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

Summary for the Epilog :

• This Epilog summarizes further thoughts on { 16 Symmetric-Lines Structures in 6-D } ,
  • arising on 2017-8-09 and 2017-8-10 .

  And our proposal here is that :

  • For each-and-every existing { 16 Symmetric-Lines Structure in 6-D } ,
    • we can always construct another { 16 Symmetric-Lines Structure in 6-D } based there-upon ,

      so that :

      • the 2 { 16-Lines Structures } will share exactly one line in common .

• And we shall give full details on the construction process below .

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Bringing-in { Structure D } and { Structure H } :

• Let us now bring-in two (2) { 16 Symmetric-Lines Structures in 6-D } for our analysis purposes here ,
  • namely :
    • { Structure D } and { Structure H } respectively ,
      • as shown in the table below ;

      noting here that :

      • { Structure D } and { Structure H } do share one line in common ,

        i.e. :

        • { Line D-1 } and { Line H-1 } are in fact one-and-the-same line .

• And we take note here that :
  • all 64 vectors in the table below are of size { square-root of [ 54 ] } .

  Structure D																	*	Structure H
  Line I.D.	* *	1st Position Vector for the Line							* *	2nd Position Vector for the Line								Line I.D.	* *	1st Position Vector for the Line							* *	2nd Position Vector for the Line
  		Vctr I.D.	Co-ordinates							Vctr I.D.	Co-ordinates									Vctr I.D.	Co-ordinates							Vctr I.D.	Co-ordinates
  *****	*	*****	**	**	**	**	**	**	*	*****	**	**	**	**	**	**	*	*****	*	*****	**	**	**	**	**	**	*	*****	**	**	**	**	**	**
  D-1	*	E-1	+3	+3	+3	+3	+3	+3	*	F-1	-3	-3	-3	-3	-3	-3	*	H-1	*	J-1	+3	+3	+3	+3	+3	+3	*	K-1	-3	-3	-3	-3	-3	-3
  D-2	*	E-2	+3	+3	+3	+3	-3	-3	*	F-2	-3	-3	-3	-3	+3	+3	*	H-2	*	J-2	-1	-1	-1	-1	+5	+5	*	K-2	+1	+1	+1	+1	-5	-5
  D-3	*	E-3	+3	+3	+3	-3	-3	+3	*	F-3	-3	-3	-3	+3	+3	-3	*	H-3	*	J-3	-1	-1	-1	+5	+5	-1	*	K-3	+1	+1	+1	-5	-5	+1
  D-4	*	E-4	+3	+3	-3	-3	+3	+3	*	F-4	-3	-3	+3	+3	-3	-3	*	H-4	*	J-4	-1	-1	+5	+5	-1	-1	*	K-4	+1	+1	-5	-5	+1	+1
  D-5	*	E-5	+3	-3	-3	+3	+3	+3	*	F-5	-3	+3	+3	-3	-3	-3	*	H-5	*	J-5	-1	+5	+5	-1	-1	-1	*	K-5	+1	-5	-5	+1	+1	+1
  D-6	*	E-6	-3	-3	+3	+3	+3	+3	*	F-6	+3	+3	-3	-3	-3	-3	*	H-6	*	J-6	+5	+5	-1	-1	-1	-1	*	K-6	-5	-5	+1	+1	+1	+1
  D-7	*	E-7	+3	+3	+3	-3	+3	-3	*	F-7	-3	-3	-3	+3	-3	+3	*	H-7	*	J-7	-1	-1	-1	+5	-1	+5	*	K-7	+1	+1	+1	-5	+1	-5
  D-8	*	E-8	+3	+3	-1	+3	-1	+1	*	F-8	-3	-3	+3	-3	+3	-3	*	H-8	*	J-8	-1	-1	+5	-1	+5	-1	*	K-8	+1	+1	-5	+1	-5	+1
  D-9	*	E-9	+3	-3	+3	-3	+3	+3	*	F-9	-3	+3	-3	+3	-3	-3	*	H-9	*	J-9	-1	+5	-1	+5	-1	-1	*	K-9	+1	-5	+1	-5	+1	+1
  D-10	*	E-10	-3	+3	-3	+3	+3	+3	*	F-10	+3	-3	+3	-3	-3	-3	*	H-10	*	J-10	+5	-1	+5	-1	-1	-1	*	K-10	-5	+1	-5	+1	+1	+1
  D-11	*	E-11	+3	+3	-3	+3	+3	-3	*	F-11	-3	-3	+3	-3	-3	+3	*	H-11	*	J-11	-1	-1	+5	-1	-1	+5	*	K-11	+1	+1	-5	+1	+1	-5
  D-12	*	E-12	+3	-3	+3	+3	-3	+3	*	F-12	-3	+3	-3	-3	+3	-3	*	H-12	*	J-12	-1	+5	-1	-1	+5	-1	*	K-12	+1	-5	+1	+1	-5	+1
  D-13	*	E-13	-3	+3	+3	-3	+3	+3	*	F-13	+3	-3	-3	+3	-3	-3	*	H-13	*	J-13	+5	-1	-1	+5	-1	-1	*	K-13	-5	+1	+1	-5	+1	+1
  D-14	*	E-14	+3	-3	+3	+3	+3	-3	*	F-14	-3	+3	-3	-3	-3	+3	*	H-14	*	J-14	-1	+5	-1	-1	-1	+5	*	K-14	+1	-5	+1	+1	+1	-5
  D-15	*	E-15	-3	+3	+3	+3	-3	+3	*	F-15	+3	-3	-3	-3	+3	-3	*	H-15	*	J-15	+5	-1	-1	-1	+5	-1	*	K-15	-5	+1	+1	+1	-5	+1
  D-16	*	E-16	-3	+3	+3	+3	+3	-3	*	F-16	+3	-3	-3	-3	-3	+3	*	H-16	*	J-16	+5	-1	-1	-1	-1	+5	*	K-16	-5	+1	+1	+1	+1	-5

