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

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

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Section XXVIII - [ Quasi Quad-Symmetric Sets ] within the [ 24 QSM-Configurations ]

Summary for the Section :

• In this Section , we shall attempt to identify the [ Quasi Quad-Symmetric Sets ] within the [ 24 QSM-Configurations ] ,
  • via the use of [ 12-elements loops ] .

  Our approach here is as follows :

  • We shall first construct 18 [ 12-elements loops ] ,

    and then :

    • chop-up each [ 12-elements loop ] into 4 equal-sections ,
      • to come up with a 4-elements [ Quasi Quad -Symmetric Set ] .

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Recalling the [ Quasi Hexa-Symmetric Sets ] from Section XII :

• Let us recall that we did identify , earlier in Section XII of this paper ,
  • 6 [ Quasi Quad-Symmetric Sets ] within the [ 24 Base Structural Patterns ] .

  And these 6 [ Quasi Quad-Symmetric Sets ] with 4 elements each are :

  • [ A ] - [ H ] - [ Z ] - [ Q ] ,

  • [ B ] - [ W ] - [ S ] - [ G ] ,

  • [ C ] - [ N ] - [ K ] - [ V ] ,

  • [ D ] - [ Y ] - [ J ] - [ T ] ,

  • [ E ] - [ M ] - [ P ] - [ X ] , and

  • [ F ] - [ R ] - [ U ] - [ L ] .

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

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Setting up the [ 12-elements loops ] for [ A ] / [ B ] / [ C ] :

• Let us bring-in again , this set of 3 [ Linkage Digrams ] , for re-familiarization purposes :

  

  Let us now identify the 3 shortest linkage-routes going from [ A ] to [ B ] :

  Name of Route		Route Path from [ A ] to [ B ]		Sequence for the Axis-of-Rotation
  ****************	*	****************	*	****	****	****	****
  Route 401		A-G-T-V-B		-X	-Y	-Z	-X
  Route 402		A-N-Y-H-B		-Y	-Z	-X	-Y
  Route 403		A-U-K-P-B		-Z	-X	-Y	-Z

  Let us also identify the 3 shortest linkage-routes going from [ B ] to [ C ] :

  Name of Route		Route Path from [ B ] to [ C ]		Sequence for the Axis-of-Rotation
  ****************	*	****************	*	****	****	****	****
  Route 404		B-V-L-Q-C		-X	-Y	-Z	-X
  Route 405		B-H-R-W-C		-Y	-Z	-X	-Y
  Route 406		B-P-Z-J-C		-Z	-X	-Y	-Z

  Let us also identify the 3 shortest linkage-routes going from [ C ] to [ A ] :

  Name of Route		Route Path from [ C ] to [ A ]		Sequence for the Axis-of-Rotation
  ****************	*	****************	*	****	****	****	****
  Route 407		C-Q-X-G-A		-X	-Y	-Z	-X
  Route 408		C-W-M-N-A		-Y	-Z	-X	-Y
  Route 409		C-J-S-U-A		-Z	-X	-Y	-Z

• Let us now proceed to construct 9 [ 12-elements loops ] using the above nine (9) route-paths .
  • We then :
    • combine [ Route 401 ] / [ Route 405 ] / [ Route 409 ] to form [ Loop 1201 ] ;
    • combine [ Route 402 ] / [ Route 406 ] / [ Route 407 ] to form [ Loop 1202 ] ;
    • combine [ Route 403 ] / [ Route 404 ] / [ Route 408 ] to form [ Loop 1203 ] ;
    • combine [ Route 404 ] / [ Route 408 ] / [ Route 403 ] to form [ Loop 1204 ] ;
    • combine [ Route 405 ] / [ Route 409 ] / [ Route 401 ] to form [ Loop 1205 ] ;
    • combine [ Route 406 ] / [ Route 407 ] / [ Route 402 ] to form [ Loop 1206 ] ;
    • combine [ Route 407 ] / [ Route 402 ] / [ Route 406 ] to form [ Loop 1207 ] ;
    • combine [ Route 408 ] / [ Route 401 ] / [ Route 404 ] to form [ Loop 1208 ] ;
    • combine [ Route 409 ] / [ Route 403 ] / [ Route 405 ] to form [ Loop 1209 ] .

