Reading “The Book of Change” by Richard Rutt.

I have always suffered from over-estimating myself.

Really, what is there difficult about reading basically any book in any language? Just take a dictionary, query the sentences one by one, word by word, and you are going to get a rough idea of any text. If you are an Indo-European speaker, try doing it with Latin.

Is it going to work? Well, not really. In Latin you have those pesky inflections, conjugations, declensions, when meaning is transmitted by word forms rather than words themselves.

But what about Chinese, especially old Chinese. Even modern Chinese does not have much grammar and has very little syntax. Surely with Chinese it should be possible?

Those were my thoughts when I bought a standard edition of “周易”, with a preface saying that it is intended to be read by parents to their children.

I spent about a weekend trying to decipher what is going on, and was overwhelmed by flying dragons (龍), speaking elephants (象), strange (magic) numbers appearing out of nowhere, superficial symmetries and a disgusting feeling that “just a little more and I will be able to make sense out of it”. Then another weekend. Then one more. Doesn’t that sound familiar to anyone trying to reverse-engineer a binary file, or even read some spaghetti code in C++?

At some point I gave up and decided that I am going to cheat and read an English translation. I just need to find a decent one, not the one done by anime fanboys which revere everything Asian and are not ready to apply critical thinking to whatever they read.

After browsing a library for a while, I found the “Book of Change” by Richard Rutt, who had been an interpreter for the U.S.~Army during the Korean war in the 1950s.

Unfortunately, even though the book uses pīnyīn for transliteration, it does not use tone marks, which makes it very difficult to find out what is going on, and sometimes allows opposing interpretations. (For example, some structures are described as “cheng”, which can mean both “chéng”(乘) and chēng (撑), which are really opposites.)

It took me months of free time to reverse engineer the book and to place the correct tone marks where they are due (and also research all the cross-references and citations). Does not this also remind you of reading an old piece of code which “somehow works”, but you need to develop and maintain it?

Well, it is a bit hard to explain.

Let us go back to the Korean flag.

The four groups of three lines, broken and solid, are coming from the Book of Changes. In fact, they are not really coming form the original text (Zhōu Yì), but from a later, Hàn dynasty, compilation, called Yì Jīng which, despite being written almost a thousand years later, became the true classic of Chinese culture.

Those groups are called “trigrams”, and represent nothing more than “machine half-words” of three bits. Yes, really, 2400 years ago people already knew about bits.

In fact, those “machine words” of six bits (twice the three-bit half-words) are what the original book is about.

The source of the original text (Zhōu Yì, not Yì Jīng), is a little bit similar to a description of a peculiar assembly language, which gives each “machine word” a short description of what it does, as well as gives implementation details for each bit, that is why exactly it has to be 1 or 0 so that the computational module as a whole produces the desired effect on the computational flow.

The list of the “words” is the following:

䷀ ䷁ ䷂ ䷃ ䷄ ䷅ ䷆ ䷇ ䷈ ䷉ ䷊ ䷋ ䷌ ䷍ ䷎ ䷏ ䷐ ䷑ ䷒ ䷓ ䷔ ䷕ ䷖ ䷗ ䷘ ䷙ ䷚ ䷛ ䷜ ䷝ ䷞ ䷟ ䷠ ䷡ ䷢ ䷣ ䷤ ䷥ ䷦ ䷧ ䷨ ䷩ ䷪ ䷫ ䷬ ䷭ ䷮ ䷯ ䷰ ䷱ ䷲ ䷳ ䷴ ䷵ ䷶ ䷷ ䷸ ䷹ ䷺ ䷻ ䷼ ䷽ ䷾ ䷿

Can you fit a lot of data into 6-bit words? Well, not really, but you could make a calculator which can count to 64.

For example, you could have the word partitioned into 3 parts, 2 bits each. Two bits give 4 instructions: +, -, \(\cup\) \(\cap\), and two two-bit operands, denoting 1 to 4 in decimal. (Why not multiply/divide? Because those are too irregular, and you need to care about carry.)

3 times 3 makes 9, which does not really fit into 8 bits, but if you sacrifice the last bit of the instruction, it works.

