On 8-Colors Combinatorics and [ Quasi 24-Symmetry ]

by Frank C. Fung ( 1st published in March, 2011. )

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Section XXIX - [ Quasi Octo-Symmetric Sets ] within the [ 24 QSM-Configurations ]

Summary for the Section :

• In this Section , we shall attempt to identify the [ Quasi Octo-Symmetric Sets ] within the [ 24 QSM-Configurations ] .

  And our approach here is rather straight-forward :

  • we shall try to combine the 4-elements [ Quasi Quad-Symmetric Sets ] from the last section , Section XXVIII ,
    • to form the 8-elements [ Quasi Octo-Symmetric Set ] here .

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Recalling the [ Quasi Octo-Symmetric Sets ] from Section XIV :

• Let us recall that we did identify , earlier in Section XIV of this paper ,
  • 3 [ Quasi Octo-Symmetric Sets ] within the [ 24 Base Structural Patterns ] , for the [ 8-colors circular-rings ] .

  And these 3 [ Quasi Octo-Symmetric Sets ] with 8 elements each are :

  • [ A ] - [ J ] - [ R ] - [ V ] - [ N ] - [ L ] - [ D ] - [ Z ] ,

  • [ B ] - [ U ] - [ M ] - [ Q ] - [ H ] - [ X ] - [ F ] - [ S ] , and

  • [ C ] - [ P ] - [ Y ] - [ G ] - [ W ] - [ T ] - [ E ] - [ K ] .

• Let us also try and see if we can identify [ Quasi Octo-Symmetric Sets ] within the set of [ 24 QSM-Configurations ] .

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Recalling the 4-elements [ Quasi Quad-Symmetric Sets ] :

• Let us now quickly recall the 18 [ Quasi Quad-Symmetric Sets ] from the last section ,
  • and see if we can use these as the [ base structural pieces ] for our construction of the 8-elements [ Quasi Octo-Symmetric Sets ] .

  Name of Group		The [ Quasi Quad-Symmetric Sets ]							Axis-of-Rotation Sequence for the Link
  									Forward Link				Reverse Link
  ****************	*	*********	*********	*********	*********	*********	*********	*	****	****	****	*	****	****	****
  1st Group of Six		A-V-R-J	B-Q-M-U	C-G-Y-P	D-L-N-Z	E-T-W-K	F-X-H-S		-X	-Y	-Z		-Z	-Y	-X
  2nd Group of Six		A-H-Z-Q	B-W-S-G	C-N-K-V	D-Y-J-T	E-M-P-X	F-R-U-L		-Y	-Z	-X		-X	-Z	-Y
  3rd Group of Six		A-P-L-W	B-J-X-N	C-U-T-H	D-S-V-M	E-Z-G-R	F-K-Q-Y		-Z	-X	-Y		-Y	-X	-Z

• And we take special notice here that :
  • In each of the 3 rows in the above table :
    • the [ lead-off configuration ] for the 6 [ Quasi Quad-Symmetric Sets ] going-across-the-row-from-left-to-right are :
      • [ A ] , [ B ] , [ C ] , [ D ] , [ E ] , and [ F ] respectively .

  We shall be now attempt to find-out :

  • if there are any specific [ Structural Mapping relations ] involving these 6 particular [ configurations ] ,
    • that may be useful to us in our construction process .

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Defining [ Bi-Polar Pairs ] and the [ Bi-Polar Structural Mapping relations ] :

• For our analysis to follow on the [ Structural Mapping relations ] , both here and in later Sections ,
  • it is necessary for us to bring-in the concept of the [ Bi-Polar Pairs ] and the [ Bi-Polar Structural Mapping relations ] .

  And we shall do so now .

• Let us now give a formal definition of the [ Bi-Polar Pairs ] and the [ Bi-Polar Structural Mapping relations ] , as follows :
  • For any two (2) given [ 8-colors configurations ] , labelled as the [ 1st Configuration ] and the [ 2nd Configuration ] respectively :

    IF :

    • the [ Structural Mapping relation ] of the [ 2nd Configuration ] relative to the [ 1st Configuration ]

      and

      the [ Structural Mapping relation ] of the [ 1st Configuration ] relative to the [ 2nd Configuration ]

      • are exactly-the-same ;

    THEN :

    • the [ 1st Configuration ] and the [ 2nd Configuration ] are deemed to be a [ Bi-Polar Pair ] ,

      and

    • the [ Structural Mapping relation ] mapping one-configuration-onto-the-other and vice versa ,
      • is deemed to be a [ Bi-Polar Structural Mapping relation ] .