  We then list-out the 9 resultant [ 12-elements loops ] in this table below :

  Name of Loop		Route Path for the [ 12-elements Loop ]		Sequence for the Axis-of-Rotation
  ****************	*	*****************************	*	****	****	****	****	****	****	****	****	****	****	****	****
  Loop 1201		A-G-T-V-B-H-R-W-C-J-S-U-A		-X	-Y	-Z	-X	-Y	-Z	-X	-Y	-Z	-X	-Y	-Z
  Loop 1202		A-N-Y-H-B-P-Z-J-C-Q-X-G-A		-Y	-Z	-X	-Y	-Z	-X	-Y	-Z	-X	-Y	-Z	-X
  Loop 1203		A-U-K-P-B-V-L-S-Q-W-M-N-A		-Z	-X	-Y	-Z	-X	-Y	-Z	-X	-Y	-Z	-X	-Y
  Loop 1204		B-V-L-Q-C-W-M-N-A-U-K-P-B		-X	-Y	-Z	-X	-Y	-Z	-X	-Y	-Z	-X	-Y	-Z
  Loop 1205		B-H-R-W-C-J-S-U-A-G-T-V-B		-Y	-Z	-X	-Y	-Z	-X	-Y	-Z	-X	-Y	-Z	-X
  Loop 1206		B-P-Z-J-C-Q-X-G-A-N-Y-H-B		-Z	-X	-Y	-Z	-X	-Y	-Z	-X	-Y	-Z	-X	-Y
  Loop 1207		C-Q-X-G-A-N-Y-H-B-P-Z-J-C		-X	-Y	-Z	-X	-Y	-Z	-X	-Y	-Z	-X	-Y	-Z
  Loop 1208		C-W-M-N-A-U-K-P-B-V-L-Q-C		-Y	-Z	-X	-Y	-Z	-X	-Y	-Z	-X	-Y	-Z	-X
  Loop 1209		C-J-S-U-A-G-T-V-B-H-R-W-C		-Z	-X	-Y	-Z	-X	-Y	-Z	-X	-Y	-Z	-X	-Y

• And we shall come back to these 9 [ 12-elements loops ] shortly ,
  • after we have constructed another set of 9 [ 12-elements loops ] for [ D ] / [ E ] / [ F ] ,
    • immediately below .

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Setting up the [ 12-elements loops ] for [ D ] / [ E ] / [ F ] :

• Let us now repeat the process above for [ D ] , [ E ] and [ F ] ;
  • to arrive at another set of 9 [ 12-elements loops ] .

  Let us now identify the 3 shortest linkage-routes going from [ D ] to [ E ] :

  Name of Route		Route Path from [ D ] to [ E ]		Sequence for the Axis-of-Rotation
  ****************	*	****************	*	****	****	****	****
  Route 451		D-X-Q-L-E		-X	-Y	-Z	-X
  Route 452		D-R-H-Y-E		-Y	-Z	-X	-Y
  Route 453		D-K-U-S-E		-Z	-X	-Y	-Z

  Let us also identify the 3 shortest linkage-routes going from [ E ] to [ F ] :

  Name of Route		Route Path from [ E ] to [ F ]		Sequence for the Axis-of-Rotation
  ****************	*	****************	*	****	****	****	****
  Route 454		E-L-V-T-F		-X	-Y	-Z	-X
  Route 455		E-Y-N-M-F		-Y	-Z	-X	-Y
  Route 456		E-S-J-Z-F		-Z	-X	-Y	-Z

  Let us also identify the 3 shortest linkage-routes going from [ F ] to [ D ] :

  Name of Route		Route Path from [ F ] to [ D ]		Sequence for the Axis-of-Rotation
  ****************	*	****************	*	****	****	****	****
  Route 457		F-T-G-X-D		-X	-Y	-Z	-X
  Route 458		F-M-W-R-D		-Y	-Z	-X	-Y
  Route 459		F-Z-P-K-D		-Z	-X	-Y	-Z

• Let us now proceed to construct 9 [ 12-elements loops ] using the above nine (9) route-paths .
  • We then :
    • combine [ Route 451 ] / [ Route 455 ] / [ Route 459 ] to form [ Loop 1251 ] ;
    • combine [ Route 452 ] / [ Route 456 ] / [ Route 457 ] to form [ Loop 1252 ] ;
    • combine [ Route 453 ] / [ Route 454 ] / [ Route 458 ] to form [ Loop 1253 ] ;
    • combine [ Route 454 ] / [ Route 458 ] / [ Route 453 ] to form [ Loop 1254 ] ;
    • combine [ Route 455 ] / [ Route 459 ] / [ Route 451 ] to form [ Loop 1255 ] ;
    • combine [ Route 456 ] / [ Route 457 ] / [ Route 452 ] to form [ Loop 1256 ] ;
    • combine [ Route 457 ] / [ Route 452 ] / [ Route 456 ] to form [ Loop 1257 ] ;
    • combine [ Route 458 ] / [ Route 451 ] / [ Route 454 ] to form [ Loop 1258 ] ;
    • combine [ Route 459 ] / [ Route 453 ] / [ Route 455 ] to form [ Loop 1259 ] .