Here are your mnemonics for 2-bit operations:

⚌ ⚍ ⚎ ⚏ 𝌁 𝌂 𝌃 𝌄 𝌅

Of course, old Chinese did not do anything this stupid. Instead of dividing one hexagram into three bigrams, to make a calculator which can count to 9, they used three-bit half-words: ☰ ☱ ☲ ☳ ☴ ☵ ☶ ☷, also called trigrams. Or at least the Hàn scholars, which were reverse-engineering Zhōu computations (算命, suàn mìng, from Chinese “calculate commands”.)

They assumed that the instruction set was the following:

1. One 6-bit word defines a command (one hexagram).
2. Two three-bit half-words (trigrams) define the operands for the command, which are unlikely to be immediate operands, because computing with numbers less than 8 is a pity, but were probably displacements relative to the address of the instruction pointer, in which case 8 is enough.
3. Two words pointed to by the operands of the command served as the arguments of the operation.

This gave them a full 6-bit arithmetic, which is enough to execute and arbitrary Turing-complete program and model any interesting phenomenon.

The ancient Chinese did not know an alphabet, but they had to represent their language somehow, which means that they had to encode their language into their primitive computational machines.

As far as I know, they did not know variable-length prefix-free coding, such as UNICODE, so instead of expanding their instruction set to the thousands of codepoints needed to encode all of Chinese, they would just strip-down their language to a bare minimum, to 64 “letters”, which had to represent any idea that a programmer (at that time called a “diviner” as a pun on the fact that computations predict results of a physical process) might want.

Their version of the ASCII table (one instruction=one symbol) is below:

1. ䷀乾 - Qian
2. ䷁坤 - Kun
3. ䷂屯 - Zhun
4. ䷃蒙 - Meng
5. ䷄需 - Xu
6. ䷅訟 - Song
7. ䷆師 - Shi
8. ䷇比 - Bi
9. ䷈小畜 - Xiao Xu
10. ䷉履 - Lu
11. ䷊泰 - Tai
12. ䷋否 - Pi
13. ䷌同人 - Tong Ren
14. ䷍大有 - Da You
15. ䷎謙 - Qian
16. ䷏豫 - Yu
17. ䷐隨 - Sui
18. ䷑蠱 - Gu
19. ䷒臨 - Lin
20. ䷓觀 - Guan
21. ䷔噬嗑 - Shi He
22. ䷕賁 - Bi
23. ䷖剝 - Bo
24. ䷗復 - Fu
25. ䷘无妄 - Wu Wang
26. ䷙大畜 - Da Xu
27. ䷚頤 - Yi
28. ䷛大過 - Da Guo
29. ䷜坎 - Kan
30. ䷝離 - Li
31. ䷞咸 - Xian
32. ䷟恆 - Heng
33. ䷠遯 - Dun
34. ䷡大壯 - Da Zhuang
35. ䷢晉 - Jin
36. ䷣明夷 - Ming Yi
37. ䷤家人 - Jia Ren
38. ䷥睽 - Kui
39. ䷦蹇 - Jian
40. ䷧解 - Jie
41. ䷨損 - Sun
42. ䷩益 - Yi
43. ䷪夬 - Guai
44. ䷫姤 - Gou
45. ䷬萃 - Cui
46. ䷭升 - Sheng
47. ䷮困 - Kun
48. ䷯井 - Jing
49. ䷰革 - Ge
50. ䷱鼎 - Ding
51. ䷲震 - Zhen
52. ䷳艮 - Gen
53. ䷴漸 - Jian
54. ䷵歸妹 - Gui Mei
55. ䷶豐 - Feng
56. ䷷旅 - Lu
57. ䷸巽 - Xun
58. ䷹兌 - Dui
59. ䷺渙 - Huan
60. ䷻節 - Jie
61. ䷼中孚 - Zhong Fu
62. ䷽小過 - Xiao Guo
63. ䷾既濟 - Ji Ji
64. ䷿未濟 - Wei Ji

Of course the ancient Chinese knew the difference between P and BPP complexity classes, but I will re-state them here for clarity:

A language L is in BPP if and only if there exists a probabilistic Turing machine M, such that:

1. M runs for polynomial time on all inputs
2. For all x in L, M outputs 1 with probability greater than or equal to 2/3
3. For all x not in L, M outputs 1 with probability less than or equal to 1/3

P contains all decision problems that can be solved by a deterministic Turing machine using a polynomial amount of computation time, or polynomial time.

