Yijing hexagram sequences


No sooner had I made a gif animation of the King Wen sequence to satisfy my curiosity than I came across I Ching Sequencer Version 1.0. This sequencer was created in Flash and compares the King Wen sequence (the order of the hexagrams in the Yijing itself), hexagram by hexagram, with the Fuxi binary sequence (i.e. Shao Yong's Xiantian ['Before Heaven'] sequence), and both with the 'mystery sequence' on pp 730–731 of the Wilhelm-Baynes translation (page numbers pertain to the 3rd edition).

Unfortunately, the Fuxi binary sequence is upside down [UPDATE: This has now been corrected in version 2.1]. In order to read the 64 gua as binary numbers with the decimal equivalents 0–63 you have to read them from the top line downwards with yang being 1 and yin 0. For example, the second hexagram in the Fuxi sequence, binary number 000001, is hexagram 23 of the King Wen sequence, not hexagram 24. Think of it like this:

rotate 90° clockwise = = 000001

However, you could equally well regard the binary number 000001 as represented by hexagram 24 reading from the bottom upwards:

This is what the animator has done. You could say this is a more sensible way of rendering hexagrams as binary numbers than what has become the convention, given that binary numbers, like decimal numbers, count up from right to left and hexagrams are formed from bottom to top. But the point is that though the animated sequence is still binary, it is not actually the Fuxi sequence as stated but the Fuxi sequence upside down. You'd have to hold a mirror above the hexagrams to see the Fuxi sequence.

Richard Rutt makes the same error, incidentally, in his book Zhouyi: The Book of Changes, p 91, when he says: 'Hexagrams in the Fuxi order, if written with 0 for broken lines and 1 for whole lines, and with the bottom line at the right, give the binary notation for 0 to 63…' No, they don't. Hexagram 23, second in Fuxi, rendered as binary on that basis is 100000, which in decimal is 32, not 1. Given this confusion, I thought it might be useful to make a simple gif animation myself of the actual Fuxi sequence:

If you are trying to work out what you are watching exactly, consider this: the top line alternates between yin and yang from hexagram to hexagram, the bottom line is yin for the first 32 hexagrams and then yang for the second 32. The fifth line (i.e. counting up from the bottom) goes two yin, then two yang, then two yin, then two yang, for the entire sequence. The fourth line goes four yin, then four yang, and repeats. The third line goes eight yin, then eight yang, and repeats. The second line: 16 yin, 16 yang, 16 yin, 16 yang. Bottom line, as stated, 32 yin, 32 yang. You can examine this more closely on a screenshot of the Fuxi sequence animation frames arranged in an 8 by 8 square, which forms:


The Shao Yong square (Fuxi sequence)

Below is the number of each hexagram as it appears in the Yijing (King Wen order):

This arrangement was set down by the Song dynasty philosopher Shao Yong (1011–1077 CE), six centuries before Wilhelm Leibniz described binary notation. Leibniz published 'De progressione dyadica' in 1679, and in 1701 the Jesuit Joachim Bouvet wrote to him enclosing a copy of Shao Yong's 'Xiantian cixu' (Before Heaven sequence).

The Xiantian diagram below reveals the simple principle behind the sequence of hexagrams, which Shao Yong then arranged in both a square and a circle.

The all-yin hexagram Kun is the all-black column on the outside left going up through six levels, forming the start of the sequence, while the all-yang hexagram Qian is the all-white column on the outside right through six divisions into yin and yang, the end of the sequence. The top row of the diagram corresponds with the top line of the 64 hexagrams. From left to right the 64 divisions are the 64 hexagrams in binary order. (I have provided some examples on another page. See also my notes on a star pattern in the Xiantian in a circle.)

The Xiantian diagram also shows graphically why the top line of the animation alternates between yin and yang, the fifth line alternates every two hexagrams, the fourth line every four hexagrams, the third line every eight hexagrams, the second line every 16 hexagrams, and the bottom line every 32.

