Binary Numbers and the I Ching
The I Ching (pronounced ye jing), or Book of Changes, is one of the world’s oldest books and also one of the most enigmatic. For more than 2,000 years it has been used in the East as a book of divination and it still is studied with awesome reverence as a rich source of Confucian and Taoist wisdom. Tens of thousands of young people in the U.S. (particularly in California), caught up in the current occult explosion and eager to know more about Eastern mysticism and early Chinese history, are now consulting the I Ching as seriously as they consult the Ouija board or the tarot cards. C. C. Jung was convinced of the I Ching’s extraordinary power to foretell the future; he even asked it about the prospects of American sales of a new English translation of itself and got an optimistic answer. Pundits who are deep into occultism—Colin Wilson, for example—have written about their experiences with the I Ching’s terrifying oracular accuracy.
The early history of the I Ching is unknown. Most likely it began as early as the eighth century B.C. as a collection of peasant omen-texts, then slowly over the centuries these documents became combined with stick divination practices. A few centuries before Christ, near the end of the Chou dynasty, it acquired its present form and became one of the five great classics of the Confucian canon.
The combinatorial foundation of the I Ching consists of 64 hexagrams. They display every possible permutation of two types of line when taken six at a time. Each hexagram has a traditional Chinese name. The two kinds of line proclaim the basic duality of Chinese metaphysics: the broken line corresponds to yin, the unbroken line to yang. Taking the lines two at a time, there are 22 = 4 ways to combine them into what are called digrams and 23 = 8 ways to form trigrams. The trigrams, with their Chinese names and symbolic meanings, are shown in the illustration below.
The eight trigrams and some of their meanings
Illustration by Ilil Arbel
There are two ancient ways of displaying the eight trigrams in a circle. The oldest, known as the Fu Hsi arrangement after the mythical founder of China’s first dynasty (the Hsia dynasty, 2205-1766 B.C.), is shown at the left in the bottom illustration on page 111 and also on the cover of this issue. Note that opposite pairs are complementary both in symbolic meaning and in the mathematical sense that each is obtained from the other by replacing yin lines with yang and yang lines with yin. This arrangement, usually surrounding the familiar yin-yang symbol, is still widely used throughout China, Japan and Korea as a good-luck charm to put over doorways and on jewelry. It is also called the “earlier heaven” or “primal arrangement.” The King Wen arrangement (after the legendary father of the founder of the Chou dynasty), shown at the right in the illustration below (also called the “later heaven” or “inner-world arrangement”), abandons the complementary positioning of the Fu Hsi sequence, so that the trigrams at the cardinal points of the compass symbolize the seasons in cyclic order. Starting at south (traditionally shown at the top) and going clockwise, the hexagrams at the cardinal points stand for summer, fall, winter and spring.
Fu Hsi arrangement of trigrams (left) and King Wen arrangement (right)
Illustration by Ilil Arbel
The oldest way of arranging the 64 hexagrams, which are known as the King Wen sequence, is the order in which they appear in the I Ching [see illustration below]. The rows are taken from right to left as the numbering indicates. Note that the hexagrams are paired in a singular way. Each odd-numbered hexagram is followed by a hexagram that is either its inverse or its complement. If the odd hexagram has twofold symmetry (is the same upside down), it is complemented to produce the next hexagram. If it lacks twofold symmetry, it is inverted.
King Wen arrangement of the 64 I Ching hexagrams
Illustration by Ilil Arbel
Is there any kind of mathematical order that determines the sequence in which the hexagram pairs follow one another? This is an unsolved problem. From time to time a student of the I Ching announces his discovery of a mathematical scheme underlying the arrangement of pairs, but on closer inspection it turns out that so many arbitrary assumptions are made that in effect the order must be assumed before it emerges from the analysis. As far as anyone knows, the pairs of the King Wen sequence are in random order, and there is no known basis for determining which member of a pair precedes the other.
Not until the 11th century did Chinese scholars discover a very simple and elegant way to order the hexagrams. This arrangement is attributed to Fu Hsi [see the illustration below]. The white space at the bottom represents the t’ai chi, the state of the universe when it was “without form, and void” (as Genesis 1:2 puts it). This undifferentiated chaos divides into the yin (colored) and yang (black) halves of the row labeled 1. In row 2 we see the yin dividing into yin and yang, and similarly the yang. These binary divisions continue upward for six steps.