• Let us now take a quick look at a schematic presentation of a [ Bi-Polar Pair ] and its [ Structural Mapping relations ] ,
  • as per this diagram below :

    

  And we can see that the [ Bi-Polar Structural Mapping relation ] above can be deemed to be symmetric ,

  • in the sense that one can always use the exact-same [ Structural Mapping relation ] :
    • to map the [ 2nd Configuration ] onto the [ 1st Configuration ] and
    • to map the [ 1st Configuration ] onto the [ 2nd Configuration ] .

  And that is to say , that :

  • the [ Bi-Polar Structural Mapping relation ] can always go either-way , both forward and backward ,

    and yield the same results .

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Identify the [ Bi-Polar Pairs ] involving [ A ] , [ B ] , [ C ] , [ D ] , [ E ] and [ F ] :

• Let us now bring-in again this set of 2 diagrams , which first appeared in Section XXVIII :

  

  • and the diagram on-the-right , above , is the original Linkage Diagram for the [ 24 QSM-Configurations ] .

• Let us now also bring-in this set of 6 diagrams , derived from the original Linkage Diagram just-afore-mentioned :
  • showing the [ linkage relations ] for the 6 configurations [ A ] , [ B ] , [ C ] , [ D ] , [ E ] and [ F ] .

  
  

• Let us simply state here that , based on these diagrams , we can now identify the following 9 pairs as [ Bi-Polar Pairs ] :
  • 1st set of 3 [ Bi-Polar Pairs ] :
    • [ A - D ] , [ B - F ] and [ C - E ] ;

  • 2nd set of 3 [ Bi-Polar Pairs ] :
    • [ A - E ] , [ B - D ] and [ C - F ] ;

  • 3rd set of 3 [ Bi-Polar Pairs ] :
    • [ A - F ] , [ B - E ] and [ C - D ] .

  And a full explanation as to why-this-is-so is to follow , next , immediately below .

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Explanation on the 9 [ Bi-Polar Pairs ] :

• Before moving-on further , let us bring-back these two (2) statements we made on the [ 2-notch rotations ] back in Section XVIII ,
  • roughly re-stated as follows :

  Statement One :

  • For any [ 2-notch counter-clockwise rotation ] in any [ Axial-Direction ] ,
    • the reverse is a [ 2-notch clockwise rotation ] in that same [ Axial-Direction ] .

    And vice versa .

  Statement Two :

  • The net result of a [ 2-notch counter-clockwise rotation ] in any [ Axial-Direction ] is the same as
    • the net result of a [ 2-notch clockwise rotation ] in that same [ Axial-Direction ] .

    And as such :

    • The 2 rotations , [ 2-notch counter-clockwise ] vs. [ 2-notch clockwise ] , may be treated as being on-par-with-one-another ;
      • and therefore freely-interchangeable for our Linkage Diagram analysis purposes here .

• Let us now proceed with the explanation , as follows :
  • For the 1st Set of 3 [ Bi-Polar Pairs ] :
    • The link from [ A ] to [ D ] is a [ 2-notch rotation-sequence in the -Z / -X / -Z Axial-Directions ] ;
      • or alternately a [ 2-notch rotation-sequence in the -X / -Z / -X Axial-Directions ] .

    • The link from [ D ] to [ A ] is a [ 2-notch rotation-sequence in the -Z / -X / -Z Axial-Directions ] ;
      • or alternately a [ 2-notch rotation-sequence in the -X / -Z / -X Axial-Directions ] .

    • The link from [ B ] to [ F ] is a [ 2-notch rotation-sequence in the -Z / -X / -Z Axial-Directions ] ;
      • or alternately a [ 2-notch rotation-sequence in the -X / -Z / -X Axial-Directions ] .

    • The link from [ F ] to [ B ] is a [ 2-notch rotation-sequence in the -Z / -X / -Z Axial-Directions ] ;
      • or alternately a [ 2-notch rotation-sequence in the -X / -Z / -X Axial-Directions ] .

    • The link from [ C ] to [ E ] is a [ 2-notch rotation-sequence in the -Z / -X / -Z Axial-Directions ] ;
      • or alternately a [ 2-notch rotation-sequence in the -X / -Z / -X Axial-Directions ] .

    • The link from [ E ] to [ C ] is a [ 2-notch rotation-sequence in the -Z / -X / -Z Axial-Directions ] ;
      • or alternately a [ 2-notch rotation-sequence in the -X / -Z / -X Axial-Directions ] .