The four two-line digrams on the second level upwards are termed greater and lesser yin and yang. Greater yin is the equivalent of old (changing) yin, lesser yang is young (unchanging) yang, lesser yin is young yin, and greater yang is old yang (see How to consult the Yijing). While greater yin and yang are obvious, there appears to be a little confusion over the two digrams made up of both a yin and a yang line. Some published sources have got them the wrong way round, thinking that because, for example, the digram composed of an upper yang and lower yin is on the 'yin side' that therefore it must be lesser yin, but this misses the point completely, namely that the second level position is actually yang (white) with yin below it, and so it is lesser yang. Similarly, lesser yin is yin with yang below it. (Notice that both halves of the diagram divided vertically are the same for the final five levels, that it is only on the first level that the two sides differ. As a result of this the 32 hexagrams on the left side contain 112 yin lines and 80 yang lines and these are called the 'yin hexagrams'; similarly the 32 hexagrams on the right side contain 112 yang lines and 80 yin lines, and are known as the 'yang hexagrams'. Clearly the difference of 32 between 112 and 80 arises from the allocation of yin and yang in the bottom layer, the other five layers being identical.)

The Xiantian diagram gave rise to Shao Yong's square. There are many things to notice about the square. For instance, the hexagrams on the 111000–000111 diagonal (bottom left to top right) are composed of complementary trigrams (eg, yin-yin-yang is complementary to yang-yang-yin). Each of the 8 hexagrams on this diagonal has three yin lines and three yang. The top row of trigrams is identical in each of the 8 horizontal hexagram octets. Going from left to right the order is: kun, gen, kan, xun, zhen, li, dui, and qian. This has the effect of making the upper trigram in each column of 8 hexagrams the same. The lower trigram in each horizontal octet is also the same, in the same order going down. So all 8 hexagrams in the uppermost horizontal octet have kun as the lower trigram, below that they all have gen as the lower trigram, and so on through kan, xun, zhen, li, dui, and qian, the lower trigram of the bottom horizontal octet. The top left to bottom right diagonal consists of hexagrams with the same upper and lower trigram, again in the same order as before. On the aforementioned 111000–000111 diagonal not only are the trigrams complementary in each hexagram but from bottom to top on the upper trigrams and from top to bottom on the lower trigrams the order is again: kun, gen, kan, xun, zhen, li, dui, and qian. This is, of course, the Fuxi order of trigrams, which when divided in half and placed in a circle forms the Xiantian bagua, or Earlier Heaven arrangement of trigrams, also known as the 'World of Thought' arrangement. You can see that the trigrams on the third division into yin and yang in the Xiantian diagram are in the same order.

Notice also the complementary symmetry about the two diagonals, just as top left (000000) is complementary to bottom right (111111) so are the two hexagrams next in towards the centre (001001 and 110110), i.e. hexagrams 52 and 58:

Just to make it clear, here are the complementary pairs on the diagonal bottom left to top right, working in towards the centre, with hexagram numbers as they appear in the Yijing (King Wen sequence) and binary numbers:

11 and 12 (111000 and 000111)

41 and 31 (110001 and 001110)

63 and 64 (101010 and 010101)

42 and 32 (100011 and 011100)

If you convert the binary numbers to decimal the sum of two complementary hexagrams is always 63. (The decimal equivalents of the above binary numbers should not be confused with the ordinal numbers of the hexagrams in the King Wen sequence. For example: 111000 = 56 in decimal and 000111 = 7, sum 63, the hexagrams being 11 and 12, respectively.) You would be correct to think I am merely scratching the surface of Shao Yong's square.

On the 'I Ching Sequencer' website it is possible to advance the hexagrams one at a time, as well as watch them loop. The three sequences run concurrently, with the 'mystery sequence' displayed largest. I'd never really taken a good look at this particular sequence at the back of Wilhelm-Baynes (pp 730–731, in the third edition), but it appears the animator is right, this is indeed a mystery. I have no idea at present what this sequence is or from where it originates (it doesn't appear in the first edition). But it is certainly a fascinating and aesthetically satisfying sequence when seen as a Flash animation. Notice that through the first 7 hexagrams a single yin line appears to rise up through the all-yang Qian hexagram. Then two yin lines appear to perform the same trick together. Then the trigram Kan rises up through Qian, which suggests that in the first two phases rather than a single yin and two yin rising up it is in fact the trigrams Li and Gen rising up, respectively. But by the 17th hexagram of the sequence this pattern is lost and one suspects that rather than a trigram rising up through a hexagram it is in fact a hexagram rising up through a hexagram, but that would require a more detailed study of the sequence, these are just a few provisional observations, first thoughts. By the 23rd in the sequence we see the familiar pattern again with three yin lines rising up together. By the end of the sequence, in the final seven hexagrams, we see a single yang line rising up through the all-yin hexagram Kun, just as a single yin rose up through all-yang Qian at the start. As the creator of the 'I Ching Sequencer' notes:

