zhou wen wang’s development of yijing (book of changes) 64 hexagrams

kevin dobson mequet 605 WATKINS STREET CONWAY, ARKANSAS 72034 501-327-9470 [email protected]

Never doubt that a small group of committed citizens can change the world: indeed its the only thing that ever has. Margaret Mead

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Zhou Wen Wangs Development of Yijing (Book of Changes) 64 Hexagrams AN INDUCTIVE RECREATION OF BINARY EXTRAPOLATIVE OPERATIONS

END OF SHANG DYNASTY, FOUNDING OF ZHOU DYNASTY 12TH/11TH CENTURIES BCE CHINA

By Kevin Mequet 2013

The Master said, Reviewing the old as a means of realizing the new

such a person can be considered a teacher.

Confucius, Analects (2, 11)

Introduction

Gottfried Wilhelm Leibniz (1646 1716) spent a significant amount of his time and creative

intellect upon the subject of China.1 While not the first western sinologistthat singular

distinction went to Li Madou/Matteo Ricci (1552 1610) a century before2he was

certainly an ardent sinophile. Whether in conversations and studies in Western Europe or

extensive correspondences with Roman Catholic missionaries in China, he investigated a

broad range of subjects from natural philosophy, theology, philosophy, politics or as is the

focus of this essay upon mathematics. As the popularizer of a base-2 binary number and

arithmetic applications system in Europe, he was intrigued and transfixed by his vicarious

encounter with the very real possibility of ancient Chinas invention and use of this very

same systemliterally several thousand years before its (re)invention in Western Europe.

Unfortunately Leibniz had two liabilities operating against him. First, unbeknownst to

him, the materials made available to him in Western Europe and the correspondences with

which he prolifically participated were teleological and imprudent by todays standards of

scholarship and academic practice. Second, also unbeknownst to him, the first emperor of

unified China in 221 BCE, Qin Shihuangdi (259 210 BCE), destroyed practically all

mathematical and literary materials in a systematic consolidation of power disrupting a long-

1 Michela Fontana, Matteo Ricci: A Jesuit in the Ming Court, Rowmann & Littlefield Publishers, Inc.,

(2011), p. 290 2 Ibid, p. 264



2

lived continuity of rich intellectual provenance.3 The importance of this cannot be overstated

as we will see.

A workable base-2 binary system was extant in Western Europe with which Leibniz was

fascinated and conversant such that Roman Catholic Church Reverend Father Joachim

Bouvet (1656 1730), a Jesuit missionary to the Imperial Capitol in Beijing for most of his

life, corresponded with him about coincident resonances with the 64 Hexagram matrix he

erroneously associated with Fu Xi First of The Three Sovereigns of Ancient China 30th

century BCE.4 The first iteration of the Yijing (Book of Changes) by Wen Wang of Zhou

occurred in late 12th century BCE. The hexagrams they were looking at were Wen Wangs

and not Fu Xis. Due to the loss of continuity precipitated by Qin Shihuangdi an inferior

turn-of-the-eras reconstruction of Wen Wangs Yijing was reformed by Shao Yung (1011

1077 CE) during the great intellectual renaissance of the Song Dynasty.5 The Jin northern

insurrection and Mongol invasion brought the Song dynasty to an end in the Late 13th

century CE. Mongol Khan successors were assimilated and alloyed into Chinese culture

founding the Yuan Dynasty that lasted into the Late 14th century CE. The collapse of the

Yuan gave birth to a golden period of civil society valorizing Confucius and ushering in Neo

Confucianism under the Ming Dynasty that lasted until 1644 CE, two years shy of Leibniz

birth. The venerated Li Madou/Matteo Ricci, a Roman Catholic Church Jesuit missionary,

opened the doors to a broader relationship between China and Western Europe at the

decline of the Ming Dynasty. It is during the advent of the turbulent and truncated

Manchurian insurrection that gave rise to the last dynastic Empire of the Great Qing. By the

3 Ibid, pp. 255 6

4 Daniel J. Cook and Henry Rosemont, Jr., Gottfried Wilhelm Leibniz: Writings on China, Open Court

Publishing Co., Chicago (1994), p. 63 [from Leibniz autograph, On the Civil Cult of Confucius, (1700/1701)] 5 James A. Ryan, Leibniz Binary System and Shao Yongs Yijing, University of Hawaii Philosophy East

& West Volume 46 (1996), pp 67 73



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late 18th century Imperial China had closed its doors to Western European meddling in its

internal affairs of state.6 This was the context that influenced Leibniz sinology.

Fu Xi never formulated nor knew of the 64 Hexagrams in 30th century BCE China. He

formulated the 8 Trigrams of Harmony Between Heaven and Earth. Leibniz worked out a

sufficient documentation of the binary system in his seminal short autograph Explication de

lArithmtique Binaire (1703Published Memoires de lAcademie Royale des Sciences) or Explanation

of Binary Arithmetic, and later recapitulated it in Remarks on Chinese Rites and Religion (1708)by

which he attempted unsuccessfully to translate Fu Xis trigrams into binary numerals.

Leibniz project was to legitimize the superiority of mathematics as the only authoritative

system of reconciliation between the Roman Catholic Church of Europe and the Natural

Philosophy and Theology of China thereby not only positioning himself as a leading

authority in the burgeoning East/West relationship but also forging a durable

accommodation between Reformation Protestantism and the same Roman Catholic Church.

It was perhaps too grand and self-congratulatory. He quite possibly was unaware of this

being Li Madou/Matteo Riccis very same project in China by then in decline and outright

repudiation by Pope Clement XI and eventually long after Leibniz death strident

nonaccommodation was permanently institutionalized as Church policy by Clement XIV.7

Leibniz thesis was that Fu Xis trigrams were binary in another format in the following

way. He took the trigrams three horizontal lines, solid signifying unity or (1) and broken or

gapped signifying aught or (0) giving 2 options for each line position to be 23 = 8 total

unique configurations: 010 (0002), 110 (0012), 210 (0102), 310 (0112), 410 (1002), 510 (1012), 610

(1102), and 710 (1112).