    Any way you look at it :

    • a { 2-notch [ -Z / -X / -Z rotation-sequence ] link } is always reversible ; and

    • a { 2-notch [ -X / -Z / -X rotation-sequence ] link } is also always reversible .

    Therefore :

    • the [ Structural Mapping relations ] between [ A and D ] / [ B and F ] / [ C and E ] do exhibit bi-symmetry and reversibility ;

      and consequently :

      • [ A - D ] , [ B - F ] and [ C -E ] can be identified as [ Bi-Polar Pairs ] .

  • For the 2nd Set of 3 [ Bi-Polar Pairs ] :
    • The link from [ A ] to [ E ] is a [ 2-notch rotation-sequence in the -Y / -Z / -Y Axial-Directions ] ;
      • or alternately a [ 2-notch rotation-sequence in the -Z / -Y / -Z Axial-Directions ] .

    • The link from [ E ] to [ A ] is a [ 2-notch rotation-sequence in the -Y / -Z / -Y Axial-Directions ] ;
      • or alternately a [ 2-notch rotation-sequence in the -Z / -Y / -Z Axial-Directions ] .

    • The link from [ B ] to [ D ] is a [ 2-notch rotation-sequence in the -Y / -Z / -Y Axial-Directions ] ;
      • or alternately a [ 2-notch rotation-sequence in the -Z / -Y / -Z Axial-Directions ] .

    • The link from [ D ] to [ B ] is a [ 2-notch rotation-sequence in the -Y / -Z / -Y Axial-Directions ] ;
      • or alternately a [ 2-notch rotation-sequence in the -Z / -Y / -Z Axial-Directions ] .

    • The link from [ C ] to [ F ] is a [ 2-notch rotation-sequence in the -Y / -Z / -Y Axial-Directions ] ;
      • or alternately a [ 2-notch rotation-sequence in the -Z / -Y / -Z Axial-Directions ] .

    • The link from [ F ] to [ C ] is a [ 2-notch rotation-sequence in the -Y / -Z / -Y Axial-Directions ] ;
      • or alternately a [ 2-notch rotation-sequence in the -Z / -Y / -Z Axial-Directions ] .

    Any way you look at it :

    • a { 2-notch [ -Y / -Z / -Y rotation-sequence ] link } is always reversible ; and

    • a { 2-notch [ -Z / -Y / -Z rotation-sequence ] link } is also always reversible .

    Therefore :

    • the [ Structural Mapping relations ] between [ A and E ] / [ B and D ] / [ C and F ] do exhibit bi-symmetry and reversibility ;

      and consequently :

      • [ A - E ] , [ B - D ] and [ C - F ] can be identified as [ Bi-Polar Pairs ] .

  • For the 3rd Set of 3 [ Bi-Polar Pairs ] :
    • The link from [ A ] to [ F ] is a [ 2-notch rotation-sequence in the -X / -Y / -X Axial-Directions ] ;
      • or alternately a [ 2-notch rotation-sequence in the -Y / -X / -Y Axial-Directions ] .

    • The link from [ F ] to [ A ] is a [ 2-notch rotation-sequence in the -X / -Y / -X Axial-Directions ] ;
      • or alternately a [ 2-notch rotation-sequence in the -Y / -X / -Y Axial-Directions ] .

    • The link from [ B ] to [ E ] is a [ 2-notch rotation-sequence in the -X / -Y / -X Axial-Directions ] ;
      • or alternately a [ 2-notch rotation-sequence in the -Y / -X / -Y Axial-Directions ] .

    • The link from [ E ] to [ B ] is a [ 2-notch rotation-sequence in the -X / -Y / -X Axial-Directions ] ;
      • or alternately a [ 2-notch rotation-sequence in the -Y / -X / -Y Axial-Directions ] .

    • The link from [ C ] to [ D ] is a [ 2-notch rotation-sequence in the -X / -Y / -X Axial-Directions ] ;
      • or alternately a [ 2-notch rotation-sequence in the -Y / -X / -Y Axial-Directions ] .

    • The link from [ D ] to [ C ] is a [ 2-notch rotation-sequence in the -X / -Y / -X Axial-Directions ] ;
      • or alternately a [ 2-notch rotation-sequence in the -Y / -X / -Y Axial-Directions ] .

    Any way you look at it :

    • a { 2-notch [ -X / -Y / -X rotation-sequence ] link } is always reversible ; and

    • a { 2-notch [ -Y / -X / -Y rotation-sequence ] link } is also always reversible .