We can use the metaphor of time-lapse photography to explain the basic purpose and function of the I Ching Sequencer. If one were to sit and watch a sunflower throughout the course of one day, it would be impossible to actually see the movement; although during the one day cycle periodic recognition of movement would occur to the viewer. If a time-lapse movie of this same sunflower were made by condensing one day into one minute we could see the 'magic' of heliotropic plants (those which follow the sun). By compressing and animating a series of changing pictures we can gain a new view of the otherwise non-linear sequence. The I Ching Sequencer can therefore be thought of as time-lapse movies of historical and significant I Ching sequences.

In Flash, by right-clicking you can zoom in on the smaller animations. At the side is a scale showing how many hexagrams of the 64 have played and position in the sequence. You can even play a techno soundtrack to accompany the animation. A fascinating and beautifully executed piece of work. An elegant hexagram sequence is like a mesmerizing dance.


Jing Fang’s ‘Eight Palaces’ arrangement

There is another important sequence, Jing Fang's 'Eight Palaces' (bagong) sequence, I will lightly touch on. I have made an animation of this as well:

Jing Fang lived from 77 to 37 BCE, when he was executed. The eight palaces arrangement (sometimes called 'eight houses') is again in eight horizontal octets. In passing left to right from one hexagram to the next in each octet or palace just one line changes, except for the last member of the octet, going from the seventh hexagram to the eighth, where the top trigram remains the same but the lines in the bottom trigram all change, giving the trigram's complement, as can be seen from a screenshot of the animation frames arranged as an 8 × 8 block. Animation-frame screenshots are useful since they unambiguously show the direction of the sequence with arrows should anyone be in doubt, but the arrangement itself is more conveniently represented below:

Notice that in each horizontal octet or palace the top line never changes. Appendix II of Wilhelm-Baynes gives the hexagrams arranged by houses, but note that though the order of the hexagrams in each house is the same as I have used, the order of the houses Wilhelm gives is different. I have used the original order, which is that of the 'family member' trigrams qian, zhen, kan, gen, kun, xun, li, and dui:

The sons are numbered by the position of the single yang line and the daughters by the position of the single yin line (they are also called eldest, middle, and youngest). You'll see that the first column on the left of the eight palaces arrangement consists of the eight hexagrams where upper and lower trigrams are the same, the name of the hexagram being the same as the trigram. These are sometimes called the 'eight pure hexagrams' (bachungua), and here are known as the 'palace hexagrams' (gonggua), being the hexagrams after which each palace is named: Palace of Qian, Palace of Zhen, and so on going down.

There are other things to notice about this arrangement, such as in the upper row of trigrams of each palace the first four are the same, while in the lower row of trigrams the four in positions 4, 5, 6, and 7 from the left are the same and going down these latter eight groups of four are in the order first of the mother and daughters then the father and sons. Also, as a corollary to this, as a single line changes from one hexagram to the next in the first four hexagrams of each palace the change occurs in the lower trigram, while in the following three the change occurs in the upper trigram.

This is all a direct consequence of the pattern of progression in each palace. Take the Palace of Qian:

Passing from hexagram to hexagram: the first line changes, then the second, then the third, then the fourth, then the fifth and with it the sixth hexagram of the palace is reached. At this point, if the sixth line were to change the hexagram would become the complement of the palace hexagram starting the octet and so a palace hexagram itself heading its own octet (in this case, Kun). Instead, the fourth line changes again, giving the seventh hexagram, known as 'the wandering soul' (youhun). Then the entire lower trigram changes to its complement to give the final hexagram, called 'the returning soul' (guihun). The lower trigram at the start and end of each octet is now the same. This pattern is followed in each of the eight palaces, though it is easiest to observe in the palaces of Qian and Kun. Here is the Palace of Xun (hexagram 57):

Given that Jing Fang's name is associated with the first appearance of 'nuclear trigrams' there are doubtless many more observations to be made about the eight palaces arrangement than these few cursory remarks can exhaust. For further information, see Harmen Mesker's well-researched PDF essay: The Eight Houses.