He committed two foundational errors that had Leibniz traveled to China would have

been revealed to him. First, there has never been, nor is there, in traditional Chinese


(2011), p. 294 7 Ibid, p. 294



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pictography a numeric representation for zero (0), which leads to the second error. The Fu

Xi 8 Trigram Sequence was numeral characters 1 8, and not 0 7. Consequently, Chinese

binary numerology could never be directly reconciled, but only tangentially analogized,

because it followed binary rules using 1, 2, and the doubling principle or 2x. The numeral

character for 1 at the beginning of the sequence as (12) stood symbolically for Harmony and

Heaven, and 10002 ended the series as 8 for Disharmony and Earth. Thus the series should

be represented as 11 = 110 = 12, 21 = 210 = 102, 310 = 112, 2

2 = 410 = 1002, 510 = 1012, 610 =

1102, 710 = 1112, and 23 = 810 = 10002. As we now can see, Leibniz could never make the Fu

Xi Trigrams sequential character line position options obey a doctrinaire adherence to solid

unity for 1 and broken/gapped aught for 0 in the Western representational binary numeral

sequence. The Fu Xi Trigram Sequence follows binary extrapolative rules, is pictographic,

but are not binary numerals in the Western sense, nor are they sequenced as such. We will

return to this line of inquiry below. QED

Why would anyone want to translate Arabic/Hindus numeralslet alone Chinese

pictographic trigrams and hexagramsinto binary in the first place? Because the binary

arithmetic system becomes advantageous in the simplification of handling large number

manipulation and also provides for the elimination of multiplication and division. 0 X 0 = 0,

0 X 1 = 0, 1 X 1 = 1; 0 / 1 = 0, 1 / 1 = 1; therefore, these two operations are rendered void.

Addition and subtraction are the only operations required as long as a durable and reliable

system for conversion from base-10 to base-2 and back is used and one makes no

computational errors. Multiplication and division are accomplished by specialized rules of

addition and subtraction, respectively. This will become important below.

Fu Xi was erroneously associated with the Yijing (Book of Changes)s 64 Hexagrams by the

Roman Catholic Church Jesuit missionaries in the 17th and 18th centuries CE. At the time, it

was impossible for Leibniz to know that Wen Wang of the 12th century BCE invented the 64

hexagrams.



5

Leibniz takes up the Fu Xi [Wen Wang] Hexagram Matrix in the fourth section of

Discourse on the Natural Theology of the Chinese (1715/1716) Concerning the Characters of Fohi,

Founder of the Chinese Empire, Used in His Writings, and Binary Arithmetic. There is

some progress made between 1708 and 1716 from his earlier work. Now he attempts to

conform the Fu Xi Trigrams into a systematic method for combining them by stacking two

at time into the Wen Wang Hexagrams. As we will see he came so very tantalizingly close to

solving it. But it seems as though he had given up any attempt at reconciliation.

Unfortunately, while it might be reasonable to extrapolate the remainder of this autograph

from Explanation of Binary Arithmetic (1703), this cant be done because it cannot be

postulated how Leibniz would have reworked it for the missing remainder ending with

than the natural numbers themselves which are their roots The documentation of this

autograph is disrupted, jarringly so, and ends here. Would Leibniz have figured out the Wen

Wang Hexagram Matrix? Would he have made a leap of logic to postulate an ancient

Chinese binary number system?

Posterity records no resolution to this conundrum because sadly Leibniz did not finish

this last work before his death.

1

It is extremely difficult to precisely establish provenances in Before Common Era China,

principally though not exclusively due to the actions taken by Qin Shihuangdi. At some

point in the late 12th century BCE Zhou Wen Wang (1152 1056 BCE), born Ji Chang,

King Wen Wang of Zhou, honored by his son Wu as the founder of the Zhou Dynasty after

the overthrow of the Shang Dynasty, extrapolated the 64 Hexagrams while he was

imprisoned as a rival political dissident by King Zhou of Shang during which he composed

the first iteration of the Yijing (Book of Changes). It is impossible to know with certainty his

was the first iteration. We can only know his was the earliest written iteration that has

survived. Hence forth we will refer to them as the Wen Wang Hexagram Matrix or 64



6

Hexagrams. For this essay we will use the traditionally accepted Yijing (Book of Changes) 64

Hexagrams format which we will see is the very matrix created by Wen Wang thirty-one-

hundred years or so ago.

Fig. 1 Credit: Wen Wang 64 Hexagrams Yijing, Late 12th Century BCE

At bottom right we find the Yijing interpretive hexagram for Heaven or the numeral 1

in the beginning position which is interestingly known as the seed position. There is a very

surprising reason why it became known as the Heaven hexagram as we will soon see. In the

top left we find the Yijing interpretive hexagram for Earth or the numeral 64 in the

completion position which is also interestingly known as the exhaustion position. Let us at

this point dispense with the extant methods of divination, its manifold iterations of evolved

prophetic, divine and/or theological interpretations, and proceed only with mathematical

designations. Once we have performed this recreation we will then return to some



7

observations on divination. From this point forward each of the 64 pictograms will be

identified with a single base-10 and base-2 numeral. While not the original purpose of the

matrix, it is my contention that it also serves an unexpected ancillary purpose. One that

might very well explain why it was invented in the first place.

The key to deciphering this arrangement requires two postulates. Postulate One: the

pictograms are idiomatic and not numerals as westerners would attempt to read them.

Postulate Two: this arrangement is not a binary number system, but a rudimentary doubling

number system rigorously following binary rules which provides for the ancillary benefit of

functioning as a rudimentary digital binary calculator for the performance of the simple

operations of addition, subtraction, multiplication and division, analogous to the binary

arithmetic system without employing binary numerals.

