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

by Frank C. Fung ( 1st published in March, 2016. )

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Section XI - Line-Symmetry Expansion via the Rectangle for 3-D

Summary for the Section :

• In this Section , we shall do the { Line-Symmetry Expansion via the Rectangle } for :
  • the { 4 Symmetric Lines in a [ 3-Dimension Vector Space ] } .

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Line-Symmetry Expansion for 3-D :

• Let us now bring-in again the { 4 Symmetric Lines } in a { 3-Dimension Vector Space } .

  

  We shall now do [ Line-Symmetry Expansion via the rectangle } to construct the 6 { Red-And-Blue Crosses } :

  

  

  Let us now do the consolidation :

  • to put the { 6 Red-Color Lines } onto one single diagram , and

  • to put the { 6 Blue-Color Lines } onto another diagram .

  

• We see here that :
  • the { 6 Red-Color Lines } are symmetric ,

    while :

  • the { 6 Blue-Color Lines } double-up in the { 3 Axial-Directions } .

  This is rather peculiar and requires further thoughts and evaluation .

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Line-Symmetry Expansion fo 2-D :

• As a safety measure , let us now do Line-Symmetry Expansion for the { 2-Dimension Plane } ,
  • before moving onto { 4-Dimension and higher-ordered Vector Spaces } .

  Let us now bring-back the { 3 Symmetric Lines } in a { 2-Dimension Plane } :

  

  We shall now do [ Line-Symmetry Expansion via the rectangle } to construct the 3 { Red-And-Blue Crosses } :

  

• Let us now do the consolidations for the { 3 Red-Color Lines } and the { 3 Blue-Color Lines } :

  

• Rather simple and straight-forward ,
  • so let us move on .

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go to the next Section : Section XII - Line-Symmetry Expansion via the Rectangle for Higher Dimensions

go to the last Section : Section X - Explaining Line-Symmetry Expansion via the Rectangle

return to the { N+1 Symmetric Lines } HomePage

Original dated 2016-3-22