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 XVI - Symmetric Lines via the Quasi-Cube for Higher Dimensions

Summary for the Section :

• In this Section , we shall do the Line-Symmetry Expansion via the Quasi-Cube for Higher-Ordered Dimensions .

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The General Procedure :

• In general , we can always create a Quasi-Cube in a { N-Dimension Vector Space } via :
  • a listing-out of the co-ordinates of the [ corners / nodal-points ] of the Quasi-Cube ,
    • allowing for { only [ +1 ] or [ -1 ] values } for each of the [ N elements ]
      • in the { Co-ordinate String } for each [ corner / nodal-point ] .

  We do take note here that :

  • for a Quasi-Cube in a { N-Dimension Vector Space } :
    • the [ Number of Corners / Nodal-Points ] is equal to :
      • [ 2 ] raised to the power [ N ] .

    • the [ Number of Sides ] is equal to :
      • [ number of Nodal-Points ] x [ number of Axial-Directions ] divided by [ 2 ] .

      i.e. :

      • each [ nodal-point ] is connected to the N [ nearest nodal-points ] via { N lines in the [ N Axial-Directions ] } .

    • the [ Number of Cross-Diagonal Symmetric Lines ] is equal to :
      • one-half of the [ Number of Nodal-Points ] .

• We then have this table below for the statistics up to [ N = 7 ] :

  Number of Dimensions for Vector Space	*	The Quasi-Cube in the { N-Dimension Space }
  		Number of Corners / Nodes	Number of sides	Number of Symmetric Cross- Diagonal Lines	*	Number of Pairs of Intersecting Lines	*	Distribution of the Intersecting Pairs
  								Angle of Intersection	Number of pairs
  2	*	4	4	2	*	2	*	arccos [ 0/2 ] = 90.00 degrees	2
  **********	*	**********	**********	**********	*	**********	*	*************************	**********
  3	*	8	12	4	*	6	*	arccos [ 1/3 ] = 70.53 degrees	6
  **********	*	**********	**********	**********	*	**********	*	*************************	**********
  4	*	16	32	8	*	28	*	arccos [ 0/4 ] = 90.00 degrees	12
  								arccos [ 2/4 ] = 60.00 degrees	16
  **********	*	**********	**********	**********	*	**********	*	*************************	**********
  5	*	32	80	16	*	120	*	arccos [ 1/5 ] = 78.46 degrees	80
  								arccos [ 3/5 ] = 53.13 degrees	40
  **********	*	**********	**********	**********	*	**********	*	*************************	**********
  6	*	64	192	32	*	496	*	arccos [ 0/6 ] = 90.00 degrees	160
  								arccos [ 2/6 ] = 70.53 degrees	240
  								arccos [ 4/6 ] = 48.19 degrees	96
  **********	*	**********	**********	**********	*	**********	*	*************************	**********
  7	*	128	448	64	*	2016	*	arccos [ 1/7 ] = 81.79 degrees	1120
  								arccos [ 3/7 ] = 64.62 degrees	672
  								arccos [ 5/7 ] = 44.42 degrees	224

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Several Observations Here :

• Obesravation One :
  • As the Number of Dimension increases :
    • the number of cross-diagonal lines also increases ,

    • the number of distinct values for the { Angles of Intersection } also increases ;
      • and this would mean more and more layers for the Structural Pattern of the Symmetric Lines as [ N ] increases .

• Observation Two :
  • The { [ 90 degrees ] angle of intersection } is only available when [ N ] is 'even' ;
    • We must therefore entertain the notion to treat [ even N's ] and [ odd N's ] seperately .

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go to the next Section : Section XVII - Summarizing on Line-Symmetry

go to the last Section : Section XV - The Quasi-Cube in a { 4-Dimension Vector Space }

return to the { N+1 Symmetric Lines } HomePage

Original dated 2016-3-22