An Approach to the 'I-CHING'

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

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

Topic I - Introduction & Brief Summary for the Section :

In this section, we shall be comparing the [ Parity of '4' ] :

• Commonly found in [ Modulo Mathematics ] , an important topic in { Number Theory } , and

• the [ Parity of '4' ] arising from the 'I-CHING'.

By the end of the Section, we would have moved onto { Non-Parity } in the 'I-CHING' arising from the { Reverse } process.

• And examples of { Non-parity } in Mathematics & Physics are also cited for the reader's reference.

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Topic II - Modulo Mathematics basics :

In general, if [ P ] is a { positive integer } and [ X ] is a { positive integer }, then the statement :

• X R modulo P

  ( read : [ X ] is congruent to [ R ] modulo [ P ] )

is the same as :

• X = N * P + R
  • with [ R ] < [ P ]

  • where : 'N' is an integer, inclusive of the integer [ 0 ] .

  And [ R ] can be viewed as the { remaider } when [ X ] is divided by [ P ] .

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Topic III - The { Additive Conjugate } & the { Multiplicative Conjugate } :

We then define the { Additive Conjugate } and the { Multiplicative Conjugate } as follows :

• If [ Y ] is the { Additive Conjugate } of [ X ] , { modulo P } , then [ Y ] satisfies the equation :

  X + Y 0 modulo P

• If [ Z ] is the { Mulitplicative Conjugate } of [ X ] , { modulo P } , then [ Z ] satisfies the equation :

  X * Z 1 modulo P

  ( [ Z ] always exists if [ P ] is a { prime number } ) .

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Topic IV - An example for { Modulo 13 } :

We then have an example here for { modulo 13 } :

• 3 + 10 0 modulo 13

  so that [ 10 ] is the { Additive Conjugate } of [ 3 ] { modulo 13 } .

• 3 * 9 1 modulo 13

  so that [ 9 ] is the { Multiplicative Conjugate } of [ 3 ] { modulo 13 } .

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Topic V - Defining the quantity [ M-O-A ] :

We now also define the quantity [ M-O-A ] as the [ { Multiplciative Conjugate } of the { Additive Conjugate } , { modulo P } ] .

And [ 4 ] is then the [ M-O-A ] of [ 3 ] { modulo 13 } , arising from :

• 3 + 10 0 modulo 13 , ( [ 10 ] is the { Additive Conjugate } of [ 3 ] ) , and

• 10 * 4 1 modulo 13 , ( [ 4 ] is the { Multiplicative Conjugate } of [ 10 ] ) .

We note here, without further proof, that :

• the [ { Multiplicative Conjugate } of the { Additive Conjugate } , { modulo P } ]

is the same as

• the [ { Additive Conjugate } of the { Multiplicative Conjugate } , { modulo P } ]

i.e. :

• 3 * 9 1 modulo 13 , ( [ 9 ] is the { Multiplicative Conjugate } of [ 3 ] ) , and

• 9 + 4 0 modulo 13 , ( [ 4 ] is the { Additive Conjugate } of [ 9 ] ) .

And we arrive here at [ 4 ] being :

• the [ { Additive Conjugate } of the { Multiplicative Conjugate } ] of [ 3 ] , { modulo 13 } .

  same as the [ { Multiplicative Conjugate } of the { Additive Conjugate } ] we have originally above.

We can then re-group the 12 digits { from [ 1 ] to [ 12 ] } for { modulo 13 } into four (4) groups :

• Group I : [ 3 , 10 , 9 , 4 ] as characterized by these { 4-quadrant diagrams } below :

  

  ( [ A-CON ] is the { Additive Conjugate } & [ M-CON ] is the { Multiplicative Conjugate } ).

• Group II : [ 2 , 11 , 7 , 6 ] as characterized by these { 4-quadrant diagrams } below :

  

• Group III : [ 5 , 8 ] as characterized by these { 4-quadrant diagrams } below :

  

• Group IV : [ 1 , 12 ] as characterized by these { 4-quadrant diagrams } below :

  

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Topic VI - The { Reverse } & the { Opposite } for Hexagrams :

The [ Reverse ] for any given Hexagram simply is the Hexagram { read-upside-down }, i.e. :

• { Line 6 } of the original Hexagram is now { Line 1 } for the new Hexagram ,
• { Line 5 } of the original Hexagram is now { Line 2 } for the new Hexagram ,
• { Line 4 } of the original Hexagram is now { Line 3 } for the new Hexagram ,
• { Line 3 } of the original Hexagram is now { Line 4 } for the new Hexagram ,
• { Line 2 } of the original Hexagram is now { Line 5 } for the new Hexagram ,
• { Line 1 } of the original Hexagram is now { Line 6 } for the new Hexagram .

