Epilog II - Multiplex Symmetry

Epilog II - On Multiplex Symmetry

by Frank C. Fung - 1st published in November, 2007.

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Introduction & Summary :

In this Epilog , we shall be concentrating on the { Cubic Close-Packed } pattern in the { lattice packing } of { crystalline solids } ,
- and the { Cubic Close-Packed } pattern does exhibit { multiplex symmetry } properties .
The 2 diagrams on-the-left , below , then show this { Cubic Close-Packed } pattern ,
- with the { Base Atom } , as marked in [ red-color ] , being surrounded by 12 { adjacent atoms } :
The 2 diagrams on-the-right , above , then show the [ SOW-14 ] arising from these 12 [ atoms ] .

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Identifying the Tetrahedra :

First , we identify the 8 { equilateral tetrahedra } ,
- formed by the { Base Atom } and each set of { 3 atoms } closest to each of the 8 corners of the { Cube } :
We notice here that there are only 2 different [ orientations ] for the 8 { tetrahedra } :
- the [ Red-Series Orientation ] Tetrahedra ,
  - consisting of the [ red-color ] , [ yellow-color ] , [ orange-color ] & [ magenta-color ] tetrahedra ;
- the [ Blue-Series Orientation ] Tetrahedra ,
  - consisting of the [ blue-color ] , [ purple-color ] , [ cyan-color ] & [ green-color ] tetrahedra .
Let us now do the following , for each of the 8 { Tetrahedra } :
- First , we identify the { centroid } for the { Tetrahedron } ,
- Next , we draw-in 4 { axial directions } running from the { centroid } towards the 4 { nodal points } of the { Tetrahedron } .
We notice here that the { 4 axial directions } thus identified for the { [ Blue-Series Orientation ] Tetrahedra } run directly opposite to the { 4 axial directions } thus identified for the { [ Red-Series Orientation ] Tetrahedra } .
The { [ Red-Series Orientation ] Tetrahedra } and the { [ Blue-Series Orientation ] Tetrahedra } then exhibit { opposite parity } in a { Bi-Polar Symmetry } situation .
We also note in passing that :
- the { 4 centroids } for the { 4 [ Blue-Series Orientation ] Tetrahedra } do form another { equilateral tetrahedron } , of the exact-same size ,
  - and { equilateral tetrahedra } are always , in themselves and by themselves , [ quad-tetrahedra symmetric ] ;
- And the same also goes for the { 4 [ Red-Series Orientation ] Tetrahedra } .

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Identify the Octahedra :

Let us now identify the { equilateral octahedra } , but first :
- let us set-up the usual { 3 orthogonal axis } , and
- multi-stack the { cubes } in these { 3 axial direction } ;
  
  as per the diagram on-the-right , below .

The next 3 diagrams above then show the positions of the { equilateral octahedra } .
Thus , the { Cubic Close-Packed } pattern does exhibit { hexa-symmetry } , and that is to say :
- If we take any { atom } here as the { Base Atom } , then , moving in any of the 6 directions ,
  - i.e. the [ 1, 0, 0 ] , [ 0, 1, 0 ] , [ 0, 0, 1 ] , [ -1, 0, 0 ] , [ 0, -1 , 0 ] and [ 0, 0, -1 ] directions , respectively ,
  will always yield us the same results .
And we also note here in passing that { octahedra } are always { Octo-Symmetric } .

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The Finding :

In the { Cubic Close-Packed } pattern construction geometry , { multiplex symmetry } is always displayed , inclusive of :
- { Bi-Polar Symmetry } ,
- { Quad-Tetrahedra Symmetry } ,
- { Hexa-Symmetry } , and
- { Octo-Symmetry } .

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An Intriguing Proposal :

Let us now use a general { rectangular box } , instead of the { cube } , for our construction :
We now notice the involvement of :
- { Congruent-Triangle Tetrahedra } ( CT-Tetrahedra ) , and
- { Congurent-Triangle Octahedra } ( CT-Octahedra ) ;
  
  in this new proposal .
It would seem then that { Bi-Polar Symmetry } , { Quad-Tetrahedra Symmetry } and { Octo-Symmetry } are maintained here ,
- but { Hexa-Symmetry } would now degenerate into a { triple set } of { Bi-Polar Symmetry } .
Rather Interesting !