  We then list-out the 9 resultant [ 12-elements loops ] in this table below :

  Name of Loop		Route Path for the [ 12-elements Loop ]		Sequence for the Axis-of-Rotation
  ****************	*	*****************************	*	****	****	****	****	****	****	****	****	****	****	****	****
  Loop 1251		D-X-Q-L-E-Y-N-M-F-Z-P-K-D		-X	-Y	-Z	-X	-Y	-Z	-X	-Y	-Z	-X	-Y	-Z
  Loop 1252		D-R-H-Y-E-S-J-Z-F-T-G-X-D		-Y	-Z	-X	-Y	-Z	-X	-Y	-Z	-X	-Y	-Z	-X
  Loop 1253		D-K-U-S-E-L-V-T-F-M-W-R-D		-Z	-X	-Y	-Z	-X	-Y	-Z	-X	-Y	-Z	-X	-Y
  Loop 1254		E-L-V-T-F-M-W-R-D-K-U-S-E		-X	-Y	-Z	-X	-Y	-Z	-X	-Y	-Z	-X	-Y	-Z
  Loop 1255		E-Y-N-M-F-Z-P-K-D-X-Q-L-E		-Y	-Z	-X	-Y	-Z	-X	-Y	-Z	-X	-Y	-Z	-X
  Loop 1256		E-S-J-Z-F-T-G-X-D-R-H-Y-E		-Z	-X	-Y	-Z	-X	-Y	-Z	-X	-Y	-Z	-X	-Y
  Loop 1257		F-T-G-X-D-R-H-Y-E-S-J-Z-F		-X	-Y	-Z	-X	-Y	-Z	-X	-Y	-Z	-X	-Y	-Z
  Loop 1258		F-M-W-R-D-K-U-S-E-L-V-T-F		-Y	-Z	-X	-Y	-Z	-X	-Y	-Z	-X	-Y	-Z	-X
  Loop 1259		F-Z-P-K-D-X-Q-L-E-Y-N-M-F		-Z	-X	-Y	-Z	-X	-Y	-Z	-X	-Y	-Z	-X	-Y

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Chopping each [ 12-elements loop ] into 4 equal-sections :

• Let us now consolidate the 18 [ 12-elements loops ] we have constructed so far ,
  • in the manner as shown in this table below .

  Name of Loop		Route Path for the [ 12-elements Loop ]		Sequence for the Axis-of-Rotation
  ****************	*	*****************************	*	****	****	****	****	****	****	****	****	****	****	****	****
  Loop 1201		A-G-T-V-B-H-R-W-C-J-S-U-A		-X	-Y	-Z	-X	-Y	-Z	-X	-Y	-Z	-X	-Y	-Z
  Loop 1204		B-V-L-Q-C-W-M-N-A-U-K-P-B		-X	-Y	-Z	-X	-Y	-Z	-X	-Y	-Z	-X	-Y	-Z
  Loop 1207		C-Q-X-G-A-N-Y-H-B-P-Z-J-C		-X	-Y	-Z	-X	-Y	-Z	-X	-Y	-Z	-X	-Y	-Z
  Loop 1251		D-X-Q-L-E-Y-N-M-F-Z-P-K-D		-X	-Y	-Z	-X	-Y	-Z	-X	-Y	-Z	-X	-Y	-Z
  Loop 1254		E-L-V-T-F-M-W-R-D-K-U-S-E		-X	-Y	-Z	-X	-Y	-Z	-X	-Y	-Z	-X	-Y	-Z
  Loop 1257		F-T-G-X-D-R-H-Y-E-S-J-Z-F		-X	-Y	-Z	-X	-Y	-Z	-X	-Y	-Z	-X	-Y	-Z
  ****************	*	*****************************	*	****	****	****	****	****	****	****	****	****	****	****	****
  Loop 1202		A-N-Y-H-B-P-Z-J-C-Q-X-G-A		-Y	-Z	-X	-Y	-Z	-X	-Y	-Z	-X	-Y	-Z	-X
  Loop 1205		B-H-R-W-C-J-S-U-A-G-T-V-B		-Y	-Z	-X	-Y	-Z	-X	-Y	-Z	-X	-Y	-Z	-X
  Loop 1208		C-W-M-N-A-U-K-P-B-V-L-Q-C		-Y	-Z	-X	-Y	-Z	-X	-Y	-Z	-X	-Y	-Z	-X
  Loop 1252		D-R-H-Y-E-S-J-Z-F-T-G-X-D		-Y	-Z	-X	-Y	-Z	-X	-Y	-Z	-X	-Y	-Z	-X
  Loop 1255		E-Y-N-M-F-Z-P-K-D-X-Q-L-E		-Y	-Z	-X	-Y	-Z	-X	-Y	-Z	-X	-Y	-Z	-X
  Loop 1258		F-M-W-R-D-K-U-S-E-L-V-T-F		-Y	-Z	-X	-Y	-Z	-X	-Y	-Z	-X	-Y	-Z	-X
  ****************	*	*****************************	*	****	****	****	****	****	****	****	****	****	****	****	****
  Loop 1203		A-U-K-P-B-V-L-S-Q-W-M-N-A		-Z	-X	-Y	-Z	-X	-Y	-Z	-X	-Y	-Z	-X	-Y
  Loop 1206		B-P-Z-J-C-Q-X-G-A-N-Y-H-B		-Z	-X	-Y	-Z	-X	-Y	-Z	-X	-Y	-Z	-X	-Y
  Loop 1209		C-J-S-U-A-G-T-V-B-H-R-W-C		-Z	-X	-Y	-Z	-X	-Y	-Z	-X	-Y	-Z	-X	-Y
  Loop 1253		D-K-U-S-E-L-V-T-F-M-W-R-D		-Z	-X	-Y	-Z	-X	-Y	-Z	-X	-Y	-Z	-X	-Y
  Loop 1256		E-S-J-Z-F-T-G-X-D-R-H-Y-E		-Z	-X	-Y	-Z	-X	-Y	-Z	-X	-Y	-Z	-X	-Y
  Loop 1259		F-Z-P-K-D-X-Q-L-E-Y-N-M-F		-Z	-X	-Y	-Z	-X	-Y	-Z	-X	-Y	-Z	-X	-Y

