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# On { N+1 Symmetric Lines } in a { N-Dimension Vector Space }

### Summary and Key Findings :

• Originally , the main purpose of this paper was to confirm the existence of :

• { N+1 Symmetric Lines } in a { N-Dimension Vector Space } ,

• for all [ N's ] greater than or equal to [ 2 ] .

But we happen to hit upon :

• a set of { 28 Symmetric Lines } in a { 7-Dimension Vector Space } .

• while doing Symmetric Lines Expansions .

#### Key Finding One :

• We were able to identify a specific linkage relation ,

• relating :

• the { N+1 Symmetric Lines } in a { N-Dimension Vector Space }

and

• the { N Symmetric Lines } in a { [ N-1 ]-Dimension Vector Space } .

And this Linkage relation is valid throughout , so that :

• the continuity in the { N-1 Symmetric Lines Linkage System } is unbroken ,

• for all values of [ N ] from [ 2 ] to [ infinity ] .

#### Key Finding Two :

• We happen to stumble upon :

• a set of [ 28 lines ] in a { 7-Dimension Space } that intersect one-another at the exact-same angle .

The set of 28 Lines therefore do display Absolute Symmetry .

#### Table of Content

Part I --- Introduction Part II --- Setting up the Symmetric Lines Section I Introduction Section II { 3 Symmetric Lines } in a { 2-Dimension Plane } Section III { 4 Symmetric Lines } in a { 3-Dimension Space } Section IV { 5 Symmetric Lines } in a { 4-Dimension Space } Section V { 6 Symmetric Lines } in a { 5-Dimension Space } Section VI Symmetric Lines in { Higher-Dimensioned Spaces } Section VII { Symmetric Lines in 4-D } vs. { Symmetric Lines in 3-D } Section VIII The Hiarchy of the { N+1 Symmetric Lines } System Section IX Defining 2 Types of Line-Symmetry Section X Explaining Line-Symmetry Expansion via the Rectangle Section XI Line-Symmetry Expansion via the Rectangle for 3-D Section XII Line-Symmetry Expansion via the Rectangle for Higher Dimensions Section XIII The { 28 Symmetric Lines } in a { 7-Dimension Space } Section XIV Structural Flexibility for the { 28 Symmetric Lines in 7-D } Section XV The Quasi-Cube in a { 4-Dimension Vector Space } Section XVI Line-Symmetry via the Quasi-Cube for Higher Dimensions Section XVII Summarizing on Line-Symmetry Section XVIII Concluding Remarks