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 X - Explaining Line-Symmetry Expansion via the Rectangle

Summary for the Section :

• In this Section , we shall explain the basis and the procedure for :
  • Line-Symmetry Expansion via the Rectangle .

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Creating the Rectangle :

• Whenever 2 lines of equal lengths intersect at their mid-points ,
  • the 4 end-points of the 2 lines will form the 4 nodal-points for a rectangle .

  And in the special case of the 2 Lines intersecting at an angle of 90 degrees ,

  • the 4 end-points of the 2 lines will form the 4 nodal-points for a square .

  

• Let us now focus our attention on the diagram on-the-right , above .

  And we see here that :

  • we can join the mid-points of each pair of [ opposite sides ] of the rectangle ,
    • to come up with two (2) new lines passing thru the original point-of-intersection .

  And we take note immediately that the 2 new lines do intersect at an angle of [ 90 degrees ] .

• And , in preparation for our discussion to follow in the next 2 Section :
  • we shall always mark-off the longer of the 2 new lines in [ red-color ] and name it the [ red-color line ] ;

    and

  • we shall also mark-off the shorter of the 2 new lines in [ blue-color ] and name it the [ blue-color line ] ;

  Additionally , we shall refer to the [ cross ] formed by the 2 new lines as the [ Red-And-Blue Cross ] .

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The Expansion Procedure :

• Let us recall that :
  • for the { N+1 Symmetric Lines } in a { N-Dimension Space } ,
    • the { N+1 lines } always display [ Absolute Symmetry ] ,

      i.e. :

      • they always intersect one-another at the exact-same angle .

  As such ,

  • we can now select and re-group the { N+1 Symmetric Lines } into pairs ,

    so that :

    • the Total Number of Pairs arising therefrom is given by :
      • [ N + 1 ] * [ N ] divided by [ 2 x 1 ] .

    Consequently , we can formulate the same number of { Red-And-Blue Crosses } from these [ pairs of lines ] ,

    • and each of the { Red-And-Blue Crosses } here may be considered as being [ on par ] with any other { Red-And-Blue Crosses } .

  Thus , we now have :

  • a new series of [ red-color lines ] and a new series of [ blue-color lines ] ,
    • which may , and may not , display some sort of symmetry .

• Let us now take a closer look at this outcome of [ Red-Color Lines ] and [ Blue-Color Lines ] , next .

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go to the next Section : Section XI - Line-Symmetry Expansion via the Rectangle for 3-D

go to the last Section : Section IX - Defining 2 Types of Line-Symmetry

return to the { N+1 Symmetric Lines } HomePage

Original dated 2016-3-22