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 XVII - Structural Analysis for the { 4-D Quasi-Octahedron }

Summary for the Section :

• In this Section , we shall do the Structural Analysis for the { 4-Dimension Quasi-Octahedron } .

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Recalling a Special Feature for the { 4-D Quasi-Octahedron } :

• Let us recall that :
  • For each of the 96 triangles of the { 4-Dimension Quasi-Octahedron } :
    • one nodal-point of the triangle will come from { Set A } ,
    • one nodal-point of the triangle will come from { Set B } , and
    • one nodal-point of the triangle will come from { Set C } .

    Line I.D.	* * *	Position Vector for 1st end-point of the Line					* * *	Position Vector for 2nd end-point of the Line
    	*	Vector I.D.	Co-ordinates				*	Vector I.D.	Co-ordinates
    *********************	*	***********	***	***	***	***	*	***********	***	***	***	***
    Cross-Diagonal Line 1	*	Vector A1	+1	+1	+1	+1	*	Vector A2	-1	-1	-1	-1
    Cross-Diagonal Line 2	*	Vector A3	+1	+1	-1	-1	*	Vector A4	-1	-1	+1	+1
    Cross-Diagonal Line 3	*	Vector A5	+1	-1	-1	+1	*	Vector A6	-1	+1	+1	-1
    Cross-Diagonal Line 4	*	Vector A7	+1	-1	+1	-1	*	Vector A8	-1	+1	-1	+1
    *********************	*	***********	***	***	***	***	*	***********	***	***	***	***
    Cross-Diagonal Line 5	*	Vector B1	+1	+1	+1	-1	*	Vector B2	-1	-1	-1	+1
    Cross-Diagonal Line 6	*	Vector B3	+1	+1	-1	+1	*	Vector B4	-1	-1	+1	-1
    Cross-Diagonal Line 7	*	Vector B5	+1	-1	+1	+1	*	Vector B6	-1	+1	-1	-1
    Cross-Diagonal Line 8	*	Vector B7	-1	+1	+1	+1	*	Vector B8	+1	-1	-1	-1
    *********************	*	***********	***	***	***	***	*	***********	***	***	***	***
    Axial-Direction Line 1	*	Vector C1	+2	0	0	0	*	Vector C2	-2	0	0	0
    Axial-Direction Line 2	*	Vector C3	0	+2	0	0	*	Vector C4	0	-2	0	0
    Axial-Direction Line 3	*	Vector C5	0	0	+2	0	*	Vector C6	0	0	-2	0
    Axial-Direction Line 4	*	Vector C7	0	0	0	+2	*	Vector C8	0	0	0	-2

  And as such , in any one of the 96 triangles :

  • the side opposite to a { [ Set C ] nodal-point } will be :
    • a line involving a { [ Set A ] nodal-point } and a { [ Set B ] nodal-point } .

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Re-Naming the 4 Axial-Directions :

• For our discussion purposes below ,
  • it is neccessary for us to re-name the 4 Axial-Directions for the { 4-Dimension Vector Space } ;
    • for ease of identification .

  And the new names for the 4 orthogonal Axial-Directions are then :

  • { C1-C2 Axial Direction } havig the same axial-direction as { Axial-Diretcion Line 1 } above ,
  • { C3-C4 Axial Direction } havig the same axial-direction as { Axial-Diretcion Line 2 } above ,
  • { C5-C6 Axial Direction } havig the same axial-direction as { Axial-Diretcion Line 3 } above ,
  • { C7-C8 Axial Direction } havig the same axial-direction as { Axial-Diretcion Line 4 } above .

• Let us also generate the 6 { 2-Dimension Planes } arising therefrom ,
  • i.e. :
    • the { C1-C2 / C3-C4 plane } ,
    • the { C5-C6 / C7-C8 plane } ,
    • the { C1-C2 / C5-C6 plane } ,
    • the { C3-C4 / C7-C8 plane } ,
    • the { C1-C2 / C7-C8 plane } , and
    • the { C3-C4 / C5-C6 plane } ;

      with each { 2-Dimension plane } here involving a pair of Axial-Directions .

