An Approach to the 'I-CHING'

by Frank C. Fung - 1st published in December, 2004.

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Section 8 - Determinants, Magic Squares, & Multiplication Tables

Topic I - Introduction & Summary for the Section :

This exploratory Section, we examine :

• Determinants in Matrix Algebra,
• [ numbers ] in the { Trigram Placement Scheme according to WEN WANG } ,
• { Multiplication Table for modulo P } where 'P' is 'prime' ;

and found some similar underlying principles.

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Topic II - A Review of the { Determinant } :

Let us now look at a set of { 3 linear equations } involving (3) variables , 'X' , 'Y' , & 'Z' , as follows :

We note that the { Determinant } for the { 3 x 3 Matrix } is given by :

{ Determinant } = { A*E*J + B*F*G + C*D*H - A*F*H - B*D*J - C*E*G }

We then have this table for the six (6) terms of the { Determinant } :

We then have this pair of diagrams below :

&

• The left-hand-side diagram then marks the { positive terms } , and
• the right-hand-side diagram then marks the { negative terms }.

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Topic III - Recalling the { Trigram Positioing Diagram } according to WEN WANG :

Let us now bring back the { Trigram Positioning Diagram } according to WEN WANG , from { Section 4 } :

&

And we shall be concentrating on the { Numbers } , as presented on the right-hand-side diagram, in this Section.

We then seperate the nine (9) digits, i.e. { 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 } , into three (3) group by size :

• the { Small Group } is { 1 , 2 , 3 } ,
• the { Medium Group } is { 4 , 5 , 6 } ,
• the { Large Group } is { 7 , 8 , 9 } .

And this shall be known as the { L-M-S Grouping Method } .

We can also seprate the nine (9) digits on a { Modulo 3 } basis :

• the { 1 mod 3 } Group is { 1 , 4 , 7 } ,
• the { 2 mod 3 } Group is { 2 , 5 , 8 } ,
• the { 3 mod 3 } Group is { 3 , 6 , 9 } .

And this shall be known as the { Modulo 3 Grouping Method } .

Let us summary these grouping methods as per the table below :

We then have this pair of the diagrams below :

&

We note, of course, that this pair of diagrams have the exact-same coloring scheme as per the pair of diagrams presented above in { Topic II } .

The underlying principle, in both, is that:

• only one & exactly one single element in each { same-color group } appears :
  • in each single row , &

  • in each single column .

This is not an un-common principle in combinatorics and we shall be looking at a similar application of this principle to the { Multiplication Table } for { modulo 13 } , next.

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Topic IV - The { Multiplication Table for Modulo 13 } :

We then have this { Multiplication Table for modulo 13 } , [ 13 ] being a { prime number } , below :

For slightly better clarity, we single-out the pair of digits [ 1 ] & [ 12 ] for the reader's review :

We note here that the digit [ 1 ] appears only once and exactly once :

• in each single row , and

• in each single column .

And this is true of { Multiplication Tables } of this type for { prime numbers } , i.e. { 'odd' prime numbers other than [ 1 ] } .

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Topic V - The { 9 Flying-Star-Positions } :

Let us now look at a special principle derived from the { WEN WANG Diagram } , i.e. the { 9 Flying-Star-Positions } , which is used extensive in FENG SHUI or Chinese Geomacy , and also in the Chinese Calander.

We now bring back this { Numbers Diagram } from above :

And in the { 9 Flying-Star-Position } principle :

• this is known as a { [ 5 ] in-the-middle } , because the number [ 5 ] is in the { middle-box } .

We then place a different [ number ] in the { middle-box } , and move the other [ numbers ] on a { matching-modulo 9-basis } .

For example, when we have a { [ 8 ] in-the-middle } :

• the original [ 5 ] ( in the middle ) is now replaced by an [ 8 ] ,
• the original [ 6 ] is now replaced by a [ 9 ] ,
• the original [ 7 ] is now replaced by a [ 1 ] ,
• the original [ 8 ] is now replaced by a [ 2 ] ,
• the original [ 9 ] is now replaced by a [ 3 ] ,
• the original [ 1 ] is now replaced by a [ 4 ] ,
• the original [ 2 ] is now replaced by a [ 5 ] ,
• the original [ 3 ] is now replaced by a [ 6 ] ,
• the original [ 4 ] is now replaced by a [ 7 ] .