Each character consists of 6 horizontal lines. Let us now dispense with Leibniz

interpretive symbolic terminology and return to traditional Chinese Yijing interpretive

symbolic terminology. The lines can be either solid symbolizing harmony 1 or

broken/gapped symbolizing disharmony 2. Since there are two options for each position of

a hexagrams 6 lines, this gives us 26 number of combinations totaling 64 individually unique

pictograms. So hence forth we will refer to each hexagram in a specific sequential position 1

64, right to left (left-reading), bottom to top, and its line positions as 1st, 2nd, 3rd, 4th, 5th, and

6th, from bottom to top. All the other characters are extrapolated from the beginning seed

character.

The formatting rules of the matrix are derived from the first hexagrams formatting rules.

It makes no difference at all the manner in which you chose the formatting convention of

the final matrix. There are 8 possibilities; left-reading, up; up-reading, right; right-reading,

down; down-reading, left; left-reading, down; up-reading, left; right-reading, up; and down-

reading, right. Once chosen, one cannot deviate from the chosen convention. One must

apply the rules to the chosen convention until the matrix is completely extrapolated. One

can easily understand why the beginning hexagram is associated with Heaven. It is



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embryonic. It gives full bloom to the final matrix of characters. The following binary

extrapolative rules open the seed hexagrams iterative developmental potential until the

exhaustion hexagram is reached completing the matrix.

The first binary extrapolative rule of the hexagrams is that there are 7 exponent numeral

character hexagrams in 7 specific locations that graphically denote periodic doubling, and

establish the matrixs geometry.

Therefore: 11 =110, 21 = 210, 2

2 = 410, 23 = 810, 2

4 = 1610, 25 = 3210, and 2

6 = 6410. In binary

the following series is true: 12 = 110, 102 = 210, 1002 = 410, 10002 = 810, 100002 = 1610, 1000002

= 3210, and 10000002 = 6410. Now that we have a beginning theoretical series we can

examine the matrix for synchronicity.

810/23/10002 410/2

2/1002 210/21/102 110/1

1/12

Fig. 2 Credit: Wen Wang 64 Hexagrams Yijing, Late 12th Century BCE Row 1 (bottom) Wen Wang Hexagram Matrix

The 1st character at the lower right of the first row bottom is 6 solid 1st, 2nd, 3rd, 4th, 5th &

6th lines depicting the numerals 110 or 12. The 2nd character to the left is 5 solid 1st, 2nd, 3rd, 4th

& 5th lines under 1 broken/gapped disharmony 6th line depicting the numerals 210 or 102. The

4th character to the left is 4 solid harmony 1st, 2nd, 3rd & 4th lines under 2 broken/gapped

disharmony 5th & 6th lines depicting the numerals 410 or 1002. The 8th character to the left end

completing the row 1 bottom is 3 solid harmony 1st, 2nd & 3rd lines under 3 broken/gapped

disharmony 4th, 5th & 6th lines depicting 810 or 10002 [Fig. 2].

1610/24/100002

Fig. 3 Credit: Wen Wang 64 Hexagrams Yijing, Late 12th Century BCE Rows 1 & 2 (from bottom) Wen Wang Hexagram Matrix



9

3210/25/1000002

Fig. 4 Wen Wang 64 Hexagrams Yijing, Late 12th Century BCE Rows 1 4 (from bottom) Wen Wang Hexagram Matrix

Now we must move to the second row from the bottom and to the eighth character to

the left end or the 16th character in the numeral series which is 2 solid harmony 1st & 2nd lines

over 4 broken/gapped disharmony 3rd, 4th, 5th & 6th lines depicting 1610 or 100002 [Fig. 3].

Next we go 2 more rows up to the 4th row from the bottom over to the eighth character to

the left end or the 32nd character in the numeral series which is 1 solid harmony 1st line under

5 broken/gapped disharmony 2nd, 3rd, 4th, 5th & 6th lines depicting 3210 or 1000002 [Fig. 4].

Lastly, we now go up another 4 rows to the 8th row to the 8th character to the left end or the

64th character in the numeral series which is no solid harmony lines at all and only 6

broken/gapped disharmony 1st, 2nd, 3rd, 4th, 5th & 6th lines depicting 6410 or 10000002 [Fig. 5].

QED



10

6410/26/10000002

Fig. 5 Credit: Wen Wang 64 Hexagrams Yijing, Late 12th Century BCE Rows 1 8 (from bottom) Wen Wang Hexagram Matrix

The doubling periodicity becomes obvious in the graphic design of the Wen Wang

Hexagram Matrix. Row 1 at the right beginning position depicts 11, double it to the left for

21, double it to the left for 22, and double it to the left end of row 1 for 23. Double row 1 to

the left end of row 2 for 24. Double these 2 rows to the left end of row 4 for 25. And finally

double these 4 rows to the left end of row 8 for 26 [Fig. 6]. QED



11

26

64 63 62 61 60 59 58 57

1000000 111111 111110 111101 111100 111011 111010 111001

56 55 54 53 52 51 50 49

111000 110111 110110 110101 110100 110011 110010 110001

48 47 46 45 44 43 42 41

110000 101111 101110 101101 101100 101011 101010 101001

40 39 38 37 36 35 34 33

101000 100111 100110 100101 100100 100011 100010 100001

25

32 31 30 29 28 27 26 25

100000 11111 11110 11101 11100 11011 11010 11001

24 23 22 21 20 19 18 17

11000 10111 10110 10101 10100 10011 10010 10001

24

16 15 14 13 12 11 10 9

10000 1111 1110 1101 1100 1011 1010 1001

23

8 7 6 5 23

4 3 21

2 11

1

1000 111 110 101 100 11 10 1 Credit: Kevin Mequet

Fig. 6 Wen Wang Hexagram Matrix by binary number theory First binary extrapolative ruleExponent Numerals

The second binary extrapolative rule of the hexagrams is that there are exactly 32 of the

same feature common to the first 1 32 numerals that must be the exact opposite for the

remaining 32 numerals 33 64, and that this feature must follow the binary principle that all

odd numerals in base-10 will end with 1 in base-2 and all even numerals will end with 0 in

base-2. The only feature common to all the numeral hexagrams 1 32 is that every

characters 1st line is solid harmony denoting 1 in binary base-2 and therefore 1 in Chinese

numerology. We will now denote the first rows 1 4, numerals 1 32 as register 1. The only

exact opposite feature common to all numerals 33 64 is that every character 1st line is

broken/gapped disharmony denoting 10 in binary base-2 and therefore 2 in Chinese



12

numerology. We will now denote the second rows 5 8, numerals 33 64 as register 2 [Fig.