How six yin-yang divisions generate the 64 hexgrams
Illustration by Ilil Arbel
The chart now automatically gives all the polygrams of orders 1 through 6. Divide rows 1 and 2 vertically into four equal parts, replace the color in each part with broken (yin) lines and you have the four digrams. Rows 1, 2 and 3, divided vertically into eight equal parts, give the eight trigrams. Rows 1, 2, 3 and 4, in 16 parts, give the 16 tetragrams, rows 1, 2, 3, 4 and 5, in 32 parts, give the 32 pentagrams, and rows 1, 2, 3, 4, 5 and 6, in 64 parts, give the 64 hexagrams. The illustration below shows the hexagrams in their traditional Fu Hsi, or “natural,” order. Taking them from right to left, starting at the bottom row, the hexagrams correspond to those provided by the Fu Hsi chart when read from left to right.
The Fu Hsi sequence that corresponds to binary numbers 0 through 63
Illustration by Ilil Arbel
We are now ready to understand why Leibniz, who thought he had invented the binary system in the late 17th century, was so staggered when he first learned of the Fu Hsi sequence from Father Joachim Bouvet, a Jesuit missionary in China. Substitute 0 for each unbroken line, 1 for each broken line, then take the hexagrams in order, reading upward on each, and you get the sequence 000000, 000001, 000010, 000011, ..., 111111. It is none other than the counting numbers from 0 through 63 expressed in binary notation!
Both Leibniz and Father Bouvet were convinced that Fu Hsi, smitten by divine inspiration, had discovered binary arithmetic, but there is not the slightest evidence for this. The 11th-century I Ching scholars had done no more than discover a natural way to arrange the hexagrams. It was not until the time of Leibniz that the Fu Hsi sequence was recognized as being isomorphic with a useful arithmetical notation.
Since the powers of 2 turn up everywhere in mathematical and physical structures, it is not surprising that Chinese scholars have been able to apply the 64 hexagrams to almost everything, from crystal structures to the solar system and the cosmos. Z. D. Sung, in his amusing little book The Symbols of Yi King (Shanghai, The China Modern Education Company, 1934), tells how he was rotating a matchbox in his hand one day (to simulate the earth’s rotation as it goes around the sun) when he suddenly perceived a natural way to generate the eight trigrams at the corners of a cube.
Let the three Cartesian coordinates of a unit cube, x, y, z, indicate the first, second and third digits of a three-digit binary number. Label the corner where the coordinates originate with 000. The other corners are labeled with three-digit binary numbers for 0 through 7, with 0 and 1 indicating the distance of the corner from the origin in each coordinate direction. The eight numbers correspond, of course, to the eight trigrams, with complementary trigrams at diametrically opposite corners of the cube [see illustration below]. By a similar procedure corners of unit hypercubes generate the higher-order polygrams. The 64 hexagrams correspond to six-digit binary numbers at the corners of a six-dimensional hypercube.
Trigrams generated by a cube
Illustration by Ilil Arbel
Instead of plunging into higher dimensions, Sung divides the cube into 64 smaller cubes that he identifies with the 64 “moods” of the classical syllogism. (The major premise, the minor premise and the conclusion of a syllogism can each be of four different forms, giving 64 possible moods.) Sung was probably unaware that this had been done earlier by C. Howard Hinton in his 1904 book The Fourth Dimension (pages 90-106). Hinton takes a curious step into hyperspace. By considering the four “figures” of each syllogism (an ancient division based on the ordering of the subject, predicate and middle terms), he obtains 256 varieties that he identifies with the 256 cells of a 4-by-4-by-4-by-4 hypercube. Cells corresponding to traditionally valid syllogisms are colored black, then the hypercube is projected onto an ordinary 4-by-4-by-4 cube. The black cells are seen to be symmetrically disposed around one corner of the cube except for one cell that should be black but is not. This led Hinton to the discovery that the anomalous syllogism is valid after all if one applies a more liberal interpretation to syllogisms, one in which the predicate is quantified as well as the subject.