The [ Opposite ] for any given Hexagram is simply changing each of the { 6 Lines } of the Hexagram to the 'Opposite Line' .

• from a { 'Yin Line' } to a { 'Yang Line' } , and/or

• from a { 'Yang Line' } to a { 'Yin Line' } .

And we also define the [ O-O-R ] Hexagram as the { [ Opposite ] of the [ Reverse ] } of any given Hexagram.

And it would be best to give an example here :

Origial Hexagram	Reverse Hexagram	Opposite Hexagram	O-O-R Hexagram

And the { 4-quadrant diagram } here is :

• ( [ OPPO ] is the 'Opposite' and [ REVER ] is the 'Reverse' ) .

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Topic VII - The Listing Order for the 64 Hexagrams :

We then have a listing of the 1st thirty (30) Hexagrams, in the order as they appear in the [ First Section ] ( ) of the 'I-CHING' :

Pair Name	Order / Rank	1st member		2nd member		Comments
Pair 1	1, 2					{ T1-T1 } & { T8-T8 }
Pair 2	3, 4					{ T6-T4 } & { T7-T6 }
Pair 3	5, 6					{ T6-T1 } & { T1-T6 }
Pair 4	7, 8					{ T8-T6 } & { T6-T8 }
Pair 5	9, 10					{ T5-T1 } & { T1-T2 }
Pair 6	11, 12					{ T8-T1 } & { T1-T8 }
Pair 7	13, 14					{ T1-T3 } & { T3-T1 }
Pair 8	15, 16					{ T8-T7 } & { T4-T8 }
Pair 9	17, 18					{ T2-T4 } & { T7-T5 }
Pair 10	19, 20					{ T8-T2 } & { T5-T8 }
Pair 11	21, 22					{ T3-T4 } & { T7-T3 }
Pair 12	23, 24					{ T7-T8 } & { T8-T4 }
Pair 13	25, 26					{ T1-T4 } & { T7-T1 }
Pair 14	27, 28					{ T7-T4 } & { T2-T5 }
Pair 15	29, 30					{ T6-T6 } & { T3-T3 }

Pair Name	Order / Rank	1st member		2nd member		Comments
Pair 16	31, 32					{ T2-T7 } & { T4-T5 }
Pair 17	33, 34					{ T1-T7 } & { T4-T1 }
Pair 18	35, 36					{ T3-T8 } & { T8-T3 }
Pair 19	37, 38					{ T5-T3 } & { T3-T2 }
Pair 20	39, 40					{ T6-T7 } & { T4-T6 }
Pair 21	41, 42					{ T7-T2 } & { T5-T4 }
Pair 22	43, 44					{ T2-T1 } & { T1-T5 }
Pair 23	45, 46					{ T2-T8 } & { T8-T5 }
Pair 24	47, 48					{ T2-T6 } & { T6-T5 }
Pair 25	49, 50					{ T2-T3 } & { T3-T5 }
Pair 26	51, 52					{ T4-T4 } & { T7-T7 }
Pair 27	53, 54					{ T5-T7 } & { T4-T2 }
Pair 28	55, 56					{ T4-T3 } & { T3-T7 }
Pair 29	57, 58					{ T5-T5 } & { T2-T2 }
Pair 30	59, 60					{ T5-T6 } & { T6-T2 }
Pair 31	61, 62					{ T5-T2 } & { T4-T7 }
Pair 32	63, 64					{ T6-T3 } & { T3-T6 }

We take note here that, in this Listing Order , the { 2nd member } of each pair of Hexagrams is :

• the { 'Reverse' } of the { 1st member } of the pair,

or :

• the { 'Opposite' } of the { 1st member } of the pair,
  • if the { 'Reverse' } of the { 1st member } is the same as the { 1st member } itself.

    And this happened for { Pair 1 } , { Pair 14 } , { Pair 15 } , & { Pair 31 } , respectively.

