An Approach to the Four Body Problem

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

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Epilog III - On { Mirror / Rotated / Counter Images }

Brief Introduction :

• In this 3rd Epilog , we shall look at the different types of { symmetry } arising from the { Tetrahedron } .

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{ Mirror / Rotated / Counter Images } :

• We then offer these "rather-loose" definitions for the { Mirror Image } , the { Rotated Image } and the { Counter Image } :

  

  • For the { Mirror Image } :
    • [ Left ] becomes [ Right ] , and [ Right ] becomes [ Left ] ;
    • [ Up ] remains [ Up ] , and [ Down ] remains [ Down ] ;
    • [ Front ] remains [ Front ] , and [ Back ] remains [ Back ] .

  • For the { Rotated Image } :
    • [ Left ] remains [ Left ] , and [ Right ] remains [ Right ] ;
    • [ Up ] becomes [ Down ] , and [ Down ] becomes [ Up ] ;
    • [ Front ] becomes [ Back ] , and [ Back ] becomes [ Front ] .

  • For the { Counter Image } :
    • Each pair of corresponding { centroid-based Position Vectors } , in the 2 Tetrahedra , are always the [ Exact Opposite ] of one-another .

• Let us now merge the 4 diagrams above into this { 2 x 2 Block Diagram } :

  
  

  The { mode of operation } here is as follows :

  • moving horizontally in this diagram then gets us the { Mirror Image } , i.e. :
    • [ A ] and [ B ] are the { Mirror Image } of one-another ,
    • [ C ] and [ D ] are the { Mirror Image } of one-another .

  • moving vertically in this diagram then gets us the { Rotated Image } , i.e. :
    • [ A ] and [ C ] are the { Rotated Image } of one-another ,
    • [ B ] and [ D ] are the { Rotated Image } of one-another .

  And , in order to arrive at the { Counter Image } , we either :

  • move horizontally and then vertically , or
  • move vertically and then horizontally .

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Other { 2 x 2 Block Diagrams } :

• We then bring back this { 2 x 2 Block Diagram } from [ Section 5 ] of { An Approach to Matrix and Linear Algebra } , from NOV / 2004 :

  SELF	T-O-S
  T-O-I	INVERSE

  where :

  • [ Matrix RHO ] and [ Matrix CHI ] are { diagonal martices } and are the { Inverse } of one-another ;
  • [ Matrix ALPHA ] is the { Rotation Martix } and [ Matrix ALPHA-T ] is its { Transpose } ;
  • [ Matrix BETA ] is the { Local Orientation Matrix } and [ Matrix Beta-T ] is its { Transpose } .

  And the { mode of operation } here is as follows :

  • when we move horizontally here , we do a [ Transpose Operation ] on the { matrix element } ;
  • when we move vertically here , we do a [ Heart-Transplant Operation ] , exchanging [ Matrix RHO ] for [ Matrix CHI ] or visa-versa .

    ( note : the { transpose } of a { diagonal matrix } is the { diagonal matrix } itself , )

  And , in order to arrive at the { Inverse } , we either :

  • move horizontally and then vertically , or
  • move vertically and then horizontally .

• We also bring back these 3 { 2 x 2 Block Diagrams } from [ Section 7 ] of { An Approach to the "I-Ching" } , from DEC / 2004 :

  

  And the { mode of operation } here is as follows :

  • when we move horizontally here , we get the [ Opposite ] of the { Hexagram } ;
  • when we move vertically here , we get the [ Reverse ] of the { Hexagram } .

  And , in order to arrive at the { Opposit-Of-the-Reverse [ O-O-R ] } , we either :

  • move horizontally and then vertically , or
  • move vertically and then horizontally .

• Very interesting similarities here , indeed .

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A passing Mention :

• We also mention , in passing , a special feature here :
  • In the { Mirror Image } ,

    we exchanged on { 1 axis } and maintain the [ status quo ] on the remaining { 2 axis } .

  • In the { Rotated Image } ,

    we exchanged on { 2 axis } and maintain the [ status quo ] on the remaining { 1 axis } .

  Splitting the { 3 axis } in a [ 1-2 split ] or a [ 2-1 split ] is also a feature to be further explored in the next paper ,

  • { An Approach to Combinatorics and its Optimization } ,

    scheduled for publication on-the-web at a later date , possibly in March, 2008 .

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go to the last section : Epilog II - On the { [ Zero ] Angular Momentum Vector }

return to The Four Body Problem HomePage

original dated 2008-1-07