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On Symmetric Structures within Symmetric Solid Objects

by Frank C. Fung ( 1st published in March, 2010. )

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Summary and Key Findings :

go to Table Of Content below

Key Finding One :

Key Finding Two :

Key Finding Three :

Key Finding Four :

Table of Content

Table Of Content
Part I --- Introduction and Preliminaries
Section IIntroduction
Section IIA Special Note on the Notations
Part II --- Symmetric Solid Object Basics
Section IIIBasic Properties of the Tetrathedron
Section IVBasic Properties of the Cube
Section VBasic Properties of the Octohedron
Section VIBasic Properties of the Dodecahedron
Section VIIBasic Properties of the Icosahedron
Part III --- The Tetrahedron, tbe Cube , and the Octahedron
Section VIIIThe Bi-Symmetric Nature of the Tetrahedron
Section IXThe Bi-Symmetric Nature of the Cube
Section XThe Tri-Symmetric Nature of the Cube
Section XIThe Bi-Symmetric Nature of the Octahedron
Section XIIThe Quad-Symmetric Nature of the Octahedron
Section XIIIBrief Summary on the Tetrahedron / Cube / Octahedron
Part IV --- The Dodecahedron
Section XIVBi-Polar Structures within the Dodecahedron
Section XVThe Bi-Symmetric Nature of the Dodecahedron
Section XVIThe Tri-Symmetric Nature of the Dodecahedron
Section XVIIThe Quad-Symmetric Nature of the Dodecahedron
Section XVIIIThe Hexa-Symmetric Nature of the Dodecahedron
Part V --- The Icosahedron
Section XIXBi-Polar Structures within the Icosahedron
Section XXThe Bi-Symmetric Nature of the Icosahedron
Section XXIThe Quad-Symmetric Nature of the Icosahedron
Section XXIIThe Penta-Symmetric Nature of the Icosahedron
Section XXIIIThe Deci-Symmetric Nature of the Icosahedron
Section XXIVThe [ 2-6-6-6-Symmetry ] co-existing within the Icosahedron
Section XXV[ Quad-Type Structures ] within the Icosahedron
Part VI --- Bringing-in the Spherical Solid Objects
Section XXVIDefining Great-Circles , Great-Cricle-Planes and Great-Circle-Arcs
Section XXVIIThe Tetrahedral Sphere ( 4-Split )
Section XXVIIIThe Quasi-Cubic Sphere ( 6-Split )
Section XXIXThe Octahedral Sphere ( 8-Split )
Section XXXThe Dodecahedral Sphere ( 12-Split )
Section XXXIThe Icosahedral Sphere ( 20-Split )
Section XXXIIRatio-Of-Areas and the Egyptian Fractions
Part VII --- Duality Considerations
Section XXXIIIDuality - Tetrahedron vs. Tetrahedron
Section XXXIVDuality Linkage via the [ TFS ( 24-Split) Sphere ]
Section XXXVDuality - Cube vs. Octahedron
Section XXXVIDuality Linkage via the [ FES ( 48-Split ) Sphere ]
Section XXXVIIDuality - Dodecahedron vs. Icosahedron
Section XXXVIIIDuality Linkage via the [ OHATS ( 120-Split ) Sphere ]
Part VIII --- Structure of the [ Quad-Type Structures ] within the Icosahedron
Section XXXIXIdentifying 2 [ Quad-Type Structures ] for Our Analysis
Section XXXXA First Look at the 2 [ Quad-Type Structures ]
Section XXXXIRotational Orientation of the [ Curved Triangular Pieces ]
Part IX --- 5-Colors Combinatorics for the [ Quad-Type Structure Coloring Schemes ]
Section XXXXIIA Special Feature of the [ Quad-Type Coloring Schemes ]
Section XXXXIII5-Colors Combinatorics for the [ ALPHA Series ]
Section XXXXIV5-Colors Combinatorics for the [ OMEGA Series ]
Part X --- Concluding Remarks
Section XXXXVConcluding Remarks

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Original dated 2010-3-31 / Updated 2010-5-08