• And we also take note here of a special feature for the above table , namely that :
  • by special arrangement :
    • { Vectors E-2 thru E-16 } do intersect { Vector E-1 } at the angle of [ 70.53 degrees ] ,

    • { Vectors F-2 thru F-16 } do intersect { Vector F-1 } at the angle of [ 70.53 degrees ] ,

    • { Vectors J-2 thru J-16 } do intersect { Vector J-1 } at the angle of [ 70.53 degrees ] ,

    • { Vectors K-2 thru K-16 } do intersect { Vector K-1 } at the angle of [ 70.53 degrees ] ;

      i.e. :

      • arccos ( 1 / 3 ) = [ 70.53 degrees ] .

• We then bring-in this set of 4 diagrams below :

  

  • showing :
    • the intersection angles for { Vectors E-2 / F-2 / J-2 / K-2 } respectively .

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The Construction Process for the new { 16 Symmetric-Lines Structure in 6-D } :

• Let us now bring-in { Structure D } as the existing { 16 Symmetric-Lines Structure in 6-D } ,
  • and try to construct another { 16 Symmetric-Lines Structure in 6-D } ,
    • which will result in { Structure H } as shown above .

  FIRST ,

  • we select { Line D-1 } as our { Flip-Flop Axis } for our construction process .

    And we take special note here that :

    • { Vector E-2 thru E-16 } do intersect { Vector E-1 } at the angle of [ 70.53 degrees ] ,

      so that :

      • the projections of { Vector E-2 thru E-16 } onto the { Flip-Flop Axis Direction } are all-the-same ,
        • namely the vector { Vector X-1 } with co-ordinates :
          • { +1 , +1 , +1 , +1 , +1 , +1 } .

    • { Vector F-2 thru F-16 } do intersect { Vector F-1 } at the angle of [ 70.53 degrees ] ,

      so that :

      • the projections of { Vector F-2 thru F-16 } onto the { Flip-Flop Axis Direction } are all-the-same ,
        • namely the vector { Vector X-2 } with co-ordinates :
          • { -1 , -1 , -1 , -1 , -1 , -1 } .

• Let us concentrate for the time-being on { Line D-2 } and { Line H-2 } , and
  • break-up the vectors { Vector E-2 } / { Vector F-2 } / { Vector J-2 } / { Vector K-2 } into 2 components ,
    • with :
      • one component being in the { Flip-Flop Axis Direction }

        and

      • the other component being in the direction orthogonal to the { Flip-Flop Axis Direction } .

