An Approach to the Four Body Problem

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

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Section III - The 5 Categories of Tetrahedra

The 5 Categories of Tetrahedra :

• Let us now list the 5 categories of { Tetrahedra } we have identified , as follows :
  • The { Equilateral Tetrahedron } , i.e. the { EQ Tetrahedron } ;
  • The { Isosceles-Triangle Tetrahedron } , i.e. the { IT Tetrahedron } ;
  • The { Congruent-Triangle Tetrahedron } , i.e. the { CT Tetrahedron } ;
  • The { General Tetrahedron } , and
  • The { Flattened Tetrahedron }.

• We then define the first 3 categories as follows :
  • The { Equilateral Tetrahedron } is a tetrahedron whose 4 triangular surfaces are { equilateral triangles } , all identical and congruent to one-another .
  • The { Isosceles-Triangle Tetrahedron } is a tetrahedron whose 4 triangular surfaces are { isosceles triangles } , all identical and congruent to one-another .
  • The { Congruent-Triangle Tetrahedron } is a tetrahedron whose 4 triangular surfaces are { general triangles } , but all identical and congruent to one-another .

  

  

• The { General Tetrahedron } is simply a tetrahedron that does not fall into the first 3 categories .
• And the { Flattened Tetrahedron } is simply a tetrahedron whose 4 { nodal points } reside on the same { plane } .

• We simply note here that the { EQ / IT / CT Tetrahedra } are always { quad-symmetric } , so that :
  • the { size and shape } of the { Triangular Surface } , with { 3 DOF's } , then fully defines the { size and shape } of these { tetrahedra } .

  ( please see the author's other paper - An Approach to the Triangle ( 2005 ) for details on Quad-Symmetry . )

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Constructing the { EQ / IT / CT Tetrahedra } :

• We shall now use the { Box Method } to construct these 3 categories of { Tetrahedra } ,
  • in an attempt to understand why we only have { 3 DOF's } here , instead of the full { 6 DOF's } .

• The { Box Method } for the construction of { EQ / IT/ CT Tetrahedra } is as follows :
  • First , we construct a { retangular box } of side-lengths [ L1 ] , [ L2 ] and [ L3 ] respectively ;
  • Then , for the 8 corners of the { Box } , we 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 } .

• For the { EQ Tetrahedron } , we have this set of 2 diagrams below :

  

• For the { IT Tetrahedron } , we have this set of 2 diagrams below :

  

• For the { CT Tetrahedron } , we have this set of 2 diagrams below :

  

• We simple note here that the { 6 faces } of the { rectangular box } are all rectangles ,
  • with each pair of { opposite faces } being identical .

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We shall , next , use { 6 parallelograms } instead of the { 6 rectangles } for our construction of the { General Tetrahedron } .

go to the next section : Section IV - Construction the { General Tetrahedron }

go to the last section : Section II - The 12 Degrees-Of-Freedom for the Tetrahedron

return to The Four Body Problem HomePage

original dated 2007-12-16