‘Waxing and waning’ bigua sequence

The bigua or 'sovereign hexagrams' sequence consists of the following 12 hexagrams: 24, 19, 11, 34, 43, 1, 44, 33, 12, 20, 23, 2. Anyone familiar with the Book of Changes will instantly see in their mind's eye what is being referred to in that sequence, but to make it quite clear I was inspired to make a little animation to show the cycle of these 12 'waning and waxing hexagrams' (xiaoxigua), at two different speeds. The one on the left is twice as fast:

The sequence waxes (xi) as yang increases upwards and wanes (xiao) as yin increases upwards. The sequence is shown below, going from top left to bottom right:

The bigua sequence directly correlates with the phases of the moon, the full moon being hexagram 1 and the new moon hexagram 2. The waxing and waning moon phases are laid out below in two rows to mirror the bigua sequence block above. The waxing crescent where yang first appears bright on the right edge of the moon is hexagram 24 where yang enters at the bottom, the crescent growing as yang rises up in hexagram 19. The first quarter is hexagram 11 (half moon), waxing gibbous through hexagrams 34 and 43 until the full moon is reached in hexagram 1. Then yin enters at the bottom in hexagram 44 just as darkness starts to overcome the full moon at the right edge, the moon wanes gibbous, yin increasing in hexagram 33, until the third quarter is reached in hexagram 12. Yin then increases through hexagrams 20 and 23 as the moon becomes a waning crescent, bright still on the left side (the single remaining yang line at the top of 23), until the new moon is reached in hexagram 2.

Notice that when the 12 hexagrams are set out in two rows of six the hexagrams above and below complement each other (yin where yang is in the other hexagram, and vice-versa). In Chinese texts complementary hexagrams are often described as 'laterally linked hexagrams' (pangtonggua). The moon phases are similarly linked. In the illustration above the top and bottom images slot together, as it were.

The sovereign hexagrams are also associated with the 12 months. The cycle is regarded as beginning (though, as I have said, a cycle doesn't begin anywhere) on hexagram 24, which is associated with December and in particular the winter solstice, going through to hexagram 2, which corresponds to November (note that in the first line hoarfrost is underfoot, a textual clue to its seasonal setting as well). Interesting too that 'Approach' (hexagram 19) is above 'Retreat' (33) – a classic reversal – and the judgment of hexagram 19 reads: 'Arrive at 8th month, have disaster.' Although the 'start' of the sequence is regarded as the 11th month, if you regard it as the 1st month of this sequence then hexagram 33 is the 8th, meaning that the disaster (reversal of fortune) referred to in hexagram 19 might be expected a month after the summer solstice (represented by hexagram 44, 7th in the sequence), rather than in mid-September, the traditional 8th month. Certainly timely retreat/withdrawal is usually the best way to avoid disaster. (The traditional hexagram of the 8th month is 20, which is of course structurally the inverse of hexagram 19 and follows it in the King Wen sequence.)

The bigua or xiaoxigua sequence has been attributed to Meng Xi (circa 90–40 BCE). Jing Fang further developed Meng Xi's work. Notice that the first six hexagrams of the first palace of Jing Fang's eight palaces arrangement followed by the first six of the fifth palace, that of father and mother, respectively, are the hexagrams of the bigua sequence in order, although the start point is hexagram 1 rather than 24. Yet, despite the fact that this sequence is a cycle so it doesn't matter where it starts, it's interesting that if you take the first six Kun Palace hexagrams followed by the first six Qian Palace hexagrams…

022419113443 Kun Palace
014433122023 Qian Palace

then you have the bigua sequence upside down, as you can see by mentally turning the above block of 12 hexagrams through 180°.