• The question we have here is this :
  • Is it possible to chop-up each [ 12-elements loop ] into 4 equal-sections ,
    • to come up with a [ Quasi Quad-Symmetric Set ] ?

    And the answer here is YES , we can !

• This is because :
  • each [ 12-elements Axis-of-Rotation Sequence ] associated with each [ 12-elements loop ]
    • is always made-up-of 4 identical-and-repeating [ 3-elements Axis-of-Rotation Sequence ]'s ;

    and as such :

    • we can now pick-off the [ 1st / 4th / 7th / 10th elements ] of the [ 12-elements loops ]
      • to form a [ Quasi Quad-Symmetric Set ] .

  And that is to say :

  • The [ Structural Mapping relations ] between :
    • the [ 1st element ] and the [ 4th element ] ,
    • the [ 4th element ] and the [ 7th element ] ,
    • the [ 7th element ] and the [ 10th element ] , and
    • the [ 10th element ] and the [ 13th / 1st element ] ;

    are always the same because the [ sequence-of-3-rotations ] linking each [ pair of elements ] is identical .

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Listing-Out the [ Quasi Quad-Symmetric Sets ] :

• Let us now list-out the 18 [ Quasi Quad-Symmetric Sets ] thus identified ,
  • as per this table below :

  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 bring back this set of 2 diagrams for the reader's easy reference and for checking purposes :

  

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A Quick Comment here :

• We simple note here that :
  • the [ 2nd Group of Six ] here matches the 6 [ Quasi Quad-Symmetric Sets ] within the [ 24 Base Strutural Patterns ]
    • that we have brought back , at the beginning of this Section , from Section XII .

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Listing-out the Structural Mapping relations :

• It is also appropriate to list-out the [ Structural Mapping relations ] for the above 3 groups of [ Quasi Quad-Symmetric Sets ] ,
  • and we shall use the [ Fu-Hei Scheme of Fixed-Positions ] as the referencing framework for the [ Fixed-Positions ] here :

    i.e. :

    • 

  And the [ Structural Mapping relations ] here are then :

  • For the 1st Group of Six :
    • The 2 colored-pieces in [ FP-1 ] and [ FP-6 ] do not move ,
    • The 2 colored-pieces in [ FP-7 ] , [ FP-4 ] exchange positions ,
    • The 4 colored-pieces in [ FP-8 ] , [ FP-2 ] , [ FP-3 ] , and [ FP-5 ] exchange positions in a rotary fashion .

  • For the 2nd Group of Six :
    • The 2 colored-pieces in [ FP-1 ] and [ FP-7 ] do not move ,
    • The 2 colored-pieces in [ FP-4 ] , [ FP-6 ] exchange positions ,
    • The 4 colored-pieces in [ FP-8 ] , [ FP-5 ] , [ FP-2 ] , and [ FP-3 ] exchange positions in a rotary fashion .

  • For the 3rd Group of Six :
    • The 2 colored-pieces in [ FP-1 ] and [ FP-4 ] do not move ,
    • The 2 colored-pieces in [ FP-6 ] , [ FP-7 ] exchange positions ,
    • The 4 colored-pieces in [ FP-8 ] , [ FP-3 ] , [ FP-5 ] , and [ FP-2 ] exchange positions in a rotary fashion .

  And [ Structural Mapping ] consistency is always maintained throughout each of the three (3) groups .

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go to the next section : Section XXIX - [ Quasi Octo-Symmetric Sets ] within the [ 24 QSM-Configurations ]

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

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

Original dated 2011-3-06