    Therefore :

    • the [ Structural Mapping relations ] between [ A and F ] / [ B and E ] / [ C and D ] do exhibit bi-symmetry and reversibility ;

      and consequently :

      • [ A - F ] , [ B - E ] and [ C - D ] can be identified as [ Bi-Polar Pairs ] .

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Listing-out the 27 [ Quasi Octo-Symmetric Sets ] :

• Now that we have established the 3 sets of [ Bi-Polar Pairs ] , i.e. :
  • 1st Set : [ A - D ] , [ B - F ] and [ C - E ] ,
  • 2nd Set : [ A - E ] , [ B - D ] and [ C - F ] ,
  • 3rd Set : [ A - F ] , [ B - E ] and [ C - D ] ;

  we are now ready to combine the 4-elements [ Quasi Quad-Symmetric sets ] into 8-elements [ Quasi Octo-Symmetric sets ] .

  • using each of the above 3 sets as a [ pairing-scheme for A-B-C-D-E-F ] .

• Let us now bring-back the [ Quasi Quad-Symmetric Sets ] from above :

  Name of Group		The [ Quasi Quad-Symmetric Sets ]							Axis-of-Rotation Sequence for the Link
  									Forward Link				Reverse Link
  ****************	*	*********	*********	*********	*********	*********	*********	*	****	****	****	*	****	****	****
  1st Group of Six		A-V-R-J	B-Q-M-U	C-G-Y-P	D-L-N-Z	E-T-W-K	F-X-H-S		-X	-Y	-Z		-Z	-Y	-X
  2nd Group of Six		A-H-Z-Q	B-W-S-G	C-N-K-V	D-Y-J-T	E-M-P-X	F-R-U-L		-Y	-Z	-X		-X	-Z	-Y
  3rd Group of Six		A-P-L-W	B-J-X-N	C-U-T-H	D-S-V-M	E-Z-G-R	F-K-Q-Y		-Z	-X	-Y		-Y	-X	-Z

  Let us now apply each of the 3 [ pairing-scheme for A-B-C-D-E-F ] we have above , one-at-a-time :

  • to each of the 3 rows in the above table ,
    • to arrive at this table below for our 27 [ Quasi Octo-Symmetric Sets ] .

  Row	The [ Quasi Octo-Symmetric Sets ]				Axis-of-Rotation Sequence for the link linking the [ Quasi Quad-Symmetric sets ]								Axis-of-Rotation Sequence for the [ Bi-Polar Link ] prevailing for the row
  					Forward Link				Reverse Link				1st Option				2nd Option
  ****	*****************	*****************	*****************	*	***	***	***	*	***	***	***	*	***	***	***	*	***	***	***
  1st	A-V-R-J-D-L-N-Z	B-Q-M-U-F-X-H-S	C-G-Y-P-E-T-W-K		-X	-Y	-Z		-Z	-Y	-X		-Z	-X	-Z		-X	-Z	-X
  2nd	A-V-R-J-E-T-W-K	B-Q-M-U-D-L-N-Z	C-G-Y-P-F-X-H-S		-X	-Y	-Z		-Z	-Y	-X		-Y	-Z	-Y		-Z	-Y	-Z
  3rd	A-V-R-J-F-X-H-S	B-Q-M-U-E-T-W-K	C-G-Y-P-D-L-N-Z		-X	-Y	-Z		-Z	-Y	-X		-X	-Y	-X		-Y	-X	-Y
  ****	*****************	*****************	*****************	*	***	***	***	*	***	***	***	*	***	***	***	*	***	***	***
  4th	A-H-Z-Q-D-Y-J-T	B-W-S-G-F-R-U-L	C-N-K-V-E-M-P-X		-Y	-Z	-X		-X	-Z	-Y		-Z	-X	-Z		-X	-Z	-X
  5th	A-H-Z-Q-E-M-P-X	B-W-S-G-D-Y-J-T	C-N-K-V-F-R-U-L		-Y	-Z	-X		-X	-Z	-Y		-Y	-Z	-Y		-Z	-Y	-Z
  6th	A-H-Z-Q-F-R-U-L	B-W-S-G-E-M-P-X	C-N-K-V-D-Y-J-T		-Y	-Z	-X		-X	-Z	-Y		-X	-Y	-X		-Y	-X	-Y
  ****	*****************	*****************	*****************	*	***	***	***	*	***	***	***	*	***	***	***	*	***	***	***
  7th	A-P-L-W-D-S-V-M	B-J-X-N-F-K-Q-Y	C-U-T-H-E-Z-G-R		-Z	-X	-Y		-Y	-X	-Z		-Z	-X	-Z		-X	-Z	-X
  8th	A-P-L-W-E-Z-G-R	B-J-X-N-D-S-V-M	C-U-T-H-F-K-Q-Y		-Z	-X	-Y		-Y	-X	-Z		-Y	-Z	-Y		-Z	-Y	-Z
  9th	A-P-L-W-F-K-Q-Y	B-J-X-N-E-Z-G-R	C-U-T-H-D-S-V-M		-Z	-X	-Y		-Y	-X	-Z		-X	-Y	-X		-Y	-X	-Y

