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On Symmetry Through-put linking 3-4-5-6-7 Dimension's ,
Octo-Symmetry for the { 4-Dimension Quasi-Octahedron } ,
and 2 Identical { 8-D Structures } oriented Symmetrically

by Frank C. Fung ( 1st published in July, 2017. )

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SUMMARY and KEY FINDINGS :

Summary :

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Key Finding One :

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Key Finding Two :

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Key Finding Three :

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Key Finding Four :

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Table of Content

Table Of Content
Part I --- Introduction and Preliminaries
Section IIntroduction
Section IIDefining the { 8-Dimension Quasi-Cube }
Part II --- Finding orthogonal Cross-Diagonal Lines in the { 8-D Quasi-Cube )
Section IIIRecalling the 28 Symmetric Lines in a { 7-Dimension Vector Space }
Section IVA Listing of the 35 Lines orthogonal the { Line-Of-Concern }
Section VClassifying the 128 Cross-Diagonal Lines of the ( 8-D Quasi-Cube }
Section VIThe Dot Product Table for the 35 Cross-Diagonal Lines
Section VII4 Sets of { 4 Lines Mutually Orthogonal to one-another } in 8-D
Part III --- Symmetric Sets arising from the { 4-D Quasi-Cube }
Section VIIISetting up the 8 Cross-Diagonal Lines of the { 4-Dimension Quasi-Cube }
Section IX24 Squares arising from the { 4-D Quasi-Cube }
Section X12 Symmetric Lines in a { 4-D Vector Space }
Section XIReconciling the { 12 Lines vs. 8 Lines } Difference
Section XIISetting up for an Analysis on Tri-Symmetry
Section XIIITri-Symmetry Analysis for { Set A / B / C }
Section XIVTri-Symmetry Analysis for { Set D / E / F }
Section XVA Special Issue on Swapping { 4 Orthogonal-Lines Structures }
Section XVIThe Octo-Symmetric Nature of the { 4-D Quasi-Octahedron }
Section XVIIStructural Analysis for the { 4-D Quasi-Octahedron }
Part IV --- 2 Identical { 8-D Structures } arising from the { 8-D Quasi-Cube }
Section XVIII2 Identical { 8-D Structures } arising from the { 8-D Quasi-Cube }
Section XIXSetting up 4 Sets of { Orthogonal Unit-Vector Basis } for our Analysis
Section XXWhy { Structure A } and { Structure B } are Identical In Shape
Section XXICreating and Segregating 256 Sets of { 8-D Orthogonal Unit-Vector Basis }
Section XXIIOrientation Symmetry for { Structure A } vs. { Structure B }
Section XXIIIUsing [ 8 x 8 Mapping Matrices ] on { Structure A } and { Structure B }
Part V --- Symmnetric Lines in 3-4-5-6-7 Dimensions's
Section XXIV2 Identical { 6-D Structures } arising from the { 6-D Quasi-Cube }
Section XXVIdentifying { 10 Symmetric Lines in 5-D }
Section XXVIIdentifying { 6 Symmetric Lines in 4-D }
Section XXVIIIdentifying { 4 Symmetric Lines in 3-D }
Section XXVIIIOrthogonal Lines arising from the { 6-D Quasi-Cube }
Section XXIXMatching Cubes residing in a { 3-D Sub-Space } and its Kernal
Section XXXLinkage System for { 6-D vs. 7-D }
Part VI --- Concluding Remarks
Section XXVIConcluding Remarks
APPENDIX
Appendix AFurther Thoughts on Symmetric Lines in 6-D and 7-D
EPILOG
Epilog IConstructing new { 16 Symmetric-Lines Structures in 6-D }

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Original dated 2017-7-23 Revised 2017-8-14