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 IX - Defining 2 Types of Line-Symmetry

Summary for the Section :

• In this Section , we shall highlight the difference between 2 types of Line-Symmetry :
  • Absolute Symmetry ,

    and

  • Relative Structural Symmetry .

  And this is done in specific preparation for our discussions in the next 3 Parts of the paper ,

  • so that we propoerly appreciate the finer difference between the 2 types of Line Symmetry .

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Defining the 2 types of Symmetry :

• We shall now define the 2 type of Line-Symmetry as follows :
  • ABSOLUTE SYMMETRY :
    • For any set of [ N lines ] of the same lengths and intersecting one-another at their mid-points :
      • the set is said to be displaying [ Absolute Symmetry ]

        IF :

        • the [ N Lines ] do intersect one-another at the exact-same angle .

  • RELATIVE STRUCTURAL SYMMETRY :
    • For any set of [ N lines ] of the same lengths and intersecting one-another at their mid-points :
      • the set is said to be displaying [ Relative Structural Symmetry ]

        IF :

        • the [ N Lines ] intersect one-another at more than one angle ;

          but :

        • If we take any one of the [ N Lines ] as our [ line-of-concern ] for the time-being ,
          • the relative structural positioning of the remaining [ N-1 Lines ] would always be the same ,
            • regardless of which of the [ N lines ] we may happen to choose as our [ line-of-concern ] .

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A Quick Example via the Cube :

• Let us now bring in the Cube in a { 3-Dimension Space } ,
  • to illustrate the finer difference between the 2 types of Line-Symmetry.

  

  And we see here that :

  • we can join the opposite corners of the Cube with straight lines ,
    • to create 4 Symmetric Lines ,
      • as per the diagram in-the-middle above .

  • we can also join the mid-points of opposite sides of the Cube with straight lines ,
    • to create 6 Symmetric Lines ,
      • as per the diagram in-the-middle below .

  

• It should be obvious that :
  • the [ 4 red-color Symmetric Lines ] above do display [ Absolute Symmetry ] ;
    • intersecting one-another at an angle of [ 70.53 degrees ] ,

      i.e. :

      • [ arccos 1/3 ] = 70.53 degrees .

  whereas :

  • the [ 6 blue-color Symmetric Lines ] above would display [ Relative Structural Symmetry ] ,
    • as explained below .

• First , we take special note here that :
  • for the 15 pairs of lines arising from pairing the [ 6 blue-color Symmetric Lines ] ,
    • 3 pairs of lines will intersect at an angle of [ 90 degrees ] :

  

  and

  • the remaining 12 pairs of lines will intersect at an angle of [ 60 degrees ] :

  

  

  

  Thus , when we select any one of the original [ 6 blue-color Symmetric Lines ] as our [ line-of-concern ] for the time-being ,

  • only one of the 5 other Lines will intersect the [ Line-of-Concern ] at an angle of [ 90 degrees ] ,

    and

  • the remaining 4 Lines will intersect the [ Line-Of-Concern ] at an angle of [ 60 degrees ] .

  And the structural pattern here will be the same regardless of which of the [ 6 Lines ] we may select as our [ line-of-concern ] .

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The Dodecaherdon vs. the Icosahedron :

• Let us now bring in the Dodecahedron and the Icosahedron :

  

• We simply take note here that :
  • for the DODECAHEDRON :
    • the [ 6 Symmetric Lines ] arising from joining the Center-Points of [ opposite pairs of pentagons ]
      • do display [ Absolute Symmetry ] ;

  whereas :

  • for the ICOSAHEDRON :
    • the [ 10 Symmetric Lines ] arising from joining the Center-Points of [ opposite pairs of equilateral triangles ]
      • will display [ Relative Structural Symmetry ] .

• A rather interesting contrast here !

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

go to the last Section : Section VIII - The Hierarchy of the { N+1 Symmetric Lines } System

return to the { N+1 Symmetric Lines } HomePage

Original dated 2016-3-22