7]. QED

26

64 63 62 61 60 59 58 57

1000000 111111 111110 111101 111100 111011 111010 111001

56 55 54 53 52 51 50 49

111000 110111 110110 110101 110100 110011 110010 110001

48 47 46 45 44 43 42 41

110000 101111 101110 101101 101100 101011 101010 101001

40 39 38 37 36 35 34 33

101000 100111 100110 100101 100100 100011 100010 100001

25

32 31 30 29 28 27 26 25

100000 11111 11110 11101 11100 11011 11010 11001

24 23 22 21 20 19 18 17

11000 10111 10110 10101 10100 10011 10010 10001

24

16 15 14 13 12 11 10 9

10000 1111 1110 1101 1100 1011 1010 1001

23

8 7 6 5 23

4 3 21

2 11

1


Fig. 7 Wen Wang Hexagram Matrix by binary number theory Second binary extrapolative ruleRegister Residency

The third binary extrapolative rule of the hexagrams is that there are exactly 32 odd and

32 even hexagram numerals, that the feature common to all odd numerals will follow the

binary principle above, and that the exact opposite feature common to all even numerals will

do the same. All 64 hexagrams 6th line follows this rule. There are 32 alternating 6th line solid

harmony 1 hexagrams in columns 1, 3, 5 & 7. Therefore columns 1, 3, 5 & 7 are odd

hexagram numerals. There are 32 alternating 6th line broken/gapped disharmony 2



13

hexagrams in columns 2, 4, 6 & 8. Therefore columns 2, 4, 6 & 8 are even hexagram

numerals [Fig. 8]. QED

26

64 63 62 61 60 59 58 57

1000000 111111 111110 111101 111100 111011 111010 111001

56 55 54 53 52 51 50 49

111000 110111 110110 110101 110100 110011 110010 110001

48 47 46 45 44 43 42 41

110000 101111 101110 101101 101100 101011 101010 101001

40 39 38 37 36 35 34 33

101000 100111 100110 100101 100100 100011 100010 100001

25

32 31 30 29 28 27 26 25

100000 11111 11110 11101 11100 11011 11010 11001

24 23 22 21 20 19 18 17

11000 10111 10110 10101 10100 10011 10010 10001

24

16 15 14 13 12 11 10 9

10000 1111 1110 1101 1100 1011 1010 1001

23

8 7 6 5 23

4 3 21

2 11

1


Fig. 8 Wen Wang Hexagram Matrix by binary number theory Third binary extrapolative ruleOdd/Even Numeral Columns

Now that we understand the overall distribution of the hexagrams we can correctly

interpret the non-exponent numerals. The 1st lines of all the hexagrams denote register

residency 1 or 2. The 6th lines of all the hexagrams denote odd or even numerals. The first

cardinal group of characters is the exponent hexagrams whose unique designs begin with the

first numeral 1 odd character, the remaining 5 are all even characters and self-organized by

the doubling principle into a total of 6 residing in register 1 with 1st lines solid harmony 1,

and 6th lines solid harmony for the first numeral 1 character and broken/gapped



14

disharmony 2 for the remainder, and a single numeral character residing in register 2 with

1st & 6th lines broken/gapped disharmony 2. The second ordinal group of hexagrams is

actually comprised of 16 distinct quadragrams sandwiched between 1st line register residency

and 6th line odd or even numeral designation.

The fourth binary extrapolative rule of the hexagrams is that all the numerals have

quadragrams sandwiched between 1st & 6th line positions, thusly: 2nd, 3rd, 4th & 5th lines gives

us 4 line positions to make 24 = 16. That means 16 different quadragrams are possible. There

are 32 hexagrams in a register. There are 16 odd/even pairs, so 1 quadragram can be used

for each odd/even pair. Then we will repeat exactly for register 2 because register 1 is 1st

solid harmony line, and register 2 is 1st broken/gapped disharmony line.

Reusing our previous binary extrapolative rules let us now extrapolate the 16

quadragrams, only now in a 4 by 4 matrix because the exhaustion character is the 16th.

Similarly to the hexagrams the beginning seed quadragram has 4 solid harmony lines. The

end exhaustion character is 4 broken/gapped disharmony lines. The first character in the

series denotes 11 = 1 and the last denotes 24 = 16. Naturally, in a 4 X 4 matrix this must

mean the first binary extrapolative rule of exponent characters makes the second character 3

solid harmony lines under 1 broken/gapped disharmony line or 21 = 2. The fourth character

therefore is 2 solid harmony lines under 2 broken/gapped disharmony lines or 22 = 4. Next

the eighth character will be 1 solid harmony line under 3 broken/gapped disharmony lines.

As we have already seen the exhaustion end character must therefore be 4 broken/gapped

disharmony lines or 24 = 16 [Fig. 9].



15

24

16 15 14 13

10000 1111 1110 1101

12 11 10 9

1100 1011 1010 1001

23

8 7 6 5

1000 111 110 101

22

4 3 21

2 11

1

100 11 10 1 Credit: Kevin Mequet

Fig. 9 Wen Wang Quadragram Matrix by binary number theory First binary extrapolative ruleExponent Numerals

The second binary extrapolative rule dictates that characters 1 8 are register 1

characters with bottom line position solid harmony lines. Therefore characters 9 16 are

register 2 characters with bottom line position broken/gapped disharmony lines [Fig. 10].