But we have strayed from the I Ching. The book (aside from its “Ten Wings,” which are appendixes by Confucian metaphysicians) consists essentially of the 64 hexagrams, each followed by a brief explanation of the symbol and six “appended judgments.” If the book is to be used as an oracle, one of the hexagrams must be randomly selected, and this must be done in such a way that the rules tell how to transform the chosen hexagram to a second one.
The oldest selection procedure, still followed by those who take the I Ching the most seriously, calls for 50 yarrow stalks, each one to two feet long. If yarrow stalks are not obtainable, 50 thin wooden sticks will serve. They should be kept in a lidded receptacle at a spot not lower than a man’s shoulders. The I Ching, carefully wrapped in clean silk, is kept alongside the sticks.
The book must never be consulted lightly. If you ask it something frivolous or in a skeptical mood, the book gives frivolous or meaningless answers. One should be completely relaxed, physically and mentally. It is essential to think of nothing, throughout the ceremony, other than the question being asked.
Let us assume that you are asking the I Ching a question and also casting the sticks. Your first step is to unwrap the book, spread the silk on a table and place the book on top. (The cloth protects the I Ching from impure surfaces.) An incense burner and the receptacle containing the sticks are placed beside the book. With your back to the south, make three kowtows, touching your forehead to the ground; then, still kneeling, pass the 50 sticks three times through the incense smoke by holding them horizontally and moving your hand in a clockwise circle. Return one stick to its container. It plays no further role in the ceremony.
Put the 49 sticks on the cloth, then with your right hand quickly divide them randomly into two piles. Call the left pile A, the other B. Take a stick from B and put it between the last two fingers of your left hand. With your right hand, push away four sticks at a time from pile A until one, two, three or four sticks remain. Place those sticks between your left second and third fingers. Next diminish pile B by pushing away four sticks at a time until one, two, three or four sticks remain. Place these between your left first and second fingers. (This last step can be shortened. Because the sum of the two remainders must be 0, modulo 4, the second remainder is easily calculated from the first.) Your left hand now holds either five or nine sticks. (The possible combinations are 1, 1, 3; 1, 2, 2; 1, 3, 1, and 1, 4, 4.) Put all these sticks to one side.
The remaining sticks are bunched together and exactly the same dividing procedure is repeated with them, beginning with the random division into two piles. At the finish your left hand will hold either four or eight sticks. (The possible combinations are 1, 1, 2; 1, 2, 1; 1, 3, 4, and 1, 4, 3.) Place them aside, next to the group put aside previously.
Bunch the remaining sticks and repeat the dividing procedure a third time. Your left hand again will hold either four or eight sticks. Put them aside, next to the two groups already there.
The number of sticks that now remain will be either 24, 28, 32 or 36. Count them by groups of four (that is, divide the total number by 4). The quotient will be 6, 7, 8 or 9. These four digits are the ritual numbers, which indicate the character of the bottom line of the hexagram. If the digit is even (6 or 8), the line is yin (broken); if it is odd (7 or 9), the line is yang (unbroken). But the ritual numbers tell you more. Seven and 8 mean that the line (whether yin or yang) is a stable line that cannot be altered. Six and 9 indicate a “moving” line that can be changed (for reasons soon to be explained) to its opposite.
All 49 sticks are now bunched together and the entire ritual is repeated to obtain the hexagram’s second line from the bottom. Four more repetitions give the remaining four lines. The entire ceremony, performed without haste, takes about 20 minutes.
Look up the chosen hexagram in the I Ching and study its accompanying text carefully. The text will answer your question and give counsel with reference to the present situation. If all six lines of the hexagram are stable, that is the end of the matter. But if one line or more are moving, change them to their opposites and look up the new hexagram. The commentary will pertain to what you can expect in the future if you follow the counsel of the first hexagram.
After the one or two hexagrams have been written down and the relevant passages in the I Ching have been read and meditated on, light another stick of incense, make three more kowtows of gratitude, put the sticks back in their box, rewrap the I Ching in its silk, then put book and sticks back in their usual high place.