  Line I.D.	* *	1st Position Vector for the Line							* *	2nd Position Vector for the Line							*	Line I.D.	* *	1st Position Vector for the Line							* *	2nd Position Vector for the Line
  		Vctr I.D.	Co-ordinates							Vctr I.D.	Co-ordinates									Vctr I.D.	Co-ordinates							Vctr I.D.	Co-ordinates
  *****	*	*****	**	**	**	**	**	**	*	*****	**	**	**	**	**	**	*	*****	*	*****	**	**	**	**	**	**	*	*****	**	**	**	**	**	**
  D-2	*	E-2	+3	+3	+3	+3	-3	-3	*	F-2	-3	-3	-3	-3	+3	+3	*	H-2	*	J-2	-1	-1	-1	-1	+5	+5	*	K-2	+1	+1	+1	+1	-5	-5
  *****	*	*****	**	**	**	**	**	**	*	*****	**	**	**	**	**	**	*	*****	*	*****	**	**	**	**	**	**	*	*****	**	**	**	**	**	**
  	*	X-1	+1	+1	+1	+1	+1	+1	*	X-2	-1	-1	-1	-1	-1	-1	*		*	X-1	+1	+1	+1	+1	+1	+1	*	X-2	-1	-1	-1	-1	-1	-1
  	*	Y-1	+2	+2	+2	+2	-4	-4	*	Y-2	-2	-2	-2	-2	+4	+4	*		*	Y-2	-2	-2	-2	-2	+4	+4	*	Y-1	+2	+2	+2	+2	-4	-4

  And we take note here that :

  • { Vector Y-1 } and { Vector Y-2 } do run in exact-opposite directions ,

    so that :

    • all vectors in the table above do reside in a 2-Dimension Plane , namely :
      • the { X-1 / Y-1 / X-2 / Y-2 } plane .

  

• Let us do the { Flip-Flop Operation } on { Line D-2 } for the construction process :
  • and we bring-in { Vector E-1 } and { Vector F-1 } as shown in the diagram on-the-left below .

  

  Let us do the Flip-Flop Operation as per the diagram in-the-middle above

  • to arrive at the vectors { Vector J-2 } and { Vector K-2 } as shown in the diagram on-the-right above .

  And the { Flip-Flop Operation } here is the same as if :

  • the entire { 2-D plane } is rotated thru an angle of [ 180 degrees ] using the { Flip-Flop Axis } as the axis-of-rotation .

• We can then perform the same type of { Flip-Flop Operation } on the remaining 14 Lines ,
  • i.e. :
    • { Line D-3 thru D-16 } respectively ,

    to arrive at the resultant { Line H-3 thru H-16 } in { Structure H } .

  And these [ 15 new lines ] together with the original { Line D-1 } now renamed as { Line H-1 } ,

  • will now give us the new { 16 Symmetric-Lines Structure in 6-D } in { Structure H } .

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Special Comment on the Intersection Angles :

• It should be obvious that :
  • the intersection angles for { Vector J-2 thru J-16 } with { Vector J-1 } will remain at [ 70.53 degrees ] ,
    • same-as-before before the { Flip-Flop Operations } ,

    with the intersection angles here being correspondent to :

    • the intersection angles for { Vector E-2 thru E-16 } with { Vector E-1 } respectively .

  But how about the other intersection angles ?

• Let us now quickly bring-in { Line D-2 / D-16 } and { Line H-2 / H-16 } for a demonstration :

  For { Line D-2 } and { Line H-2 } , we have :

  Line I.D.	* *	1st Position Vector for the Line							* *	2nd Position Vector for the Line							*	Line I.D.	* *	1st Position Vector for the Line							* *	2nd Position Vector for the Line
  		Vctr I.D.	Co-ordinates							Vctr I.D.	Co-ordinates									Vctr I.D.	Co-ordinates							Vctr I.D.	Co-ordinates
  *****	*	*****	**	**	**	**	**	**	*	*****	**	**	**	**	**	**	*	*****	*	*****	**	**	**	**	**	**	*	*****	**	**	**	**	**	**
  D-2	*	E-2	+3	+3	+3	+3	-3	-3	*	F-2	-3	-3	-3	-3	+3	+3	*	H-2	*	J-2	-1	-1	-1	-1	+5	+5	*	K-2	+1	+1	+1	+1	-5	-5
  *****	*	*****	**	**	**	**	**	**	*	*****	**	**	**	**	**	**	*	*****	*	*****	**	**	**	**	**	**	*	*****	**	**	**	**	**	**
  	*	X-1	+1	+1	+1	+1	+1	+1	*	X-2	-1	-1	-1	-1	-1	-1	*		*	X-1	+1	+1	+1	+1	+1	+1	*	X-2	-1	-1	-1	-1	-1	-1
  	*	Y-1	+2	+2	+2	+2	-4	-4	*	Y-2	-2	-2	-2	-2	+4	+4	*		*	Y-2	-2	-2	-2	-2	+4	+4	*	Y-1	+2	+2	+2	+2	-4	-4