And this is the strongest confirmation that the { 'Reverse' } and the { 'Opposite' } are very much an integral part of the 'I-CHING' .

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Topic VIII - Re-Grouping the 64 Hexagrams :

We can then re-group the 64 Hexagrams into three (3) groups :

• Group I - 8 members ( the SYMMETRIC group ) :

  Name of Sub-group	Origial Hexagram	Reverse Hexagram	Opposite Hexagram	O-O-R Hexagram	Comments
  SYM-1					Pair 1
  SYM-2					Pair 14
  SYM-3					Pair 15
  SYM-4					Pair 31

  These 8 members of this group are then characterized by :

  • the { 'Reverse' } is the same as the Hexagram itself.

  And the format of the { 4-quadrant diagram } here is :

  

• Group II - 4 members ( the OPPOSITE group ) :

  Name of Sub-group	Origial Hexagram	Reverse Hexagram	Opposite Hexagram	O-O-R Hexagram	Comments
  OPP-1					Pair 6
  OPP-2					Pair 32

  These 4 members of the group are then characterized by :

  • the { 'Reverse' } is the same as the { 'Opposite' } ,

  And the format of the { 4-quadrant diagram } here is :

  

• Group III - 48 members ( the ORDINARY group ) :

  Name of Sub-group	Origial Hexagram	Reverse Hexagram	Opposite Hexagram	O-O-R Hexagram	Comments
  ORD-1					Pair 2 & Pair 25
  ORD-2					Pair 3 & Pair 18
  ORD-3					Pair 4 & Pair 7
  ORD-4					Pair 5 & Pair 8
  ORD-5					Pair 10 & Pair 17
  ORD-6					Pair 11 & Pair 24
  ORD-7					Pair 12 & pair 22
  ORD-8					Pair 13 & Pair 23
  ORD-9					Pair 16 & Pair 21
  ORD-10					Pair 19 & Pair 20
  ORD-11					Pair 26 & Pair 29
  ORD-12					Pair 28 & Pair 30

  These 48 members of the group are then characterized by :

  • the { 'Reverse' } , the { 'Opposite' } , the { 'O-O-R' } and the { 'SELF' } are all different from one-another.

  And the format of the { 4-quadrant diagram } here is :

  

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Topic IX - Observing the similarities :

We now make an observation that :

• the { 4-quadrant diagrams } for { Modulo Mathematics } , i.e. :

  , , and

and

• the { 4-quadrant diagrams } for the Hexagrams, i.e. :

  , , and

are rather similar.

We simply note here that the { 4-quadrant diagrams } are also used as an analysis tool in the author's other paper, "An Approach to Matrix & Linear Algebra" .

The { Parity-of-'4' } is clearly a very interestsing subject requiring further studies.

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Topic X - An Analysis of the { Reverse } Process :

We shall now look more closely at the { Reverse } process, using the { ICG Hexagram Placement Scheme } from Section 5 as a guideline for our analysis.

Let us now break-up the six (6) Lines of the Hexagram into two (2) groups :

• The 1st group, containing { 'Line 2' } & { 'Line 5' } , shall then be known as { Group L-2-5 } ;

• The 2nd group, containing { 'Line 1' } , { 'Line 3' } , { 'Line 4' } & { 'Line 6' } , respectively, shall then be known as { Group L-1-3-4-6 } .

We simply note here that, in the { Reverse } process :

• [ Line 2 ] flips onto [ Line 5 ] , and

• [ Line 5 ] flips onto [ Line 2 ] .

And [ Line 2 ] & [ Line 5 ] are the [ Y-Low-Line ] and the [ Y-Hign-Line ] , respectively, in the { ICG Hexagram Placement Scheme } .

Thus,

• { Group L-2-5 } exhibits [ Y-onto-Y ] in the flipping process, whereas,

• { Group L-1-3-4-6 } exhibits [ X-onto-Z ] and [ Z-onto-X ] in the flipping process.

For { Group L-2-5 } , we have this table for the [ SELF ] & the [ Reverse ] :

SELF	Y=0 mod 4	Y=1 mod 4	Y=2 mod 4	Y=3 mod 4
Reverse
	Y=0 mod 4	Y=3 mod 4	Y=2 mod 4	Y=1 mod 4

We note here that :

• [ Y = 0 mod 4 ] flips onto [ Y = 0 mod 4 ] ,
• [ Y = 1 mod 4 ] flips onto [ Y = 3 mod 4 ] ,
• [ Y = 2 mod 4 ] flips onto [ Y = 2 mod 4 ] ,
• [ Y = 3 mod 4 ] flips onto [ Y = 1 mod 4 ] .