24

16 15 14 13

10000 1111 1110 1101

12 11 10 9

1100 1011 1010 1001

23

8 7 6 5

1000 111 110 101

22

4 3 21

2 11

1


Fig. 10 Wen Wang Quadragram Matrix by binary number theory Second binary extrapolative ruleRegister Residency

The third binary extrapolative rule dictates that the first and third columns of characters

must have top line position solid harmony lines denoting odd numeral characters. Therefore

the second and fourth columns of characters must have top line position broken/gapped

disharmony lines denoting even numeral characters [Fig. 11].



16

24

16 15 14 13

10000 1111 1110 1101

12 11 10 9

1100 1011 1010 1001

23

8 7 6 5

1000 111 110 101

22

4 3 21

2 11

1


Fig. 11 Wen Wang Quadragram Matrix by binary number theory Third binary extrapolative ruleOdd/Even Numeral Columns

Using a variation on the fourth binary extrapolative rule we can complete the

quadragram matrix. The middle two line positions must follow the exponent characters lead

and have one configuration that applies to each odd/even pair of numeral characters, thusly:

numerals 1 & 2 have both middle line positions solid harmony lines, numerals 3 & 4 have 2nd

line position solid harmony lines under 3rd line position broken/gapped disharmony lines,

numerals 5 & 6 have 2nd line position broken/gapped disharmony lines under 3rd line

position solid harmony lines, and numerals 7 & 8 must therefore end the series at exhaustion

both middle lines broken/gapped disharmony line, completing the register 1 numerals 1 8.

Repeat exactly for register 2 numerals 9 16 [Fig. 12].

24

16 15 14 13

10000 1111 1110 1101

12 11 10 9

1100 1011 1010 1001

23

8 7 6 5

1000 111 110 101

22

4 3 21

2 11

1




17

Fig. 12 Wen Wang Quadragram Matrix by binary number theory Fourth binary extrapolative rule16 Quadragrams Sandwiched & Completion

Now we can confidently complete the Wen Wang Hexagram Matrix. Insert sequentially

each quadragram between each hexagram bottom and top lines, 1 quadragram for every

odd/even numeral pair to complete the register 1 numerals 1 32; then repeat exactly for

the register 2 numerals 33 64 [Fig. 13].

We have now correctly derived the Wen Wang Hexagram Matrix using binary number

theory.

26

64 63 62 61 60 59 58 57

1000000 111111 111110 111101 111100 111011 111010 111001

56 55 54 53 52 51 50 49

111000 110111 110110 110101 110100 110011 110010 110001

48 47 46 45 44 43 42 41

110000 101111 101110 101101 101100 101011 101010 101001

40 39 38 37 36 35 34 33

101000 100111 100110 100101 100100 100011 100010 100001

25

32 31 30 29 28 27 26 25

100000 11111 11110 11101 11100 11011 11010 11001

24 23 22 21 20 19 18 17

11000 10111 10110 10101 10100 10011 10010 10001

24

16 15 14 13 12 11 10 9

10000 1111 1110 1101 1100 1011 1010 1001

23

8 7 6 5 23

4 3 21

2 11

1


Fig. 13 Wen Wang Hexagram Matrix by binary number theory Fourth binary extrapolative rule16 Quadragrams Sandwiched & Completion



18

2

As compelling and beautiful as this recreation of the Wen Wang Hexagram Matrix by

extrapolation from binary number theory is, regretfully, it is not how he devised his matrix

from Fu Xis trigrams. Now using our new eyes developed from extrapolating the matrix

from binary number theory lets recreate what Wen Wang did some thirty-one-hundred or so

years ago.

At the beginning of this essay I said we would return to the Fu Xi Trigram Sequence. Let

us use the same formatting rules and conventions as we did for the Wen Wang Hexagram

Matrix. Therefore, each trigram consists of 3 line positions with 2 line options for each

giving 23 = 8 total distinct characters in this sequence. Using our binary extrapolative rules

we have 4 exponent characters 11 = 1, 21 = 2, 22 = 4 and 23 = 8. So the beginning first

trigram seed character is 3 solid harmony lines, the second trigram character is 2 solid

harmony lines under 1 broken/gapped disharmony line, the fourth character is 1 solid

harmony line under 2 broken/gapped disharmony lines, and the ending final exhaustion

character is 3 broken/gapped disharmony lines. The register 1 characters consist of numerals

1 4 with bottom solid harmony lines, and the register 2 characters consist of the numerals

5 8 with broken/gapped disharmony lines. The odd numeral characters have top solid

harmony lines, and the even numeral characters have broken/gapped disharmony lines.

Therefore, the middle line position binary extrapolative rule must give the third character to

be bottom solid harmony line, middle broken/gapped disharmony line, and top solid

harmony line; the fifth character must be bottom broken/gapped disharmony line, middle

solid harmony line, and top solid harmony line; the sixth character must be bottom

broken/gapped disharmony line, middle solid harmony line, and top broken/gapped

disharmony line; and the seventh character must be bottom broken/gapped disharmony line,

middle broken/gapped disharmony line, and top solid harmony line [Fig. 14]. QED

23

22

21

11

x8 x7 x6 x5 x4 x3 x2 x1 Credit: Kevin Mequet



19

Fig. 14 Fu Xi Trigram Sequence by binary number theory First Fourth binary extrapolative rulesComplete Sequence

Wen Wang had this much venerated and valorized sequence to mediate Between Heaven

and Earth in his mind as he worked. Having extrapolated his matrix from binary number

theory we can see how easily he extrapolated from the Fu Xi Trigram Sequence to make a

more compelling matrix matching the complexities of Heaven and Earth that would be the

most rational arrangement of character divination. All he didliterallywas multiply the 8

characters by the 8 characters to make a graphic depiction of that multiplication in an 8 X 8

= 64 character matrix.