The bigua sequence is most often seen as a circle of hexagrams in Chinese diagrams.


The Mawangdui sequence

In the sequence of hexagrams in the Mawangdui manuscript (168 BCE) the top trigram remains the same for each horizontal octet, whereas in the Fuxi sequence the bottom trigram remains the same. The Mawangdui sequence, however, probably has no great mathematical or symbolic significance. Richard Rutt has suggested it may simply be a 'finding order' for the hexagrams. If not that, then an attempt to order by trigram that was thought sufficiently important to ditch the received King Wen order, which on textual evidence appears to be the earliest (see below). Each octet begins with one of the eight pure hexagrams, where the upper and lower trigrams are the same. It is interesting to compare the starting hexagrams in the first column on the left with those of the 'Eight Palaces' arrangement. The 1st, 3rd, 5th, and 7th are the same; the 2nd, 4th, 6th, and 8th are inverses (fangua).

See also the animation-frame screenshot. Below are the hexagram numbers in the order they appear in the Mawangdui manuscript, with the standard hexagram numbers (King Wen order) in brackets:

An explanation of the Mawangdui sequence can be found in Richard Rutt's review article: Opening a new field for dragons. (A PDF chart is available giving the hexagrams in King Wen order with corresponding Mawangdui number.)


The King Wen sequence

The King Wen sequence is the order the hexagrams appear in the Yijing:

The reasoning, if any, that informs this sequence is unknown. The hexagrams proceed in pairs, each the inverse of the other, except for the eight hexagrams that are the same both ways up, when the hexagrams of the pair complement each other (yin where yang was and vice-versa), namely hexagrams 1 and 2, 27 and 28, 29 and 30, 61 and 62. There are also four pairs that are both the inverse and complement of each other: 11 and 12, 17 and 18, 53 and 54, 63 and 64.

In the 32 pairs there doesn't appear to be a discernible rule for determining which of the pair should come first. Similarly, the reason for the order of the 32 pairs is obscure, although clearly the start and finish of the sequence is not random, in that it goes from all yang (hexagram 1) and all yin (hexagram 2) to the two hexagrams that have an even distribution of yin and yang throughout the six lines (63 and 64).

I've created a further diagram of the King Wen sequence in a form to aid pattern recognition, after Shao Yong's Xiantian diagram.

Note that there is no evidence for the philosophy of yin and yang before the 4th century BCE, so it could be a mistake to presume too much upon this school of thought when attempting to understand the informing principle of a sequence likely to have been created at least 700 years earlier. It should be emphasised that the actual origin of the 64 hexagram diagrams (as opposed to the aetiological story of King Wen doubling the trigrams) is also a complete unknown. I have already stated, in 'The Mandate of Heaven', that the 64 hexagram figures (not the text) must be the product of a single mind, since they must all be taken together as one concept. Whether that mind was King Wen's we may discover should his tomb ever be found.

This appears to be the earliest known sequence, which you can deduce from textual evidence such as the pattern of paired lines 11/1 and 12/1, 41/5 and 42/2, 43/4 and 44/3, 63/3 and 64/4 (and also 56/4 and 57/6, although this is not an odd-even structural pair). Although there are also textual pairings that appear in hexagrams that do not follow each other, such as 11/5 and 54/5, and 10/3 with 54/1 and 54/2 (10/2 is also connected to 54/2, though this is disguised in Wilhelm-Baynes – the 'dark man' of 10/2 is the same as the 'solitary man' of 54/2). See also my original larger animation of this sequence. The hexagrams, incidentally, have never been numbered in Chinese editions, this is a practice originating from translation.

The mystery of the King Wen sequence has driven many people to attempt to solve it down the centuries, and in the past few decades in particular there have been more than a few making staunch claims to have solved it in various ways. But all of this effort begins to look clumsy in the light of József Drasny's Yi-globe.


For further study: see the archive of Chinese diagrams. An interesting modern sequence relying on a pangtong switch followed by the change of a single line was proposed in 'Philosophy East and West' 29.4 (1979) by Stephen E McKenna and Victor H Mair: A reordering of the hexagrams of the I Ching. [PDF]