Quasi 24-Symmetry

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 :

go to Table Of Content below

Key Findings :

go to Table Of Content below

go to Table Of Content below

Table Of Content
Part I --- Introduction and Preliminaries
Section IIntroduction
Section IIBasic Combinatorics for [ 8-Colors Circular-Rings ]
Part II --- [ Quasi 24-Symmetry ] in action
Section IIISetting up the [ 24 Base Structural Patterns ]
Section IVDemonstrating [ Quasi 24-Symmetry ] - Phase I
Section VDemonstrating [ Quasi 24-Symmetry ] - Phase II
Section VIDemonstrating [ Quasi 24-Symmetry ] - Phase III
Section VIISummarizing on the [ Qusai 24-Symmetry ] Criteria
Part III --- Splitting the 5,040 Circular-Sequences into 210 x [ sets of 24 elements ]
Section VIIIDemonstrating [ Quasi 24-Symmetry ] - Phase IV
Section IXDemonstrating [ Quasi 24-Symmetry ] - Phase V
Section XSplitting 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 XIQuasi Tri-Symmetry within the [ 24 Base Structural Patterns ]
Section XIIQuasi Quad-Symmetry within the [ 24 Base Structural Patterns ]
Section XIIIQuasi Hexa-Symmetry within the [ 24 Base Structural Patterns ]
Section XIVQuasi Octo-Symmetry within the [ 24 Base Structural Patterns ]
Section XVStructural Considerations in Quasi Symmetry
Part V --- The [ 8-Colors Cube ] capable of Half-on-Half Rotations
Section XVISetting up the [ 8-Colors Cube ]
Section XVIIThe Global Set of 40,320 [ 8-colors configurations ]
Section XVIIIThe Reduced Set of 5,040 [ 8-colors configurations ]
Section XIXConverting the [ 8-colors configurations ] into [ 8-numbers strings ]
Section XXQuasi 24-Symmetry arising from the Half-on-Half Rotations
Section XXIUnderstanding Physical [ Quasi 24-Symmetry ] - Phase I
Section XXIIUnderstanding Physical [ Quasi 24-Symmetry ] - Phase II
Section XXIIIUnderstanding Physical [ Quasi 24-Symmetry ] - Phase III
Section XXIVPhysical [ Tri-Symmetric Sets ] within the [ 24 QSM-Configuartions ]
Section XXVThe 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 XXXRevisiting the Concept of [ QI ] and [ OU ] ( / )
Section XXXIThe [ 6-elements Loops ] for the Single Rolls
Section XXXIIRevisiting Mirror Images
Section XXXIIIThe Tetrathedron and Full-Body Rolls
Part VII --- Phyical Symmetry vs. Logical Symmetry
Section XXXIVSetting up the 210 [ sets-of-24 ] with Physical Symmetries
Section XXXVThe Effects of variations in the [ Read-Off Order ]
Section XXXVIThe Selection Crtieria for re-grouping [ Read-Off Orders ]
Section XXXVIIDemonstrating the different sets of [ 24 Base Structural Patterns ]
Part VIII --- Concluding Remarks
Section XXXVIIIConcluding Remarks

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Original dated 2011-3-08