This is where Leibniz was truly and tragically blind. Had he eschewed a default

hierarchical Western colonial overlay upon the East he could have seen that the characters

could never give themselves to a Western representationalist binary numeral systembut

were instead, hidden plainly in sight, the pictographic depiction of 8 X 8 = 64. If you take

every character and divide it into a lower and upper trigram then the operation becomes

glaringly obvious. The lower trigram for all 8 characters in the bottom first row is the first

Fu Xi Trigram Sequence character. Then the second character for all 8 characters in the

second row, and so on through the top eighth row [Fig. 15].



20

23

23

64 23

63 23

62 23

61 23

60 23

59 23

58 23

57

8x 8x 8x 8x 8x 8x 8x 8x 8x

56 55 54 53 52 51 50 49

7x 7x 7x 7x 7x 7x 7x 7x 7x

48 47 46 45 44 43 42 41

6x 6x 6x 6x 6x 6x 6x 6x 6x

40 39 38 37 36 35 34 33

5x 5x 5x 5x 5x 5x 5x 5x 5x

22

22

32 22

31 22

30 22

29 22

28 22

27 22

26 22

25

4x 4x 4x 4x 4x 4x 4x 4x 4x

24 23 22 21 20 19 18 17

3x 3x 3x 3x 3x 3x 3x 3x 3x

21

21

16 21

15 21

14 21

13 21

12 21

11 21

10 21

9

2x 2x 2x 2x 2x 2x 2x 2x 2x

11

11

8 11

7 11

6 11

5 11

4 11

3 11

2 11

1

1x 1x 1x 1x 1x 1x 1x 1x 1x Credit: Kevin Mequet

Fig. 15 Wen Wang Hexagram Matrix by Fu Xi Trigram Sequence Lower Half of HexagramsTrigram the same for each character in a row

Every single characters upper trigram in all 8 rows is the same Fu Xi Trigram Sequence

character distributed sequentially left-reading [Fig. 16]. Or congruently put another way, each

column of characters has the same upper trigram.



21

23

22

21

11

x8 x7 x6 x5 x4 x3 x2 x1

23

64 63 62 61 22

60 59 21

58 11

57

x8 x7 x6 x5 x4 x3 x2 x1

23

56 55 54 53 22

52 51 21

50 11

49

x8 x7 x6 x5 x4 x3 x2 x1

23

48 47 46 45 22

44 43 21

42 11

41

x8 x7 x6 x5 x4 x3 x2 x1

23

40 39 38 37 22

36 35 21

34 11

33

x8 x7 x6 x5 x4 x3 x2 x1

23

32 31 30 29 22

28 27 21

26 11

25

x8 x7 x6 x5 x4 x3 x2 x1

23

24 23 22 21 22

20 19 21

18 17

x8 x7 x6 x5 x4 x3 x2 x1

23

16 15 14 13 22

12 11 21

10 11

9

x8 x7 x6 x5 x4 x3 x2 x1

23

8 7 6 5 22

4 3 21

2 11

1

x8 x7 x6 x5 x4 x3 x2 x1 Credit: Kevin Mequet

Fig. 16 Wen Wang Hexagram Matrix by Fu Xi Trigram Sequence Upper Half of HexagramsTrigram the same for each character in a column

A pictographic depiction of 8 X 8 = 64 [Fig. 17]. QED



22

23

22

21

11

x8 x7 x6 x5 x4 x3 x2 x1

23

26

64 63 62 61 60 59 58 57

8x 8x8 8x7 8x6 8x5 8x4 8x3 8x2 8x1

56 55 54 53 52 51 50 49

7x 7x8 7x7 7x6 7x5 7x4 7x3 7x2 7x1

48 47 46 45 44 43 42 41

6x 6x8 6x7 6x6 6x5 6x4 6x3 6x2 6x1

40 39 38 37 36 35 34 33

5x 5x8 5x7 5x6 5x5 5x4 5x3 5x2 5x1

22

25

32 31 30 29 28 27 26 25

4x 4x8 4x7 4x6 4x5 4x4 4x3 4x2 4x1

24 23 22 21 20 19 18 17

3x 3x8 3x7 3x6 3x5 3x4 3x3 3x2 3x1

21

24

16 15 14 13 12 11 10 9

2x 2x8 2x7 2x6 2x5 2x4 2x3 2x2 2x1

11

23

8 7 6 5 23

4 3 21

2 11

1

1x 1x8 1x7 1x6 1x5 1x4 1x3 1x2 1x1 Credit: Kevin Mequet

Fig. 17 Wen Wang Hexagram Matrix by Fu Xi Trigram Sequence Lower Half & Upper Half of HexagramsLower Trigram by Upper Trigram gives 64 unique hexagrams by pictographic depiction

Now anyone can easily extrapolate the Wen Wang 64 Hexagrams Yijing (Book of Changes)

matrix as was done some thirty-one-hundred or so years ago.

3

Did you by any chance notice the characters in the Wen Wang Hexagram Matrix that are red

on the bias from character position 1 through 64? At numerals 1, 10, 19, 28, 37, 46,

55, and 64 [Fig. 1]? As Western sequential numerals they make no sense. But as a



23

pictographic depiction of numeral operations they do. Look closely at each character. Notice

anything distinctive? If youre paying attention to this essay it should now stand out for you.

Each character is a graphic depiction of Fu Xi Trigram Sequence numeral character squares

or x2. Each one has a lower and upper trigram that is the same.