For { Group L-1-3-4-6 } , we have the following four (4) tables for the [ SELF ] & the [ Reverse ] :

• The 1st table has 4 members :

  SELF	X=0 mod 4 Z=0 mod 4	X=1 mod 4 Z=3 mod 4	X=2 mod 4 Z=2 mod 4	X=3 mod 4 Z=1 mod 4
  Reverse
  	X=0 mod 4 Z=0 mod 4	X=1 mod 4 Z=3 mod 4	X=2 mod 4 Z=2 mod 4	X=3 mod 4 Z=1 mod 4

  We note here that each of the { 4 members } flips exactly onto itself.

• the 2nd table also has 4 members :

  SELF	X=0 mod 4 Z=2 mod 4	X=1 mod 4 Z=1 mod 4	X=2 mod 4 Z=0 mod 4	X=3 mod 4 Z=3 mod 4
  Reverse
  	X=2 mod 4 Z=0 mod 4	X=3 mod 4 Z=3 mod 4	X=0 mod 4 Z=2 mod 4	X=1 mod 4 Z=1 mod 4

  We note here that :

  • each member contains 2 { 'Yang Lines' } and 2 { 'Yin Lines' } ,

  • each member flips onto a different member in the group.

• the 3rd table has 4 members :

  SELF	X=1 mod 4 Z=0 mod 4	X=0 mod 4 Z=1 mod 4	X=3 mod 4 Z=0 mod 4	X=0 mod 4 Z=3 mod 4
  Reverse
  	X=0 mod 4 Z=3 mod 4	X=3 mod 4 Z=0 mod 4	X=0 mod 4 Z=1 mod 4	X=1 mod 4 Z=0 mod 4

  We note here that :

  • each member contains 1 { 'Yang Line' } and 3 { 'Yin Lines' } ,

  • each member flips onto a different member in the group.

• the 4th & final table has 4 members :

  SELF	X=2 mod 4 Z=1 mod 4	X=1 mod 4 Z=2 mod 4	X=2 mod 4 Z=3 mod 4	X=3 mod 4 Z=2 mod 4
  Reverse
  	X=3 mod 4 Z=2 mod 4	X=2 mod 4 Z=3 mod 4	X=1 mod 4 Z=2 mod 4	X=2 mod 4 Z=1 mod 4

  We note here that :

  • each member contains 3 { 'Yang Lines' } and 1 { 'Yin Line' } ,

  • each member flips onto a different member in the group.

Let us now summarize these findings via this set of four (4) diagrams below , i.e. :

• the 1st pair of diagrams for :
  • the { X-Z Plane } at { Y = 0 } and the { X-Z Plane } at { Y = 2 } ,

• the 2nd pair of diagrams for :
  • the { X-Z Plane } at { Y = 1 } and the { X-Z Plane } at { Y = 3 } .

Then, firstly, for the { X-Z planes } at [ Y = 0 mod 4 ] and at [ Y = 2 mod 4 ] :

• each of the { red-color Hexagrams } flips onto itself.

• each of the { purple-color Hexagrams } flips onto a different { purple-color Hexagram } on the same { X-Z plane } .

• each of the { light-green-color Hexagrams } flips onto a { light-blue-color Hexagram } on the same { X-Z plane } , and visa-versa.

And secondly, for the { X-Z planes } at [ Y = 1 mod 4 ] and at [ Y = 3 mod 4 ] :

• each of the { orange-color Hexagrams } on the { X-Z plane } at [ Y = 1 mod 3 ]
  flips onto a { orange-color Hexagram } on the { X-Z plane } at [ Y = 3 mod 4 ] , and visa-versa.

• each of the { pink-color Hexagrams } on the { X-Z plane } at [ Y = 1 mod 4 ]
  flips onto a { pink-color Hexagram } on the { X-Z plane } at [ Y = 3 mod 4 ] , and visa-versa.

• each of the { dark-green-color Hexagrams } on the { X-Z plane } at [ Y = 1 mod 4 ]
  flips onto a { dark-blue-color Hexagram } on the { X-Z plane } at [ Y = 3 mod 4 ] , and visa-versa.