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Recalling the 12 Triangles associated with an { Anchor Nodal-Point } :

• Let us now bring-in again this table below from the last Section :
  • identifying the 12 { pairs of Nodal-Points } for each of the 8 { Anchor Nodal-Points in [ Set C ] } ,
    • which gave rise to the 12 equilateral triangles associated with the said { Anchor Nodal-Point in-question } .

  Anchor Nodal-Point in Question		* * *	32 Pairs of Nodal-Points with the Position Vectors for each pair of Nodal-Points intersecting one-another at the angle of [ 60 degrees ]																																			* * *
  Nodal- Point I.D.	Position Vector for Nodal-Point		A 1 * B 7	A 2 * B 8	A 3 * B 6	A 4 * B 5	A 5 * B 2	A 6 * B 1	A 7 * B 4	A 8 * B 3	* * *	A 1 * B 5	A 2 * B 6	A 3 * B 8	A 4 * B 7	A 5 * B 3	A 6 * B 4	A 7 * B 1	A 8 * B 2	* * *	A 1 * B 3	A 2 * B 4	A 3 * B 1	A 4 * B 2	A 5 * B 5	A 6 * B 6	A 7 * B 8	A 8 * B 7	* * *	A 1 * B 1	A 2 * B 2	A 3 * B 3	A 4 * B 4	A 5 * B 8	A 6 * B 7	A 7 * B 5	A 8 * B 6
  ********	**********	*	**	**	**	**	**	**	**	**	*	**	**	**	**	**	**	**	**	*	**	**	**	**	**	**	**	**	*	**	**	**	**	**	**	**	**	*
  Point C1	Vector C1	*									*	X		X		X		X		*	X		X		X		X		*	X		X		X		X		*
  Point C2	Vector C2	*									*		X		X		X		X	*		X		X		X		X	*		X		X		X		X	*
  ********	**********	*	**	**	**	**	**	**	**	**	*	**	**	**	**	**	**	**	**	*	**	**	**	**	**	**	**	**	*	**	**	**	**	**	**	**	**	*
  Point C3	Vector C3	*	X		X			X		X	*									*	X		X			X		X	*	X		X			X		X	*
  Point C4	Vector C4	*		X		X	X		X		*									*		X		X	X		X		*		X		X	X		X		*
  ********	**********	*	**	**	**	**	**	**	**	**	*	**	**	**	**	**	**	**	**	*	**	**	**	**	**	**	**	**	*	**	**	**	**	**	**	**	**	*
  Point C5	Vector C5	*	X			X		X	X		*	X			X		X	X		*									*	X			X		X	X		*
  Point C6	Vector C6	*		X	X		X			X	*		X	X		X			X	*									*		X	X		X			X	*
  ********	**********	*	**	**	**	**	**	**	**	**	*	**	**	**	**	**	**	**	**	*	**	**	**	**	**	**	**	**	*	**	**	**	**	**	**	**	**	*
  Point C7	Vector C7	*	X			X	X			X	*	X			X	X			X	*	X			X	X			X	*									*
  Point C8	Vector C8	*		X	X			X	X		*		X	X			X	X		*		X	X			X	X		*									*

• Let us do a quick study on :
  • the 12 lines arising from each set of { 12 pairs of nodal-points associated with a [ Anchor Nodal-Point ] } ,
    • i.e. :
      • the 12 lines each with a { nodal-point from Series A } and a { nodal-point from Series B } .

  And we see here that :

  • each set of 12 lines is comprised of :
    • 4 lines parallel to one of the { 4 Axial-Directions } ,
    • 4 lines parallel to another of the { 4 Axial-Directions } ,
    • 4 lines parallel to yet another one of the { 4 Axial-Directions } ;

      as shown in this table below .