This is graphic depiction of Fu Xi Trigram Sequence numerals 1 X 1 = 1 giving a single

hexagram character, the seed hexagram symbolizing Heaven. Fu Xi Trigram Sequence

numerals 2 X 2 = 4 giving a 2 X 2 hexagram matrix comprising 4 characters. Fu Xi Trigram

Sequence numerals 3 X 3 = 9 giving a 3 X 3 hexagram matrix comprising 9 characters. Fu Xi

Trigram Sequence numerals 4 X 4 = 16 giving a 4 X 4 hexagram matrix comprising 16

characters. Fu Xi Trigram Sequence numerals 5 X 5 = 25 giving a 5 X 5 hexagram matrix

comprising 25 characters. Fu Xi Trigram Sequence numerals 6 X 6 = 36 giving a 6 X 6

hexagram matrix comprising 36 characters. Fu Xi Trigram Sequence numerals 7 X 7 = 49

giving a 7 X 7 hexagram matrix comprising 49 characters. And finally, Fu Xi Trigram

Sequence numerals 8 X 8 = 64 giving an 8 X 8 hexagram matrix comprising 64 characters

and the last exhaustion character symbolizing Earth [Fig. 17].

The appropriate convention is to refer to these combinations relationally as lower

trigram by upper trigram, not arithmetically as lower trigram times upper trigram. In the

figure diagrams the convention used is lower trigram 1 8 by upper trigram 1 8. So the

hexagram 3 X 7 = 21 denotes the sequential 23rd hexagram with lower 3rd trigram by the

upper 7th trigram delineating a matrix of 21 characters bounded 3 up by 7 across to the left.

4

To my western eyes this matrix at first looked like a digital binary calculator because I saw

the characters sequentially, not as Wen Wang or his contemporaries would have seen them. I

now understand this was a serendipitous happenstance of its simultaneity with binary

number theory. While not intended as its primary function, because it follows binary number

theory laws, it can be used as a sequential digital binary calculator in the following way.



24

If we exchange sequential Arabic/Hindus numerals for the hexagram characters we get 1

64 in an 8 X 8 numeral matrix. Now lets flip and rotate this matrix for western convention

upper left corner start position, right reading top to bottom. You manipulate this manual

calculator digitally using your index finger following binary arithmetic operations [Fig. 18].

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

Credit: Kevin Mequet

Fig. 18 Wen Wang Yijing Digital Binary Calculator (Patent Pending) Interpretive info: right-reading; red, exponent numerals; blue, odd numerals; black, even numerals; yellow, register 1 numerals; and green, register 2 numerals.

Place your finger or a stylus on the character that you wish to begin the calculation, and

then count sequentially to get the result. If adding, count forward to the right from the first

addend numeral the number of the second addend to achieve the sum. If subtracting, count

backward to the left from the first subtractend numeral the number of the second

subtractend to achieve the resultant subtend. If the second subtractend is larger when you

reach numeral 1 tap your figure or stylus again, count once againbecause there is no (0) in

this calculatorand continue counting backward to the left, the resultant subtend is a



25

negative number or a deficit. If multiplying, count the multiple forward to the right, such as

4 X 5, start at numeral 1 and count 5 sets of 4 characters till you reach numeral 20. The

answer is 20. You could also correctly count 4 (5s) forward to the left. If dividing, do the

reverse of multiplying, such as 20 / 5, start at the numeral 20 count the divisor 5 backwards

to the left, thats 1 (5), another 5, thats 2 (5s), count another 5, thats 3 (5s), another 5 which

brings us to numeral 1, thats 4 (5s), the answer is 4 with no remainder. If theres a remainder

note it with the resultant dividend.

What happens if your calculation exceeds 64? Start adding or subtracting (64s). Lets

suppose an answer of 103 wants to be obtained. How do you deal with it? You count to 64

and then start from the beginning numeral 1 again and keep going, such as 64 and 39. The

answer is 64 and 39. So running debt or surplus totals would be formatted in multiples of 64

and a remainder under 64, if any. Extraordinarily large numberswith carecan be

manipulated.

Conveniently, the numerals and design tells you what to do; its self-embodied and

complete. By using very simple rules it does the work of addition, subtraction, multiplication

and division very similar to the binary numeral system. But this isnt what Wen Wang

invented his matrix to do in his time. He invent this singular matrix to produce what he said

would become the most rational system of divination.

5

Lets now turn to an important consideration. Why do human beings regardless of disparate

evolutionary cultural development strive to embody existence into systematic mathematical

objects? The posing of this question seems to be at the heart of this essay and might provide

a methodology for finding common ground cross-culturally for promoting profound dialog

and mutually illuminating inquiry.



26

Leibniz and Western Europe held the view that mathematics is at the heart of all

creation and human endeavor. It is the perfect unchanging language of G-d in Western

Europe and of the Son of Heaven in China. Through ancient antecedents, both in the

European and East Asian continents, Li Madou reintroduced China to this view

reinvigorated and exhorted by Christopher Clavius, Matteo Riccis mathematics teacher at

university.

As was noted at the beginning of this essay Qin Shihuangdi precipitated a titanic

discontinuity in Chinas intellectual continuum. It was a breach so vast that it could not be

mitigated internally, but awaited a chance encounter from outside her borders in the late 16th

century Ming Dynasty by Li Madou from the Great West. That chance encounter would set

in motion events of unforeseen consequence. Until Li Madou wrote in Mandarin a treatise

on mathematics translating Euclids Elements into Jihe yuanben (1607)literally translated as

the origin of quantity but more-widely known as Elementary Treatise on Geometrya

small rivulet of reconnection could not develop into a flood of reclaimed arithmetic

expertise. Lauded by the Chinese literati as an exemplary homage to Confucius Li Madous

preface quoted from Christopher Clavius translation of Euclids Elements. It was an oversell

of mathematics to the Chinese to bolster a skeptical Chinese Imperium of its value and

virtue. Ironically it was the same oversell as was Clavius preface to bolster a skeptical

Roman Catholic Church of the integral value and virtue of arithmetic in the syllabuses of

their universities. Clavius compared mathematics to a fountain from which the other

branches of science gush; Ricci compared it to a ladder leading up to the peaks of

knowledge.8 After an arithmetic drought of 1,800 years the importance of mathematics to

Chinas well-being and promotion of good societal order fell upon receptive ears.

The downside is that this view falls prey to an overdominant condition, whereby creative

speculation, language and the arts become secondary endeavors. One can so easily mistake

8 Ibid, p. 254



27

mathematical objects as interchangeable, even replaceable, with reality. Unfortunately, this is

not so.