Those too lazy to go through the ancient stick ritual can use a simpler method of casting that has been popular in China for several centuries. It calls for six identical coins, preferably old Chinese coins with square holes. They should be kept polished and should never be removed from their container except when the I Ching is being consulted. Observe the same beginning ritual followed for the sticks: kowtowing, kneeling, passing the coins through the incense and so on. Shake the coins in your cupped hands and let them drop simultaneously to the cloth. Having previously decided on which sides of the coins are yin and which yang, consult the following chart to determine whether the throw gives you 6, 7, 8 or 9.
Three yins = 6 (a moving yin line).
Two yins, one yang = 7 (static yin line).
Two yangs, one yin = 8 (static yangline).
Three yangs = 9 (moving yang line).
(If one thinks of the yin side as 2 and the yang side as 3, the sum of the three values will be the desired ritual number.)
Working out the probabilities provided by the stick and coin procedures reveals a subtle difference between the two divination methods. As far as picking the initial hexagram is concerned, the methods are virtually the same, but the probabilities are not the same in choosing the second hexagram. It is not hard to show that in both procedures the probability of choosing a broken line at each of the six steps is 1/2: the same as that of choosing an unbroken line. (This assumes that each time the sticks are randomly divided into piles A and B and A is reduced to one, two, three or four sticks, the probabilities for each of the four outcomes are equal. This is not strictly true, but the deviations from equality are so slight that they have a negligible effect on the final results.) Thus any hexagram has the same probability of being selected as any other. The two procedures are also alike in giving a probability of 1/4 that a given line will be moving. Since there are six lines, 6/4, or 1 1/2, lines of a hexagram, on the average, will be moving.
When coins are used, the probability that a broken line will change is the same (1/4) as the probability that an unbroken line will change, and similarly the probability is 3/4 that each type of line will remain stable. But when sticks are used, this is not the case. The probability that a broken line will change is 1/16 as compared to 3/16 for an unbroken line (or respective probabilities of 7/16 versus 5/16 that the lines will remain stable). In other words, when sticks are cast, it is three times more likely that an unbroken line will change than a broken one. It is true that any hexagram is as likely to be chosen first as any other, but the more broken lines a hexagram has, the more likely it is that it will appear as the second hexagram. Purists who object to coin-casting have sound mathematical support. Not only does the stick ritual discourage frivolous consultation but also its asymmetry produces a more interesting set of probabilities. We shall say nothing about such impious corruptions as the practice of obtaining the ritual numbers from dollar bills, license plates, telephone numbers and so on.
To readers who may wish to experiment with the I Ching, my first recommendation is the Richard Wilhelm and C. F. Baynes translation, rendered into English from the German.
The Wilhelm-Baynes volume includes the famous foreword by Jung in which he explains the oracular power of the I Ching by his theory of “synchronicity," a theory defended by Arthur Koestler in his recent book The Roots of Coincidence. According to Jung, the I Ching’s predictions, and relevant events that actually happen, are not causally linked in the Western scientific sense. They are “acausally” related in the Eastern metaphysical sense of being parts of a vast cosmic design that lies beyond the reach of science but is partially accessible to the subconscious mind of the person who casts the sticks. The 64 hexagrams and their meanings are Jungian archetypes, deeply engraved on the collective unconscious of humanity.
Tough-minded skeptics who test the I Ching realize at once why the book seems to work. The text is so ambiguous that, no matter what hexagrams are selected, it is always possible to interpret them so that they seem to apply to the question. Indeed, the scope for intuitive interpretation is so great that in China before Mao (I do not know how it is today) there was a large class of professional I Ching interpreters whose services were available for a fee on street corners, at fairs and in marketplaces. Surely one reason for the popularity of coin-casting was that it maximized the profits of these fortune-tellers by speeding up their readings.
And if the I Ching’s predictions fail to materialize? Well, perhaps the text was not correctly interpreted, or maybe you were not in the right frame of mind when you were tossing the sticks or coins. Besides, the future is not completely determined. The I Ching, like the stars of astrology, does no more than indicate probable trends.
Tender-minded believers in the occult, who have not yet consulted the I Ching and who long for powerful, mysterious magic, are hereby forewarned. This ancient book’s advice can be far more shattering psychologically than the advice of any mere astrologer, palmist, crystal gazer or tea-leaf reader.
--Originally published: Scientific American 230(1);108-113. (January, 1974)