• We then bring-in this set of 3 diagrams :
  • for the [ Schematic Presentation ] of the 3 [ Quasi Octo-Symmetric Sets ] in the [ 1st row ] ;

  

  • showing the relative positions of the 8 elements in terms of the [ Structural Mapping relations ] .

• But [ Schematic Presentation Diagrams ] can deceptive occassionally ,
  • it is therefore appropriate to bring-in also the Linkage Diagrams for further reference .

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Bringing-in again the [ 12-elements loops ] :

• Let us now concentrate , for the moment , on the very-first [ Quasi Octo-Symmetric Set ] in the above table ,
  • i.e. : the [ A-V-R-J-D-L-N-Z ] set ,
    • constructed via combining the 2 [ Quasi Quad-Symmetric Sets ] :
      • the [ A-V-R-J ] set and the [ D-L-N-Z ] set respectively .

  We can then bring-in again the [ 12-elements loops ] associated with these 2 [ Quasi Quad-Symmetric Sets ] ,

  • as per this diagram below .

  

• And we see here that the 2 [ 12-elements loops ] here do account for 24 links ,
  • out of a total of 36 links for the [ 24 QSM-Configurations ] .

  And the 12 links missing here are then made-up-of :

  • 4 [ brown-color links ] for the [ 2-notch -X axis rotations ] ,
  • 4 [ green-color links ] for the [ 2-notch -Y axis rotations ] , and
  • 4 [ blue-color links ] for the [ 2-notch -Z axis rotations ] .

• Let us now fill-in the links :

  

  Several observations here :

  • In the diagram on-the-left :
    • We can now identify the 4 [ Bi-Polar Links ] involving the [ -Z / -X / -Z axis rotation-sequence ] ,
      • i.e : the [ A-U-K-D ] / [ V-T-F-Z ] / [ R-H-Y-N ] / [ J-C-Q-L ] links .

  • In the diagram in-the-middle :
    • We can now identify the single { 2-notch [ -Y Axis rotation ] } links ,
      • linking [ A - N ] / [ V - L ] / [ R - D ] / [ J - Z ] .

  • In the diagram on-the-right :
    • We can now identify the 4 [ Bi-Polar Links ] involving the [ -X / -Z / -X axis rotation-sequence ] ,
      • i.e. : the [ A-G-X-D ] / [ V-B-P-Z ] / [ R-W-M-N ] / [ J-S-E-L ] links .

• But there is a special [ catch ] here :
  • Each-and-every single [ 2-notch rotation in any Axial-Direction ] is in-itself-and-by-itself a [ Bi-Polar Link ] .

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Using the single [ 2-notch rotation ] as the [ Bi-Polar Link ] :

• Let us now bring-back again the 2 [ 12-elements loops ] for the [ A-V-R-J-D-L-N-Z ] set :
  • as per the diagram on-the-left , below .

  

  Let us now rotate the { [ 12-elements loop ] at the bottom } thru [ an angle of 180 degrees ] ,

  • to arrive at the diagram in-the-middle , above .

  We then fill in the 4 single { 2-notch [ -Y Axis rotation ] }'s :

  • to arrive at the diagram on-the-right , above .

• And based on this very-last diagram on-the-right , it is now clearly possible for us :
  • to use the [ { 2-notch [ -Y Axis rotation ] } link as the [ Bi-Polar Link ] for our [ A-V-R-J-D-L-N-Z ] set here .

  And , upon using this [ single 2-notch rotation ] link as the [ Bi-Polar Link ] for the entire [ 1st row ] ,

  • our [ Schematic Presentation ] for the 3 [ Quasi Octo-Symmetric Sets ] in this [ 1st row ] would then become :

  

• Isn't that really neat !
  • You bet .

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go to the next section : Section XXX - Revisiting the Concept of QI and OU

go to the last section : Section XXVIII -[ Quasi Quad-Symmetric Sets ] within the [ 24 QSM-Configurations ]

return to the 8-Colors Combinatorics / Quasi 24-Symmetry HomePage

original dated 2011-3-06