → 
¦¦||||
 ← 34 
䷡  大壯 dà zhuàng
The Power of the Great


䷢  晉 jìn
Progress
35 → 
¦¦¦|¦|
 ← 36 
䷣  明夷 míng yí
Brilliance Injured


䷤  家人 jiā rén
The Family (The Clan)
37 → 
|¦|¦||
 ← 38 
䷥  睽 kuí
Opposition

䷦  蹇 jiǎn
Obstruction
39 → 
¦¦|¦|¦
 ← 40 
䷧  解 xiè
Deliverance


䷨  損 sǔn
Decrease
41 → 
||¦¦¦|
 ← 42 
䷩  益
Increase

䷪  夬 guài
Break-Through (Resoluteness)
43 → 
|||||¦
 ← 44 
䷫  姤 gòu
Coming To Meet


䷬  萃 cuì
Gathering Together (Massing)
45 → 
¦¦¦||¦
 ← 46 
䷭  升 shēng
Pushing Upward


䷮  困 kùn
Oppression (Exhaustion)
47 → 
¦|¦||¦
 ← 48 
䷯  井 jǐng
The Well


䷰  革
Revolution (Molting)
49 → 
|¦|||¦
 ← 50 
䷱  鼎 dǐng
The Cauldron


䷲  震 zhèn
The Arousing (Shock, Thunder)
51 → 
|¦¦|¦¦
 ← 52 
䷳  艮 gèn
Keeping Still, Mountain


䷴  漸 jiàn
Development (Gradual Progress)
53 → 
¦¦|¦||
 ← 54 
䷵  歸妹 guī mèi
The Marrying Maiden


䷶  豐 fēng
Abundance
55 → 
|¦||¦¦
 ← 56 
䷷  旅
The Wanderer


䷸  巽 xùn
The Gentle (The Penetrating, Wind)
57 → 
¦||¦||
 ← 58 
䷹  兌 duì
The Joyous, Lake


䷺  渙 huàn
Dispersion (Dissolution)
59 → 
¦|¦¦||
 ← 60 
䷻  節 jié
Limitation

䷼  中孚 zhōng fú
Inner Truth
61 → 
||¦¦||




䷽  小過 xiǎo guò
Preponderance of the Small
62 → 
¦¦||¦¦


䷾  既濟 jì jì
After Completion
63 → 
|¦|¦|¦
 ← 64 
䷿  未濟 wèi jì
Before Completion




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
Combinatorics
Combinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Aspects of combinatorics include counting the structures of a given kind and size , deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria ,...

 basis was reportedly explained for the first time in 2006.

Other hexagram sequences

  • Binary sequence, also known as Fu Xi sequence or Shao Yong
    Shao Yong
    Shao Yong , courtesy name Yaofu , named Shào Kāngjié after death, was a Song Dynasty Chinese philosopher, cosmologist, poet and historian who greatly influenced the development of Neo-Confucianism in China....

     sequence
  • Mawangdui sequence
  • Eight Palaces sequence (attributed to Jing Fang
    Jing Fang
    Jing Fang , born Li Fang , courtesy name Junming , was a Chinese music theorist, mathematician and astrologer. Born in present-day Puyang, Henan during the Han Dynasty , he was the first to notice how closely a succession of 53 just fifths approximates 31 octaves...

    ).

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