We have reached a philosophical and scientific impasse that must be surmounted. The

case is laid out quite clearly and succinctly by Lee Smolin in his latest book, Time Reborn

(2013), where we must accept the reality of time suffusing the universe, events as real in the

moment, time as the passage of successive moments with directionality, novelty and change

in an unbounded, open-ended, evolving Leibnizian Universe of relationality, fundamentally

unpredictable and uncertain, but extravagantly diverse and unique. He says Leibnizian

because he takes as a given two important principles of Leibniz: The Principles of Sufficient

Reason and Identity of the Indiscernibles,9 or Univocity of Singular Uniqueness.10

The problem to be mitigated is summed up quite nicely by Smolin:

Mathematics will continue to be a handmaiden to science, but she can no longer be

the Queen.11

Mathematics and sciencethe same as the arts/humanities and the sciencesmust be

interrelated and interplayed the way Deleuze formatted his two numerical series interacting

in intensity and extensity giving rise to his formulation of the virtual and actual, both/and,

neither preferenced or annihilated, distinguishable but indivisible.12

And so the same must be said of this extrapolation of Wen Wangs Yijing Hexagrams

which culminates Leibniz unrequited inquiry begun three centuries ago.

9 Lee Smolin, Time Reborn, Houghton Mifflin Harcourt Boston, NYC (2103), pp. 114 - 7

10 Gilles Deleuze, Difference & Repetition, Columbia University Press (1968 French, 1994 English), p. 37;

the author is synthesizing Deleuzes thought which he synthesized from Leibniz and Spinoza. 11

Lee Smolin, Time Reborn, Houghton Mifflin Harcourt Boston, NYC (2103), p. 251 12

Clayton Crockett, Deleuze Beyond Badiou, Columbia University Press, NYC (2103), p. 40



28

Li Madou had no patience, let alone a spirit of accommodation, with what he termed the

several thousands of charlatans peddling false fortune-telling in the Imperial Capitol.13 He

was undoubtedly referring to Yijing divination. This was an unfortunate colonial

misunderstanding. I would desire correcting it and welcome an open dialog to deepen my

understanding of this long-beloved Chinese tradition in the Confucian ideal:

The Master said, Reviewing the old as a means of realizing the newsuch a

person can be considered a teacher. 14

We seem, regardless of differing evolutionary cultural development, to be extraordinarily

concerned with the avoidance of risk and detriment by attempting to transform unruly,

unpredictable, uncertain existence into orderly, controllable life with the benefit of certainty.

This is at root a zero-sum game. We must eschew this illusory wish for an authentic

relationship with reality that includes time and acclimatizes us to becoming comfortable with

uncertainty. Wishing mathematics to remain the Queen of science in the West is not so

dissimilar with wishing the Yijing to give some certainty in an uncertain world. Mathematical

objects can only approximate processes in time in the world and never perfectly model or

replace them. I could confidently say Heideggers Being the World would fit nicely here. So

like everything else the interpretive project must extend to mathematical objects: useful

handmaidens but never Queens, as is demonstrated in this essay. We must come to a place

where we realize that to live fully we must live joyfully into uncertainty. Perhaps we can

more effectively connect with each other cross-culturally if we share our mutuality in

certainty and together venture beyond it?

Coming full circle, lets reconsider Wen Wangs Hexagrams. If we perform the same

operation upon the 8 X 8 = 64 matrix as we did to produce the Yijing Digital Binary

Calculator we get a relational matrix of Fu Xi trigrams multiplied by themselves [Fig. 19].


(2011), p. 262 14

Ibid, p. 243; Confucius, Analects (2, 11)



29

1x1 1x2 1x3 1x4 1x5 1x6 1x7 1x8

2x1 2x2 2x3 2x4 2x5 2x6 2x7 2x8

3x1 3x2 3x3 3x4 3x5 3x6 3x7 3x8

4x1 4x2 4x3 4x4 4x5 4x6 4x7 7x8

5x1 5x2 5x3 5x4 5x5 5x6 5x7 5x8

6x1 6x2 6x3 6x4 6x5 6x6 6x7 6x8

7x1 7x2 7x3 7x4 7x5 7x6 7x7 7x8

8x1 8x2 8x3 8x4 8x5 8x6 8x7 8x8

Credit: Kevin Mequet Fig. 19 Relational Wen Wang Yijing Hexagrams from Fu Xis Trigrams Interpretive info: right-reading; red, x2 characters; blue, odd column characters; black, even column characters; yellow, register 1 characters; and green, register 2 characters.

We cannot know with certainty how Wen Wang used his Yijing Hexagram Matrix for the

most rational method of divination. It is lost to us now by the actions commanded by Qin

Shihuangdi. But just as we have reconstructed here how the matrix was originally created, it

might be possible to inductively reason how the methodology of divination might have been

conceived by Wen Wang. An idea of Stuart Kaufmanns might help us here. It posits that in

a relational world one must look for what can happen next, or the processes of the world

follow the Principle of the Adjacent Possible.15 In a relational Yijing Hexagram Matrix

devised from multiplying Fu Xis Trigrams by themselves relationally, might we look for

relational operations of divination based upon Kaufmanns Principle of the Adjacent

Possible? I would like very much to investigate this further with Chinese scholars.

15 Lee Smolin, Time Reborn, Houghton Mifflin Harcourt Boston, NYC (2103), p. xvi



30

Copyright 2013 Kevin Dobson Mequet; Zhou Wen Wangs Development of Yijing (Book Of Changes) 64 Hexagrams: An Inductive Recreation of Binary Extrapolative Operations. It is intended for strictly individual personal use. All rights are reserved. It may not be reprinted without obtaining prior written permission from Kevin Mequet. It may not be altered, misquoted, truncated, misattributed or plagiarized. Please adhere strictly to competent academic citation and attribution.

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