Ancient Chinese computers were much more primitive than we have now, and needed an external supply of random bits, usually in the sets of 9.

It is really easy: 6 bits are needed to select one hexagram out of 64, and 3 bits are needed to select a particular line out of the 6 available in a hexagram. In fact, we need 2.5 bits, because 6 is not a power of 2, but the remaining half-bit would be still useful, because sometimes there is a need to apply randomness not just throughout the computational process (the BPP model), but also in the process of decoding the text. The last half-bit is thus used to select the codeword or the “extranumerary” bits in the codewords 1 and 2.

It is not entirely clear what most of the codewords meant in that ISA, but we can be certain about the codewords 1 (䷀乾) and 2 (䷁坤): “call a procedure” (干, old Chinese Qián, modern Chinese Gān, which means “to do”, so very naturally “do” a procedure) and “yield a value” (坤, both Old and Modern Chinese Kūn, “to yield”).

It is easy to guess that ䷇比 is a comparison operator ( A < B), ䷖剝 is a noop operator, and ䷗復 is a copying operator because these words still mean the same thing in modern Chinese. Generally, most single-bit operators are easy to guess.

The evaluation was performed in a series of calls to “eval” and “apply”, which is epitomised in the famous “Yīn/Yàng” symbol:

Specifically, it was probably carried out as some kind of a fluidic solver:

1. https://pmc.ncbi.nlm.nih.gov/articles/PMC6553589/
2. https://theconversation.com/researchers-built-an-analogue-computer-that-uses-water-waves-to-forecast-the-chaotic-future-205842

If the “hexagrams” are actually bone plates with punched holes, it makes sense that in addition to performing calculations, the programmer also has to care about sufficient and not excessive pressure in the machine, hence paying particular attention to the first and the fourth “carrier” bits (holes). Alternatively, if one looks at the earliest depictions of the machine words:

it starts to make sense that the machine words were not punched bone cards, but rather something like clay tablets with troughs and/or bumps, which either impeded or diverted the flow of water.

The ancient Chinese had a very good understanding of binary arithmetic, as we know from other sources, not just the assembly manual:

When everyone in the world knows beauty as beauty, ugliness appears. When everyone knows good as good, not-good arrives. Therefore being and non-being give birth to one another; Difficult and easy give completion to one another; Long and short form one another; High and low fill? one another; Sound and voice harmonize with one another; Ahead and behind follow after one another. Therefore the sage accomplishes things by doing nothing, Furthering a teaching that is without words. All things arise, and he does not leave them. He gives them life but without possessing them. He acts but without relying on his own ability. He succeeds but without dwelling on his success. And because he does not dwell on it, it does not leave him.

This is one of the first self-reflections of a programmer in the world. The especially noteworthy part is “He gives them life but without possessing them”, that is, he is writing programs who appear to be alive, but he publishes them in the open-source form to not possess them, and also “Therefore the sage accomplishes things by doing nothing”, where the sage (the programmer) is relaxing while his program is doing things for him, making him rich his goals.

Ah, at the end of the brief introduction to the Book of Change, I must explain why and how exactly the word “change” comes into play.

In modern terms, the “change”, “yì” (易) would rather be correctly called “bit flipping”. When a bit changes, it switches from a state to an opposite state, there is nothing really happening there. But through the process of flipping bits, the encoding transforms from a set of codewords encoding the input and the program into the set of codewords encoding the answer, and this is how the programmer’s manual came to be known as a “manual of change”.

