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 XII - Line-Symmetry Expansion via the Rectangle for Higher Dimensions

Summary for the Section :

• In this Section , we shall do { Line-Symmetry Expansion via the rectangle } for { Higher-Ordered Dimensions } .

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Doing the Expansions :

• We see here that :
  • we can follow the procedure we have established previously to create
    • more { Red-And-Blue Crosses } in { higher-ordered Dimension Vector Spaces }

      to come up with :

      • matching sets of { Red-Color Symmetric Lines } and { Blue-Color Symmetric Lines }.

  We then have this table below summarizing the results :

  Number of Dimensions for Vector Space	Number of Symmetry Lines	*	Number of Red-And-Blue Crosses formed upon pairing of the Symmetric Lines	Number of Intersecting Pairs for the derived Red Set or Blue Set of Lines	*	Red-Color Series		*	Blue-Color Series
  						Angle of Intersection	Number of pairs		Angle of Intersection	Number of pairs
  ( N )	( N + 1 )	*	( P )	( Q )	*	*************************	********	*	*************************	********
  **********	**********	*	**********	**********	*	*************************	********	*	*************************	********
  2	3	*	3	3	*	arccos [ 1/2 ] = 60.00 degrees	3	*	arccos [ 1/2 ] = 60.00 degrees	3
  **********	**********	*	**********	**********	*	*************************	********	*	*************************	********
  3	4	*	6	15	* *	arccos [ 0/2 ] = 90.00 degrees	3	* *	arccos [ 4/4 ] = 0.00 degrees	3
  						arccos [ 1/2 ] = 60.00 degrees	12		arccos [ 0/4 ] = 90.00 degrees	12
  **********	**********	*	**********	**********	*	*************************	********	*	*************************	********
  4	5	*	10	45	* *	arccos [ 0/2 ] = 90.00 degrees	15	* *	arccos [ 4/6 ] = 48.19 degrees	15
  						arccos [ 1/2 ] = 60.00 degrees	30		arccos [ 1/6 ] = 80.41 degrees	30
  **********	**********	*	**********	**********	*	*************************	********	*	*************************	********
  5	6	*	15	105	* *	arccos [ 0/2 ] = 90.00 degrees	45	* *	arccos [ 4/8 ] = 60.00 degrees	45
  						arccos [ 1/2 ] = 60.00 degrees	60		arccos [ 2/8 ] = 75.52 degrees	60
  **********	**********	*	**********	**********	*	*************************	********	*	*************************	********
  6	7	*	21	210	* *	arccos [ 0/2 ] = 90.00 degrees	105	* *	arccos [ 4/10 ] = 66.42 degrees	105
  						arccos [ 1/2 ] = 60.00 degrees	105		arccos [ 3/10 ] = 72.54 degrees	105
  **********	**********	*	**********	**********	*	*************************	********	*	*************************	********
  7	8	*	28	378	* *	arccos [ 0/2 ] = 90.00 degrees	210	* *	arccos [ 4/12 ] = 70.53 degrees	210
  						arccos [ 1/2 ] = 60.00 degrees	168		arccos [ 4/12 ] = 70.53 degrees	168
  **********	**********	*	**********	**********	*	*************************	********	*	*************************	********
  8	9	*	36	630	* *	arccos [ 0/2 ] = 90.00 degrees	378	* *	arccos [ 4/14 ] = 73.40 degrees	378
  						arccos [ 1/2 ] = 60.00 degrees	252		arccos [ 5/14 ] = 69.08 degrees	252

  with :

  • [ P ] = [ N + 1 ] x [ N ] divided by [ 2 x 1 ]
  • [ Q ] = [ P ] x [ P - 1 ] divided by [ 2 x 1 ]

• Several quick observations here :
  • Observation One :
    • the { Red-Color Lines } will spot only 2 values for the { Angle of Intersection } , namely :
      • [ 90 degrees ] and [ 60 degrees ] .

  • Observation Two :
    • the { Blue-Color Lines } will usually also only spot 2 values for the { Angle of Intersection } , and :
      • the 2 values do vary as the { Number-of-Dimensions [ N ] } increases .

  • Observation Three :
    • for the { Blue-Color Lines } , the 2 values of the { Angle of Intersection }
      • collided and merged at 7-D and LO AND BEHOLD :
        • we have just found [ a set of { 28 Symmetric Lines } in a { 7-Dimension Vector Space } ] !

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The { 28 Symmetric Lines in 7-D } :

• But LO AND BEHOLD :
  • The { 28 Blue-Color Lines } do intersect one-another at the same angle , namely :
    • arccos [ 4/12 ] = 70.53 degrees .

• We therefore have a set of { 28 Symmetric Lines } in a { 7-Dimension Space }
  • displaying [ absolute symmetry ] .

  This is intriguing and we need to confirm :

  • and we shall do the confirmation next .

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go to the next Section : Section XIII - The { 28 Symmetric Lines } in a { 7-Dimension Space }

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

return to the { N+1 Symmetric Lines } HomePage

Original dated 2016-3-22