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

by Frank C. Fung ( 1st published in July, 2017. )

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Section XXXI - Concluding Remarks

Concluding Remarks :

• This paper started on 2017-5-8 and finished on 2017-7-23 ,
  • a toatl span of 77 days .

  Given the vast amount of Tables presented throughout the paper ,

  • we do entertain the possibility of a typo-error here-and-there .

• But this should not slow-down the overall presentation of ideas here .

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Some Thoughts on { Base 60 } :

• { Base 60 } is a system used :
  • in Babylonian mathematics ,

    and

  • extensively in the Chinese Astrological Calander .

• In Section XXIX of this paper ,
  • we started-off with :
    • the { 6-Dimension Quasi-Cube } with 64 nodal-points ,

    and end-up with :

    • identifying 60 matching pairs of { 4-lines Structures in the formation of the 4 Cross-Diagonal Lines of the Cube } ,
      • with one structure residing in { Structure A } and the other residing in { Structure B } ;

      so that :

      • the 2 { 4-lines Structures } are orthogonal .

• And this is liken onto :
  • starting-off with the I-CHING's 64 Hexagrams of 6-Lines each , and
    • moving therefrom onto the [ Base 60 ] counting system .

• How did we move from [ 64 ] to [ 60 ] ?

  Some thoughts given here is that :

  • May be we should be looking at the 2 tetrahedra within the Cube ,
    • instead of the Cube itself ?

  And if we do this , we would be focusing on :

  • pairs of vectors intersecting at the angle of [ 109.47 degrees ] ;

    because :

    • 2 such { pairs of vectors } intersecting one-another at the angle of [ 109.47 degrees ]
      • would give us 4 vectors in the formation of a tetrahedron ,

        i.e. :

        • 4 vectors intersecting one-another at the angle of [ 109.47 degrees ] .

• And for a { 16-lines structure } such as { Structure A } or { Structure B } :
  • the Line-Pairing mechanism would give us [ 120 pairs ] to start-off with :
    • i.e. :
      • [ 120 ] = 16 x 15 / 2 x 1

      and

      • [ 120 ] is double [ 60 ] .

• Just a thought !

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go to Appendix A : Further Thoughts on Symmetric Lines in 6-D and 7-D

go to Epilog I : Constructing new { 16 Symmetric-Lines Structures in 6-D }

go to the last Section : Section XXX - Linkage System for { 6-D vs. 7-D }

return to the Symmetry Through-put HomePage

Original dated 2017-7-23 Revised 2017-8-14