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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

### Summary :

• This paper started out as an attempt for a better understanding of :

• the { 28 Symmetric Lines in 7-Dimension } intersecting one-another at the angle of [ 70.53 degrees ] ,

• i.e. :

• arccos [ 1 / 3 ] = [ 70.53 degrees ] .

We came around full-circles to establish that there are :

• { 16 Symmetric Lines in 6-Dimension } intersecting one-another at the angle of [ 70.53 degrees ] ,

• { 10 Symmetric Lines in 5-Dimension } intersecting one-another at the angle of [ 70.53 degrees ] ,

• { 6 Symmetric Lines in 4-Dimension } intersecting one-another at the angle of [ 70.53 degrees ] ,

• { 4 Symmetric Lines in 3-Dimension } intersecting one-another at the angle of [ 70.53 degrees ] ,

with :

• each { set of symmetric lines in a lower-Dimension } being a sub-set of the { set of symmetric lines in the next-higher-Dimension } .

#### Key Finding One :

• We were able to split up the 32 Cross-Diagonal Lines of the { 6-Dimension Quasi-Cube } into 2 identical structures :

• { Structure A } with 16 Lines intersecting one-another at the angle of [ 70.53 degrees ] , and
• { Structure B } with 16 Lines intersecting one-another at the angle of [ 70.53 degrees ] .

Interesting enough , we were able to identify :

• a specifing set of 4 Lines in { Structure A } in the formation of { the 4 Cross-Diagonal Lines of a 3-D Cube } ,

and

• a matching set of 4 Lines in { Structure B } in the formation of { the 4 Cross-Diagonal Lines of a 3-D Cube } ;

so that :

• the 1st [ set of 4 Lines in Structure A ] and the 2nd [ set of 4 Lines in Structure B ] are orthogonal .

60 pairs of such matching { sets of 4-Lines } were found .

#### Key Finding Two :

• We were able to split up the 128 Cross-Diagonal Lines of the { 8-Dimension Quasi-Cube } into 2 identical Structures ,

• and the 2 { 64-Lines Strutctures } are oriented symmetrically .

#### Key Finding Three :

• We were able to verify that the 96-triangles { 4-D Quasi-Octahedron } is also Octo-Symmetric in nature .

#### Key Finding Four :

• For each-and-every { 16 Symmetric-Lines Structure in 6-D } :

• we can always construct another { 16 Symmetric-Lines Structure in 6-D } based there-upon ,

so that :

• the 2 { 16-Lines Structures } will share exactly one Line in common .

#### Table of Content

Part I --- Introduction and Preliminaries Part II --- Finding orthogonal Cross-Diagonal Lines in the { 8-D Quasi-Cube ) Section I Introduction Section II Defining the { 8-Dimension Quasi-Cube } Section III Recalling the 28 Symmetric Lines in a { 7-Dimension Vector Space } Section IV A Listing of the 35 Lines orthogonal the { Line-Of-Concern } Section V Classifying the 128 Cross-Diagonal Lines of the ( 8-D Quasi-Cube } Section VI The Dot Product Table for the 35 Cross-Diagonal Lines Section VII 4 Sets of { 4 Lines Mutually Orthogonal to one-another } in 8-D Section VIII Setting up the 8 Cross-Diagonal Lines of the { 4-Dimension Quasi-Cube } Section IX 24 Squares arising from the { 4-D Quasi-Cube } Section X 12 Symmetric Lines in a { 4-D Vector Space } Section XI Reconciling the { 12 Lines vs. 8 Lines } Difference Section XII Setting up for an Analysis on Tri-Symmetry Section XIII Tri-Symmetry Analysis for { Set A / B / C } Section XIV Tri-Symmetry Analysis for { Set D / E / F } Section XV A Special Issue on Swapping { 4 Orthogonal-Lines Structures } Section XVI The Octo-Symmetric Nature of the { 4-D Quasi-Octahedron } Section XVII Structural Analysis for the { 4-D Quasi-Octahedron } Section XVIII 2 Identical { 8-D Structures } arising from the { 8-D Quasi-Cube } Section XIX Setting up 4 Sets of { Orthogonal Unit-Vector Basis } for our Analysis Section XX Why { Structure A } and { Structure B } are Identical In Shape Section XXI Creating and Segregating 256 Sets of { 8-D Orthogonal Unit-Vector Basis } Section XXII Orientation Symmetry for { Structure A } vs. { Structure B } Section XXIII Using [ 8 x 8 Mapping Matrices ] on { Structure A } and { Structure B } Section XXIV 2 Identical { 6-D Structures } arising from the { 6-D Quasi-Cube } Section XXV Identifying { 10 Symmetric Lines in 5-D } Section XXVI Identifying { 6 Symmetric Lines in 4-D } Section XXVII Identifying { 4 Symmetric Lines in 3-D } Section XXVIII Orthogonal Lines arising from the { 6-D Quasi-Cube } Section XXIX Matching Cubes residing in a { 3-D Sub-Space } and its Kernal Section XXX Linkage System for { 6-D vs. 7-D } Section XXVI Concluding Remarks Appendix A Further Thoughts on Symmetric Lines in 6-D and 7-D Epilog I Constructing new { 16 Symmetric-Lines Structures in 6-D }