An Approach to the Four Body Problem

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

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Section IV - Constructing the { General Tetrahedron }

Constructing the { General Tetrahedron } :

• We note , first-of-all , that the { Box Method } for the construction of the { EQ / IT / CT Tetrahedra } involves only { retangular surfaces } ,
  • so that all angles are always { Right-Angles } , i.e. { angles of 90 degrees } .

  We shall now use { angles other-than 90 degrees } in our construction of the { General Tetrahedron } .

• The construction method for the { General Tetrahedron } is as follows :
  • First , we select any 3 arbitrary lengths , [ L1 ] , [ L2 ] and [ L3 ] , respectively .

  • Next , we select any 3 arbitrary angles , [ Theta ] , [ Psi ] and [ Gamma ] respectively , all greater than zero but less than [ 180 degrees ] ,
    • so that the sum of the 3 angles is always less than or equal to [ 360 degrees ] .

      ( see the next Section on { Flattened Tetrahedron } for a fuller explanation on this restriction ) .

  • We then construct 3 pairs of { identical parallelograms } , using :
    • [ L1 ] , [ L2 ] and [ Gamma ] for the 1st set of 2 { identical parallelograms } ;
    • [ L2 ] , [ L3 ] and [ Theta ] for the 2nd set of 2 { identical parallelograms } ;
    • [ L3 ] , [ L1 ] and [ Psi ] for the 3rd set of 2 { identical parallelograms } .

  • We then combine the 6 { parallelogram surfaces } to form a { parallelogram squashed-box } , as shown below on-the-left .

  • For the 8 corners of the { parallelogram squashed-box } , we then mark-off one set of { alternate corners } [ black-color ] and the other set [ white-color ] .

  • We then join the 4 [ black-color ] corners to form the { Tetrahedron } .

  

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The { Mid-Points } on the { 6 sides } :

• Let us now mark-off the { mid-point } on each of the { 6 sides } of the { Tetrahedron } ,
  • and join the { mid-points } on each pair of { non-touching sides } , as shown below .

  

• Combining the 3 diagrams then yield this diagram on-the-left , below .

  

  And we note here that the intersection of the { 3 Lines } is always the { centroid } of the { Tetrahedron } .

• We also note 2 very special properties for each of these { 3 Lines } :
  • each { Line } is always parallel to a corresponding set of { 4 parallel sides } on the { parallelogram squashed-box } ,
  • the length of the { Line } is always equal to the length of the said-corresponding { 4 parallel sides } .

    as per the diagram on-the-right , above .

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An Alternate Method of Construction :

• We then propose this alternate method of construction for the { General Tetrahedron } :
  • We start-off with { 3 sticks } of arbitrary lengths ,

  • We simple orient the { 3 sticks } in { 3-D space } so that the { 3 sticks } bisect one-another .
    • We can then label the 3 { angles of intersection } for { each pair of sticks } as the angles [ Theta ] , [ Psi ] and [ Gamma ] .

  • We can then follow the instructions we have first listed above for the construction of the { Tetrahedron } ,
    • using the [ 3 lengths ] and the [ 3 angles ] we have developed here .

• There is a key point here :
  • We have now separated the original { 6 DOF's } into :
    • [ 3 length variables ] , and
    • [ 3 DOF's associated with the orientation of the 3 sticks ] .

  Could be rather convenient .

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Constructing 2 Tetrahedra in the same Box :

• But we also note here that actually 2 { tetrahedra } may be constructed using the same { parallelogram squashed-box } :
  • one arising from the [ Black-Color Corners ] and the other arising from the [ White-Color Corners ] .

  

  The 2 { tetrahedra } are then complimentary to one-another , in the sense that :

  • each { position vector } in one { tetrahedron } is always matched by a { position vetcor } in the other , with :
    • both { position vectors } being always [ of-the-exact-same-size ] , and
    • the 2 { position vectors } always running in the [ exactly-opposite directions ] .

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go to the next section : Section V - The { Flattened Tetrahedron }

go to the last section : Section III - The 5 Categories of Tetrahedra

return to The Four Body Problem HomePage

original dated 2007-12-16