  Anchor Nodal-Point in Question		* * *	8 Lines parallel to the C1-C2 Axis								* * *	8 Lines parallel to the C3-C4 Axis								* * *	8 Lines parallel to the C5-C6 Axis								* * *	8 Lines parallel to the C7-C8 Axis								* * *
  Nodal- Point I.D.	Position Vector for Nodal-Point		A 1 * B 7	A 2 * B 8	A 3 * B 6	A 4 * B 5	A 5 * B 2	A 6 * B 1	A 7 * B 4	A 8 * B 3		A 1 * B 5	A 2 * B 6	A 3 * B 8	A 4 * B 7	A 5 * B 3	A 6 * B 4	A 7 * B 1	A 8 * B 2		A 1 * B 3	A 2 * B 4	A 3 * B 1	A 4 * B 2	A 5 * B 5	A 6 * B 6	A 7 * B 8	A 8 * B 7		A 1 * B 1	A 2 * B 2	A 3 * B 3	A 4 * B 4	A 5 * B 8	A 6 * B 7	A 7 * B 5	A 8 * B 6
  ********	**********	*	**	**	**	**	**	**	**	**	*	**	**	**	**	**	**	**	**	*	**	**	**	**	**	**	**	**	*	**	**	**	**	**	**	**	**	*
  Point C1	Vector C1	*									*	X		X		X		X		*	X		X		X		X		*	X		X		X		X		*
  Point C2	Vector C2	*									*		X		X		X		X	*		X		X		X		X	*		X		X		X		X	*
  ********	**********	*	**	**	**	**	**	**	**	**	*	**	**	**	**	**	**	**	**	*	**	**	**	**	**	**	**	**	*	**	**	**	**	**	**	**	**	*
  Point C3	Vector C3	*	X		X			X		X	*									*	X		X			X		X	*	X		X			X		X	*
  Point C4	Vector C4	*		X		X	X		X		*									*		X		X	X		X		*		X		X	X		X		*
  ********	**********	*	**	**	**	**	**	**	**	**	*	**	**	**	**	**	**	**	**	*	**	**	**	**	**	**	**	**	*	**	**	**	**	**	**	**	**	*
  Point C5	Vector C5	*	X			X		X	X		*	X			X		X	X		*									*	X			X		X	X		*
  Point C6	Vector C6	*		X	X		X			X	*		X	X		X			X	*									*		X	X		X			X	*
  ********	**********	*	**	**	**	**	**	**	**	**	*	**	**	**	**	**	**	**	**	*	**	**	**	**	**	**	**	**	*	**	**	**	**	**	**	**	**	*
  Point C7	Vector C7	*	X			X	X			X	*	X			X	X			X	*	X			X	X			X	*									*
  Point C8	Vector C8	*		X	X			X	X		*		X	X			X	X		*		X	X			X	X		*									*

• It makes one wonder whether the 12 lines are indeed the 12 sides of a { 3-Dimension Cube } .

  And indeed they are :

  • as per this table below .

  Anchor Nodal- Point in Question	* * *	Characteristics of the Cube associated with the Anchor Nodal-Point in Question																													* * *
  		Triple Set of 4 sides parallel to 3 of 4 Axial- Direction				* * *	8 Corners / Nodal-Points of the Cube																	* * *	4 Cross-Diagonal Lines of the Cube	* * *	Center -point of the Cube
  		C1 - C2	C3 - C4	C5 - C6	C7 - C8		A 1	A 2	A 3	A 4	A 5	A 6	A 7	A 8	* *	B 1	B 2	B 3	B 4	B 5	B 6	B 7	B 8		4 pairs of nodal-points		co-ordinates
  Point C1	*		X	X	X	*	X		X		X		X		*	X		X		X			X	*	A1-B8 / A3-B5 / A5-B1 / A7-B3	*	+1	0	0	0	*
  Point C2	*		X	X	X	*		X		X		X		X	*		X		X		X	X		*	A2-B7 / A4-B6 / A6-B2 / A8-B4	*	-1	0	0	0	*
  ********	*	***	***	***	***	*	**	**	**	**	**	**	**	**	*	**	**	**	**	**	**	**	**	*	***************************	*	**	**	**	**	*
  Point C3	*	X		X	X	*	X		X			X		X	*	X		X			X	X		*	A1-B6 / A3-B7 / A6-B3 / A8-B1	*	0	+1	0	0	*
  Point C4	*	X		X	X	*		X		X	X		X		*		X		X	X			X	*	A2-B5 / A4-B8 / A5-B4 / A7-B2	*	0	-1	0	0	*
  ********	*	***	***	***	***	*	**	**	**	**	**	**	**	**	*	**	**	**	**	**	**	**	**	*	***************************	*	**	**	**	**	*
  Point C5	*	X	X		X	*	X			X		X	X		*	X			X	X		X		*	A1-B4 / A4-B1 / A6-B5 / A7-B7	*	0	0	+1	0	*
  Point C6	*	X	X		X	*		X	X		X			X	*		X	X			X	X		*	A2-B3 / A3-B2 / A5-B6 / A8-B8	*	0	0	-1	0	*
  ********	*	***	***	***	***	*	**	**	**	**	**	**	**	**	*	**	**	**	**	**	**	**	**	*	***************************	*	**	**	**	**	*
  Point C7	*	X	X	X		*	X			X	X			X	*		X	X		X		X		*	A1-B2 / A4-B3 / A5-B7 / A8-B5	*	0	0	0	+1	*
  Point C8	*	X	X	X		*		X	X			X	X		*	X			X		X		X	*	A2-B1 / A3-B4 / A6-B8 / A7-B6	*	0	0	0	-1	*