  For { Line D-16 } and { Line H-16 } , we have :

  Line I.D.	* *	1st Position Vector for the Line							* *	2nd Position Vector for the Line							*	Line I.D.	* *	1st Position Vector for the Line							* *	2nd Position Vector for the Line
  		Vctr I.D.	Co-ordinates							Vctr I.D.	Co-ordinates									Vctr I.D.	Co-ordinates							Vctr I.D.	Co-ordinates
  *****	*	*****	**	**	**	**	**	**	*	*****	**	**	**	**	**	**	*	*****	*	*****	**	**	**	**	**	**	*	*****	**	**	**	**	**	**
  D-16	*	E-16	-3	+3	+3	+3	+3	-3	*	F-16	+3	-3	-3	-3	-3	+3	*	H-16	*	J-16	+5	-1	-1	-1	-1	+5	*	K-16	-5	+1	+1	+1	+1	-5
  *****	*	*****	**	**	**	**	**	**	*	*****	**	**	**	**	**	**	*	*****	*	*****	**	**	**	**	**	**	*	*****	**	**	**	**	**	**
  	*	X-1	+1	+1	+1	+1	+1	+1	*	X-2	-1	-1	-1	-1	-1	-1	*		*	X-1	+1	+1	+1	+1	+1	+1	*	X-2	-1	-1	-1	-1	-1	-1
  	*	Q-1	-4	+2	+2	+2	+2	-4	*	Q-2	+4	-2	-2	-2	-2	+4	*		*	Q-2	+4	-2	-2	-2	-2	+4	*	Q-1	-4	+2	+2	+2	+2	-4

• Based on the above , we can then write :
  • { Vector E-2 } = { Vector X-1 } plus { Vector Y-1 }
  • { Vector E-16 } = { Vector X-1 } plus { Vector Q-1 }

  • { Vector J-2 } = { Vector X-1 } minus { Vector Y-1 }
  • { Vector J-16 } = { Vector X-1 } minus { Vector Q-1 }

• And noting here that :
  • { Vector X-1 } and { Vector Y-1 } are orthogonal to one-another so that their { Dot-Product } is zero , and
  • { Vector X-1 } and { Vector Q-1 } are orthogonal to one-another so that their { Dot-Product } is also zero ;

  we can now write :

  • Dot-Product of { Vector E-2 } and { Vector E-16 } is equal to :
    • Dot-Product of { Vector X-1 } and { Vector X-1 }

      plus

    • Dot-Product of { Vector Y-1 } and { Vector Q-1 } .

  • Dot-Product of { Vector J-2 } and { Vector J-16 } is equal to :
    • Dot-Product of { Vector X-1 } and { Vector X-1 }

      plus

    • Dot-Product of { Vector Y-1 } and { Vector Q-1 } .

  The 2 Dot-Products above will therefore evaluate to the exact-same value , always .

• This then demonstrate why the Dot-Products will always remain the-same-as-before
  • before the Flip-Flop Operations to arrive at the 15 new Lines for { Structure H } .

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The Proprosal for constructing new { 16 Symmetric-Lines Structure in 6-D } :

• We have therefore this proposal for constructing new { 16 Symmetric-Lines Structure in 6-D }

  STEP ONE :

  • Bring-in an existing { 16 Symmetric-Lines Structure in 6-D } and
    • select any one of the 16 Lines as the { Flip-Flop Axis } for the construction process .

  STEP TWO :

  • Perform the 15 { Flip-Flop Operations } on the remaining 15 Lines ,
    • using the Line selected in STEP ONE above as the { Flip-Flop Axis } for the operations ;

    to arrive at a new set of 15 Lines .

  The new set of 15 Lines together with the original Line selected in STEP ONE

  • will then give us the new { 16 Symmetric-Line Structure in 6-D } .

• Isn't that interesting !

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go to the last Section : Section XXXI - Concluding Remarks

return to the Symmetry Through-put HomePage

Original dated 2017-8-14