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

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

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Summary and Key Findings :

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• While we started-out with [ 8-colors rings ] , the main thrust came in Part V of the paper :
  • When we looked at the [ 8-colors Cube ] capable [ half-on-half rotations ] in the 6 Axial-Directions .

    

  And when we do [ 180-degrees rotations ] but does not allow for the [ red-color piece ] to move ,

  • we can only reach 24 Configurations as a result .

  

  The [ 24 configurations ] then form a self-contained set and :

  • are ranked on-par with one-another because the [ Structural Mapping relations ] here are well-defined

    and symmetric in the sense that :

    • If we put any of the [ 24 configurations ] into [ Position A ] in the Linkage Diagram on-the-right above ,
      • the other 23 configurations would fall-neatly-into-place in accordance to :

        their relative [ Structural Mapping relations ] to the [ Configuration-in-Position A ] .

  And thus the notion of [ Quasi 24 Symmetry ] .

Key Findings :

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• We then found [ Quasi Tri / Quad / Hexa / Octo sub-symmetries ] within the set-of-24 -configurations :
  • and the [ structural framework ] here for the [ Quasi-sub-symmeties ] are rather interesting .

  

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Table Of Content

	Part I --- Introduction and Preliminaries
Section I	Introduction
Section II	Basic Combinatorics for [ 8-Colors Circular-Rings ]
	Part II --- [ Quasi 24-Symmetry ] in action
Section III	Setting up the [ 24 Base Structural Patterns ]
Section IV	Demonstrating [ Quasi 24-Symmetry ] - Phase I
Section V	Demonstrating [ Quasi 24-Symmetry ] - Phase II
Section VI	Demonstrating [ Quasi 24-Symmetry ] - Phase III
Section VII	Summarizing on the [ Qusai 24-Symmetry ] Criteria
	Part III --- Splitting the 5,040 Circular-Sequences into 210 x [ sets of 24 elements ]
Section VIII	Demonstrating [ Quasi 24-Symmetry ] - Phase IV
Section IX	Demonstrating [ Quasi 24-Symmetry ] - Phase V
Section X	Splitting the 5,040 [ Circular-Ring Patterns ] into 210 x [ sets of 24 ]
	Part IV --- [ Quasi Sub-Symmetries ] found within the [ 24 Base Structural Patterns ]
Section XI	Quasi Tri-Symmetry within the [ 24 Base Structural Patterns ]
Section XII	Quasi Quad-Symmetry within the [ 24 Base Structural Patterns ]
Section XIII	Quasi Hexa-Symmetry within the [ 24 Base Structural Patterns ]
Section XIV	Quasi Octo-Symmetry within the [ 24 Base Structural Patterns ]
Section XV	Structural Considerations in Quasi Symmetry
	Part V --- The [ 8-Colors Cube ] capable of Half-on-Half Rotations
Section XVI	Setting up the [ 8-Colors Cube ]
Section XVII	The Global Set of 40,320 [ 8-colors configurations ]
Section XVIII	The Reduced Set of 5,040 [ 8-colors configurations ]
Section XIX	Converting the [ 8-colors configurations ] into [ 8-numbers strings ]
Section XX	Quasi 24-Symmetry arising from the Half-on-Half Rotations
Section XXI	Understanding Physical [ Quasi 24-Symmetry ] - Phase I
Section XXII	Understanding Physical [ Quasi 24-Symmetry ] - Phase II
Section XXIII	Understanding Physical [ Quasi 24-Symmetry ] - Phase III
Section XXIV	Physical [ Tri-Symmetric Sets ] within the [ 24 QSM-Configuartions ]
Section XXV	The Linkage Diagram for the [ 24 QSM-Configurations ]
Section XXVI	[ Quasi Hexa-Symmetric Sets ] with the [ 24 QSM-Configurations ]
Section XXVII	[ Quasi Tri-Symmetric Sets ] within the [ 24 QSM-Codnfigurations ]
Section XXVIII	[ Quasi Quad-Symmetric Sets ] within the [ 24 QSM-Configurations ]
Section XXIX	[ Quasi Octo-Symmetric Sets ] within the [ 24 QSM-Configurations ]
	Part VI --- Miscellaneous Related Topics
Section XXX	Revisiting the Concept of [ QI ] and [ OU ] ( / )
Section XXXI	The [ 6-elements Loops ] for the Single Rolls
Section XXXII	Revisiting Mirror Images
Section XXXIII	The Tetrathedron and Full-Body Rolls
	Part VII --- Phyical Symmetry vs. Logical Symmetry
Section XXXIV	Setting up the 210 [ sets-of-24 ] with Physical Symmetries
Section XXXV	The Effects of variations in the [ Read-Off Order ]
Section XXXVI	The Selection Crtieria for re-grouping [ Read-Off Orders ]
Section XXXVII	Demonstrating the different sets of [ 24 Base Structural Patterns ]
	Part VIII --- Concluding Remarks
Section XXXVIII	Concluding Remarks

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Original dated 2011-3-08