• The above table is then of great interest to us , because :
  • if we take the [ center-point of the cube ] on the very-right column as our point-of-reference for the entire ROW ,
    • we would be able to construct 8 identical Structures that are bi-symmetric in nature .

  And that is to say :

  • IF :
    • we take the [ center-point of the cube ] as the point-of-reference for each of the 8 rows ,

    THEN :

    • we would have for each of the 8 rows :
      • 2 vectors of size [ 1 ] ,
        • one pointing at the Point-Of-Origin

          and

        • the other pointing in the exact-opposite direction at the { Anchor Nodal-Point in-question } ;

      • 8 vectors of size { square-root of [ 3 ] } ,
        • pointing at the 8 nodal-points of the Cube ;
          • with the 8 vectors here being orthogonal to the { 2 vectors of size [ 1 ] } immediately above .

• But that is not all :
  • If we were to chop the 8 Structures above in half and throw away the outer half with the { Anchor Nodal-Points } ,
    • we would be left with :
      • the Point-Of-Origin at the center ,

        and

      • 16 nodal-points from { Series A } and {Series B } in the formation of the 24-Squares { 4-D Quasi-Cube } .

  WOW !

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Sharing Faces :

• It makes one wonder as to whether the 8 Cubes above do share common faces .

  And in fact thery do ,

  • as per this table below .

  Anchor Nodal- Point	* * *	4 Squares in C1-C2 / C3-C4 plane				* * *	4 Squares in C5-C6 / C7-C8 plane				* * *	4 Squares in C1-C2 / C5-C6 plane				* * *	4 Squares in C3-C4 / C7-C8 plane				* * *	4 Squares in C1-C2 / C7-C8 plane				* * *	4 Squares in C3-C4 / C5-C6 plane				* * *
  		A1 * B5 * A4 * B7	A2 * B6 * A3 * B8	A5 * B2 * A8 * B3	A6 * B1 * A7 * B4		A1 * B1 * A3 * B3	A2 * B2 * A4 * B4	A5 * B5 * A7 * B8	A6 * B6 * A8 * B7		A1 * B3 * A8 * B7	A2 * B4 * A7 * B8	A3 * B1 * A6 * B6	A4 * B2 * A5 * B5		A1 * B1 * A7 * B5	A2 * B2 * A8 * B6	A3 * B3 * A5 * B8	A4 * B4 * A6 * B7		A1 * B1 * A6 * B7	A2 * B2 * A5 * B8	A3 * B3 * A8 * B6	A4 * B4 * A7 * B5		A1 * B3 * A5 * B5	A2 * B4 * A6 * B6	A3 * B1 * A7 * B8	A4 * B2 * A8 * B7
  ********	*	***	***	***	***	*	***	***	***	***	*	***	***	***	***	*	***	***	***	***	*	***	***	***	***	*	***	***	***	***	*
  Point C1	*					*	X		X		*					*	X		X		*					*	X		X		*
  Point C2	*					*		X		X	*					*		X		X	*					*		X		X	*
  ********	*	***	***	***	***	*	***	***	***	***	*	***	***	***	***	*	***	***	***	***	*	***	***	***	***	*	***	***	***	***	*
  Point C3	*					*	X			X	*	X		X		*					*	X		X		*					*
  Point C4	*					*		X	X		*		X		X	*					*		X		X	*					*
  ********	*	***	***	***	***	*	***	***	***	***	*	***	***	***	***	*	***	***	***	***	*	***	***	***	***	*	***	***	***	***	*
  Point C5	*	X			X	*					*					*	X			X	*	X			X	*					*
  Point C6	*		X	X		*					*					*		X	X		*		X	X		*					*
  ********	*	***	***	***	***	*	***	***	***	***	*	***	***	***	***	*	***	***	***	***	*	***	***	***	***	*	***	***	***	***	*
  Point C7	*	X		X		*					*	X			X	*					*					*	X			X	*
  Point C8	*		X		X	*					*		X	X		*					*					*		X	X		*