I should further note that the problem of constantly changing computational frameworks was clearly very painful for the Zhōu programmers, as Dào Dé Jīng testifies:

A good cook changes his knife once a year - because he cuts. A mediocre cook changes his knife once a month - because he hacks. I’ve had this knife of mine for nineteen years and I’ve cut up thousands of oxen with it, and yet the blade is as good as though it had just come from the grindstone. There are spaces between the joints, and the blade of the knife has really no thickness. If you insert what has no thickness into such spaces, then there’s plenty of room more than enough for the blade to play about it. That’s why after nineteen years the blade of my knife is still as good as when it first came from the grindstone.

Nowadays we can see exactly the same situation, the old knife (UNIX, C, and GNU Bash) are still as sharp as in the 70s, whereas Rust and JavaScript keep reinventing themselves every month, because they hack. But I digress.

In addition to the handbook on the instruction set architecture and the implementation, the Yì Jīng, a much later compilation (which includes Zhōu Yì), also includes some documents which were not written by the machine designers.

1. Programmers review (彖传, tuán zhuàn, meaning “the one who infers”, programmer) on the new ISA
2. Circuit/performance analysis profiles (象传, xiàng zhuàn, literally “scheme analysis”) which analyses how signal flows through each of the circuits implementing the instruction
3. Programmer’s manual (繫辭, xì cí), literally “connecting words”, meaning “writing programs”.
4. Short history of the ISA (说卦, shuōguà)
5. A mnemonic for memorising the ISA , (序卦，Xùguà)
6. An attempt to simplify the instruction set by dividing it into two parts, unfinished and seemingly later abandoned (杂卦, záguà, literally “your instruction set is in disarray”)

There is actually no mention of 無爲/无为 in 周易, but it is actually closely related to the ancient Chinese skill of cybernetics, so deeply understood by the Sages.

Wúwéi manifests in the fact that fighting with a machine is of no use (无用). To explain it, imagine the following loop:

while true ; do
      sleep 60
      xscreensaver-command -lock
done

No matter how strong you are, how much willpower you possess, or how skilful you are in typing your password, your work will still be interrupted every minute, unless you follow the principles of 道, and pkill xscreensaver.

In the case of the part of cybernetics dedicated to managing humans, well, the principle is the same – if your management process is draining your power, you will never guide the company into the direction intended.

So, 道 is essentially “the graph of computation”, which the physicists call a “world line”, and 易 is its differential with respect to time in the Minkowski space.

The book stresses quite a few times that spending “action” to achieve what you want to achieve is unlikely to succeed, because fighting against the world line it like fighting against the machine. However, sages manage to apply their “action” to redirect the world line slightly, so that eventually it comes into the point the programmer needs.

Of course, this essay is a game, do not take it too seriously.

However, there are a few things to consider.

Bits

Bits, as far as we understand them, have to intrinsic meaning, and the meaning of ASCII 01000000 (@) is not composed of two independent 0100 and 0000.

Thinking about a-priori meaningless structured things

STEM people often find it enjoyable in thinking about mechanistic things, such as chess, tabletop games, even Rubik’s cube.

The Book of Changes and divination in general seems to be lying in the same domain, but taking this notion to the extreme. There is no meaning in the hexagrams whatsoever, yet they are structured in a fairly advanced way, at least if the tradition (Fu Xi order, Luo River Document, He Map) is studied carefully.

Presumably, thinking about structured meaningless things, that is, iterating over patterns presented to the thinker in pure forms (patterns which have no meaning), allow one to infer some regularities, consistent behaviours in things around him, which do have meaning, things the thinker is actually considering.

In this thought the Book of Changes is consistent not just with the Daoist tradition, but also with the Confucian tradition, and with modern psychology, which is quite aware of patterns in human behaviour, which sometimes overbear normal human lives and bring unhappiness, but also sometimes allow breaking away from the patterns which are not reflected by the person.

Seemingly, the Chinese knew the concept of cavemen even during the Han dynasty: (大传2.III)

上古穴居而野處

I have never heard about European ancient texts mentioning living in caves. The Bible mentions living in a “Garden of Eden”.

Wikipedia and Baidupedia have quite a few references to various memes connected to the Book of Changes: 64 hexagrams, 8 trigrams, Early Chinese mythology, Early Chinese constellations, Heavenly Stems, Earthly Branches, and such.