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Topic XI - The Rules for 'Flipping' :

We then summarize the { Rules for 'flipping' } as follows :

For the [ SELF ] situated at and its [ REVERSE ] situated at , the following rules are applicable :

• Rule #1-A :
  • For { } :

    

• Rule #1-B :
  • For { } :

    

• Rule #2-A :
  • For { } :

    and

    ( the { red-color and orange-color Hexagrams } )

• Rule #2-B :
  • For { } :

    and

    ( the { purple-color & pink-color Hexagrams } )

• Rule #2-C :
  • For { } :

    

    ( the { light-green/light-blue-color & dark-green/dark-blue-color Hexagrams } )

This then demonstrate the { Non-Parity } arising from the { Reverse } process :

• The 'movements' are different :
  • when [ Y is 'even' ] , as against
  • when [ Y is 'odd' ] .

• The 'movements' are different :
  • when { X + Z } is congruent to { 0 mod 4 } , as against
  • when { X + Z } is congruent to { 2 mod 4 } , as against
  • when { X + Z } is congruent to { 1 mod 4 } or { 3 mod 4 } .

A very interesting branch onto { Non-parity } from a rather simple-looking procedure, i.e. the { Reverse } process.

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Topic XII - { Non-Parity } in the 'I-CHING' :

While there may be many different principles of { Non-parity } , we shall now take this one for our purposes here :

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

In our { Reverse } ( 'flipping' ) context here, it is :

• take 1 pair of [ Y-onto-Y ] , i.e. :
  • flipping [ Line-2 ] onto [ Line-5 ] --- [ Y-on-Y ] ,
  • flipping [ Line-5 ] onto [ Line-2 ] --- [ Y-on-Y ] ,

• and take 2 pairs of [ X-onto-Z ] & [ Z-onto-X ] , i.e. :
  • flipping [ Line-1 ] onto [ Line-6 ] --- [ X-on-Z ] ,
  • flipping [ Line-3 ] onto [ Line-4 ] --- [ Z-on-X ] ,
  • flipping [ Line-4 ] onto [ Line-3 ] --- [ X-on-Z ] ,
  • flipping [ Line-6 ] onto [ Line-1 ] --- [ Z-on-X ] .

Let us now look at an example here in Mathematics exhibiting this principle :

• The expansion of the Pythagorus Equation involving :
  • a { pure power-of-'2' } , and
  • two { positive odd integers } .

  We then have this example involving [ 8 ] , the { pure power-of-'2' } and [ 11 ] & [ 17 ] , the two { positive odd integers } :

  • A = 2 * { 8 * 11 * 17 } = 2,992
  • B = 8 * 8 * 11 * 11 - 17 * 17 = 7,455
  • C = 8 * 8 * 11 * 11 + 17 * 17 = 8,033

  And we have :

  • [ 8, 033 ]^2 = [ 2,992 ]^2 + [ 7,455 ]^2

• the same type of expansion involving a { pure power-of-'2' } & two { positive odd integers } is also found in the expansion on the duality of the { Fermat Sum-Of-Two-Squares Equation } , i.e. the equation :
  • C^2 = D^2 + E^2 = F^2 + G^2

  Interested readers may go to "An Approach to the Characteristics of Numbers" by the author for further details.

And { Non-parity } is also a phenomenon in Physics :

• The { non-Conservation of Parity } in Physics :

  In 1956, the late Dr. Chien-Shiung Wu ( ) , of Columbia University , disproved the { Conservation of Parity } in Physics, in an experiment involving [ radioactive cobalt-60 ] .

  ( the author does not fully understand this one. )

And other types of { Non-Parity } are also exhibited :

• e.g., in the factorization of [ Fermat Numbers ] , using the { Augmented Binary format } :

  

  { Non-parity } is found on :

  • [ Switching-on-a-broken-pair ] vs. [ Switching-on-a-completed-pair ] for the { Richochet-in-the-Box } .

  Interested readers may go to "An Approach to the Characteristics of Numbers" by the author for further details.

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go to the next Section : Section 8 - Determinants, Magic Squares, & Multiplication Tables

go to the last Section : Section 6 - { Combinatorial Mathematics } in the 'I-CHING'

return to the ICG HomePage

original dated 2004-12-18