Binary Logic, Strictly Octal and Amazingly Visual

Binary Logic

14 10 11 Yin YangThe polarities of Yin [Earth, Feminine] and Yang [Heaven, Masculine] lend themselves to the binary logic of

  • 0 for Yin the broken line and
  • 1 for Yang, the solid line.

Strictly Octal

3 Yin or Yang lines form 8 possible trigrams:

14 11 03 Octal Trigrams BY

3 0‘s or 1‘s form 8 octal numbers between 0 and 7. 

14 11 17 Octal Trigrams

In the decimal number system, every position has the value of a power of 10.

Similarly, each line position has a numerical value:

  • the bottom line: 1 (20)
  • the middle line: 2 (21)
  • the upper line: 4 (22).

Adding the lines up, every trigram has an the octal value between 0 and 7.

00 000 = 0 01 001 = 1 02 010 = 2 03 011 = 3
04 100 = 4 05 101 = 5 06 110 = 6 07 111
= 7

Amazingly Visual

Colourful Diagrams illustrate:

  • diagonal patterns of symmetry
  • orthogonal patterns of reflection
  • gradual transitions from trigrams to hexagrams through changing lines.

Chinese Polarities, Indian Zero and Arabic Numerals through Western Digital Eyes

Numerical interpretations of the 64 hexagrams encompass all events and phenomena between Heaven and Earth. They were used for oracles and future divining, before they became a book of wisdom.

My understanding is based on Richard Wilhelm’s German translation (1923) – the first Western language – and “The Book of Changes and the Unchanging Truth” by Taoist Master Ni, Hua Ching (1983) who brought Chinese teachings to California.

All things between Heaven and Earth were expressed in 64 symbols or hexagrams. The 64 symbols are combined from 8 * 8 images or trigrams as below.

14 11 03 Octal Trigrams 2

At the time, zero had not been invented and hexagrams were numbered from 1 to 64, according to the King Wen sequence of hexagrams.

My most important innovation is the attribution of an octal value to each trigram as above.

The King Wen sequence is based on the combination of trigrams into ‘polar pairs‘ of hexagrams, as can be seen in this diagram:

King Wen Sequence PairsI Ching hexagrams 00-77Since then, Gottfried Leibniz invented the binary number system in 1679.

It consists of 0 and 1 and is the foundation for digital electronics.

Binary and Unicode values have been attributed to the 64 hexagrams as illustrated in the Wikipedia entry and elsewhere.

This graph shows the Tao in the place of 00 and the attribution of the numbers 1 to 64 to the hexagrams. 

I.e. this diagram helps to find the visual image, if you have received the number of a hexagram through divination.

It demonstrates the use of our decimal number system which has been including zero for centuries.

M1 77 33 Top Left Bottom RightBut the other way round, to find the number of a hexagram when you see the image, especially as the result of a divination, is a challenge:

  • Richard Wilhelm who translated the I Ching into German as the first Western language, produced a list and the matrix on the left.
  • I added octal numbers and my arbitrary choice of colouring the diagonal hexagrams.

Octal numbers turn out to be a new principle for

  • a logically binary order of visual Yin and Yang lines;
  • strictly octal sequence of trigrams: 

14 11 03 Octal Trigrams BY

This leads to sequencing the hexagrams numerically :

  • from 00 to 77 rather than 1 to 64;
  • thus telling a different story when using each hexagram as a chapter:
  • when the I Ching is a book about the Unchanging Truth rather than an oracle that describes the transitions caused by lines changing between two hexagrams.

4 77 to 33: Top Left to Bottom RightIn this diagram of the original Richard Wilhelm matrix, I have added

1) a single digit octal number for each trigram: from 0 to 7;

  • this results in the order 7, 0, 1, 2, 4, 6, 5, 3;

2) a double digit octal number for each hexagram;

  • this results in numbers from 00 to 77;

3) links to the King Wen hexagram number on Wikipedia.

Here is a pdf file of the above diagram so that the links to the original hexagrams on Wikipedia are clickable.

The Organisation of 2 Lines into 8 Trigrams and 64 Hexagrams

1.solid and a broken line are the fundamental building blocks:

  • 0 as the broken and 1 for the solid line result in octal values for trigrams from 0 to 7;
  • in this example, hexagram no 54 is made up from the lower trigram of 3 and the upper trigram of 1;
  • the octal value for the above hexagram is 31 which you find in the coloured diagram above in the 8th line and third column, associated with King Wen no. 54.

2. Eight trigrams are made from 3 lines, that can be described by the octal value of three binary numbers:

  • a line is either 0 or 1 – like in the binary number system;
  • in the octal system,threebinarydigitsare added up to one:
    • the first line is either 0 or 1: 20
    • the second one is either 0 or 2: 21
    • the third one is either 0 or 4: 22.
  • The octal number system uses the digits 0, 1, 2, 3, 4, 5, 6, 7
    and the decimal system uses 8 and 9 on top of 0 to 7.

3.hexagram is an arrangement of lines composed from 2 trigrams:

  • pairs of hexagrams mirror each other visually;
  • Richard Wilhelm produced a matrix and a list which followed the movement of the broken lines, so that visual images could be associated with the numerical identifiers from 1 to 64;
  • pairs of hexagrams also mirror each other numerically: eg. 01 and 10 or 34 and 43;
  • the diagonal of ‘doubled trigrams‘ consists of the hexagrams with octal values of 00, 11, 22, 33, 44, 55, 66, 77 which don’t occur in pairs.

As the positions of lines change to form one hexagram from another one, there is an intricate relationship between hexagrams, based on the binary values 0 and 1 of Yin and Yang lines.

It is the intricacy of these line changes that make the richness of the I Ching.

It is the ordering quality of numbers that allows for

  • numbering trigrams from 0 to 7
  • and sequencing hexagrams from 00 to 77, based on the octal value of their lower and upper trigrams.

I have used octal numbers

1. to number strictly and unambiguously trigrams from 0 to 7:

  • into different matrices of hexagrams
  • and use colouring effects to illustrate patterns of change and symmetry.

2. to re-number equally unambiguously hexagrams from 00 to 77:

The symmetries show

  • pathways of growth and decay in ascending and descending numerical orders;
  • avenues of inward and outward directions;
  • diagonals and centre lines as critical lines of reflection;
  • movement inwards or outwards from diagonal end points;
  • patterns of balance between even and odd numbers.

In these first four diagrams [with four Diagonals 1, 2, 3, 4] the hexagrams move from 00 to 77, i.e. numerically ascending from Earth to Heaven or the dark outside to the light centre:

The same diagrams ordered  ‘backwards’ [Diagonals 4, 3, 2, 1] result in this pattern:

The dark inside radiates to the lighter outside above, numerically ascending.

The next four diagrams [Diagonals 5, 6, 7, 8] show movement from 77 to 00, i.e. from Heaven to Earth: numerically descending, visually radiating from the dark centre towards the outside.

And now we perceive new possible interrelationships between hexagrams in one arrangement and another. This is particularly relevant when we reflect upon the lines that change to turn one hexagram into another one, or a longer term trend that is represented by a matrix.

This arrangement consists of diagonals 8, 7, 6, 5 and radiates from its light core to the darker outside.

The significance of this added ‘visual realm of interpretation’ is discussed in Prose.

Here is another visual sequence: in this lookup table, the author Paul O’Brien chose what translates into 7, 1, 2, 4, 0, 6, 5, 3 as the octal sequence of trigrams.


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