And so-on-and-so-forth, to bring us to this set of 9 diagrams below :

We now make three (3) observations :

• for { [ 2 ] in-the-middle } , { [ 5 ] in-the-middle } & { [ 8 ] in-the-middle } , we have :
  • each of the 3 rows adding-up to [ 15 ] ; and

  • each of the 3 columns adding-up to [ 15 ] .

• for { [ 3 ] in-the-middle } , { [ 6 ] in-the-middle } & { [ 9 ] in-the-middle } , we have :
  • 2 rows each adding-up to [ 18 ] & the remaining row adding-up to [ 9 ] ; and

  • 2 columns each adding-up to [ 18 ] & the remaining column adding-up to [ 9 ] .

• for { [ 1 ] in-the-middle } , { [ 4 ] in-the-middle } & { [ 7 ] in-the-middle } , we have :
  • 2 rows each adding-up to [ 12 ] & the remaining row adding-up to [ 21 ] ; and

  • 2 rows each adding-up to [ 12 ] & the remaining row adding-up to [ 21 ] .

We then summarized these findings as follows :

• 15 + 15 + 15 = 45

• 18 + 18 + 9 = 45

• 12 + 12 + 21 = 45

  arising from : ( 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 = 45 )

Taking out the H.C.F. ( Highest Common Factor ) then yields :

• 1 + 1 + 1 = 3

• 2 + 2 + 1 = 5

• 4 + 4 + 7 = 15

We note that these three (3) equations are of the format :

• A + A + B = C ,

  or alternately, the format :

• 2*A + B = C ,

with :

• { A = 1 } & { B = 1 } for the 1st equation { 1 + 1 + 1 = 3 } ;

• { A = 2 } & { B = 1 } for the 2nd equation { 2 + 2 + 1 = 5 } ;

• { A = 4 } & { B = 7 } for the 3rd equation { 4 + 4 + 7 = 15 } .

The 1st & 2nd of these equations is an illustration of expressing an { 'odd' prime number } in the format :

• P = 2 * N + 1

  And the importance of 'N' in { Number Theory } is that :

  • When { 'N' is 'odd' } , 'P' is of the format { 4*K + 3 } and

    when { 'N' is 'even' } , 'P' is of the format { 4*K + 1 } .

    And this separation of { 4*K +1 } vs. { 4*K + 3 } { 'odd' numbers } is central / primitive to { Number Theory } .

  • The Fermat Little Theorem is also based on the value of 'N', and the factorization of 'N' ;

    ( see also 'An Approach to the Characteristics of Numbers' by the author on :

    • { Multiplication Tables modulo P } , { Squares-Of-Squares } , etc. )

The 3rd equation is an equation for a { composite number }, i.e. { 3 x 5 = 15 } .

• And it is baffling.

But, anyways, these equations do illustrate the principle of { Non-Parity } previous discussed Section 8, i.e. :

• Take 1 of [ something ] and take 2 of [ something-else ] .

  And in our case here, it is :

  • Take 1 of [ B ] and take 2 of [ A ] ,

    to make up the { 'odd' integer } .

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Topic VI - Brief Summary :

Many principles / observations arising from the 'I-CHING' have found amazing similarities in Mathematics / Combinatorics / Linear Algebra , etc.

In this section here, we've examined two (2) underlying principles :

• only one & exactly one element of each color-group in each row / each column .

• mapping relations arising from the { 9 Flying-Star-Positions } and the consequences arising thereof .

Further studies / refinements may well be warranted.

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go to the next Section : Section 9 - The { 'Tao' of the 'I' }

go to the last Section : Section 7 - The { Parity-of-'4' } in the 'I-CHING' & Non-parity

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original dated 2004-12-18