  We do observe here that :

  • The { Cube associated with Anchor Nodal-Point C1 } :
    • does not share a face with the { Cube associated with Anchor Nodal-Point C2 } ,

      but

    • do share a face , each different , with the { 6 Cubes associated with Anchor Nodal-Point C3 / C4 / C5 / C6 / C7 / C8 } .

  • The { Cube associated with Anchor Nodal-Point C2 } :
    • does not share a face with the { Cube associated with Anchor Nodal-Point C1 } ,

      but

    • do share a face , each different , with the { 6 Cubes associated with Anchor Nodal-Point C3 / C4 / C5 / C6 / C7 / C8 } .

  • The { Cube associated with Anchor Nodal-Point C3 } :
    • does not share a face with the { Cube associated with Anchor Nodal-Point C4 }

      but

    • do share a face , each different , with the { 6 Cubes associated with Anchor Nodal-Point C1 / C2 / C5 / C6 / C7 / C8 } .

  • The { Cube associated with Anchor Nodal-Point C4 } :
    • does not share a face with the { Cube associated with Anchor Nodal-Point C3 }

      but

    • do share a face , each different , with the { 6 Cubes associated with Anchor Nodal-Point C1 / C2 / C5 / C6 / C7 / C8 } .

  • The { Cube associated with Anchor Nodal-Point C5 } :
    • does not share a face with the { Cube associated with Anchor Nodal-Point C6 } ,

      but

    • do share a face , each different , with the { 6 Cubes associated with Anchor Nodal-Point C1 / C2 / C3 / C4 / C7 / C8 } .

  • The { Cube associated with Anchor Nodal-Point C6 } :
    • does not share a face with the { Cube associated with Anchor Nodal-Point C5 } ,

      but

    • do share a face , each different , with the { 6 Cubes associated with Anchor Nodal-Point C1 / C2 / C3 / C4 / C7 / C8 } .

  • The { Cube associated with Anchor Nodal-Point C7 } :
    • does not share a face with the { Cube associated with Anchor Nodal-Point C8 } ,

      but

    • do share a face , each different , with the { 6 Cubes associated with Anchor Nodal-Point C1 / C2 / C3 / C4 / C5 / C6 } .

  • The { Cube associated with Anchor Nodal-Point C8 } :
    • does not share a face with the { Cube associated with Anchor Nodal-Point C7 } ,

      but

    • do share a face , each different , with the { 6 Cubes associated with Anchor Nodal-Point C1 / C2 / C3 / C4 / C5 / C6 } .

  Isn't that wonderful !

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What's Next :

• Let us now move on to Part IV on the { 8-Dimension Quasi-Cube } , next .

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go to the next Section : Section XVIII - 2 Identical Structures arising from the { 8-D Quasi-Cube }

go to the last Section : Section XVI - The Octa-Symmetric Nature of the { 4-D Quasi-Octahedron }

return to the Symmetry Through-put HomePage

Original dated 2017-7-23