Why are there 64 hexagrams in the I Ching?

There are 64 hexagrams because a hexagram is six lines, and each line can only be one of two kinds, solid (yang) or broken (yin). Two choices at each of six positions gives 2 × 2 × 2 × 2 × 2 × 2 = 64 possible figures, no more and no fewer. The same number appears if you stack one of the eight trigrams on top of another: 8 × 8 = 64.

The arithmetic, step by step

Start with a single line: two possibilities. Two lines give four combinations, three lines give eight. Those eight three-line figures are the trigrams (八卦): heaven ☰, lake ☱, fire ☲, thunder ☳, wind ☴, water ☵, mountain ☶ and earth ☷. Our guide to the eight trigrams and the trigram reference cover each.

A hexagram is two trigrams, one above the other. Any of the eight can sit below and any of the eight above, so there are 8 × 8 = 64 pairings. Eight of them are a trigram doubled, such as Hexagram 1, Qian (heaven over heaven) and Hexagram 29, Kan (water over water); the other 56 combine two different trigrams.

That is the whole mathematical reason. The 64 are not a selection chosen from some larger set; they are every six-line figure that can exist. You can see all of them in the hexagram library.

The commentaries' version: doubling

The Great Commentary (繫辭) gives the classic account of how the figures unfold:

易有太極，是生兩儀，兩儀生四象，四象生八卦: "In the Changes there is the Supreme Ultimate. It generates the two modes; the two modes generate the four images; the four images generate the eight trigrams."

Each step doubles: one, two, four, eight. In the Song dynasty, Shao Yong (邵雍) carried the doubling on to sixteen, thirty-two and sixty-four, and arranged the hexagrams in that order, now called the Fu Xi or "Early Heaven" sequence. Zhu Xi printed Shao Yong's diagrams at the front of his own commentary on the Zhouyi, which is a large part of why they became famous. Around 1701 the Jesuit Joachim Bouvet sent one of these diagrams to Leibniz, who saw in it the binary arithmetic he had been working on; that story is told in Leibniz and the I Ching.

Note that this doubling order is a later, systematic arrangement. The order of the received text, the King Wen sequence, follows a different logic: it pairs each hexagram with its inverse or opposite, giving 32 pairs.

Why six lines and not five or seven?

The tradition answers with cosmology. The Discussion of the Trigrams (說卦) says that heaven, earth and humanity each have two aspects (yin and yang for heaven, soft and firm for earth, benevolence and righteousness for humanity), and so 兼三才而兩之，故易六畫而成卦: "it combines the three powers and doubles them; therefore the Changes forms each hexagram with six strokes." The Great Commentary adds 六爻之動，三極之道也, "the movement of the six lines is the way of the three ultimates."

On this reading, the bottom two lines are earth, the middle two humanity, the top two heaven. Six is not arbitrary: it is the smallest figure that gives each of the three realms both a yin and a yang place. This is a philosophical explanation added by the commentaries, not a record of how the figures were first invented, and modern scholars treat it that way.

Who made 64 from 8?

Here the tradition openly disagrees. The Tang scholar Kong Yingda (孔穎達), in the preface to his official subcommentary, lists the competing views: Wang Bi and others held that Fu Xi himself doubled the trigrams; Zheng Xuan's followers said Shennong; Sun Sheng said Yu of Xia; Sima Qian and others said King Wen. No one can settle it from the texts.

What is certain is that 64 was not unique to the Zhouyi. The Rites of Zhou (周禮) says the Grand Diviner kept three systems of Changes, the Lianshan, the Guicang and the Zhouyi, and 其經卦皆八，其別皆六十有四: "their basic trigrams are all eight, and their derived hexagrams all number sixty-four." See Lianshan, Guicang and Zhouyi. Archaeology has also turned up early groups of numbers in sets of three and six on bronzes, bones and bamboo, which many scholars read as records of divination results before the solid and broken line notation became standard; our guide to excavated I Ching texts discusses them.

What the number came to mean

Once the set is complete, it invites being read as a picture of everything. The Sequence of the Hexagrams (序卦) opens with heaven and earth, 有天地然後萬物生焉, "there are heaven and earth, and then the ten thousand things are born", and ends not with completion but with Hexagram 64, Wei Ji, Before Completion, because 物不可窮也, "things cannot be exhausted". The 64 are meant as a closed set of types of situation that still opens onto endless change.

The number also multiplies. Each hexagram has six lines, giving 384 line texts. Counting which lines are changing, a cast can produce 4⁶ = 4,096 distinct results, one for every pairing of a primary and a changed hexagram; the Han-dynasty Jiaoshi Yilin (焦氏易林) supplies a verse for each of those 4,096 pairings. Jing Fang's system of eight palaces reorganises the same 64 into eight families of eight.

Using this when you read

• When a hexagram appears, look at its two trigrams first. The name and the Image usually come straight from that pairing: water over thunder, fire under heaven.
• Remember that every possible six-line figure is in the book. There is no "missing" hexagram for an unusual situation; the art is in finding which time you are in.
• If you want to learn the set, start from the 8 × 8 grid rather than the list; see how to memorise the 64 hexagrams.
• Then cast a reading and identify its trigrams before you read a word of text.