Four Body Problem

# An Approach to the Four Body Problem

### Summary & Brief Introduction :

• The { Tetrahedron } is core and central to the { Four Body Problem } ,

• and 6 Degrees-Of-Freedom ( DOF's ) are usually involved in defining the { size and shape } of the { Tetrahedron } .

• We noticed that the { line } joining the mid-points on each pair of { non-touching / opposite sides } always passes thru the { centroid } of the { Tetrahedron } ,

• and this set of { 3 lines } may then be used to fully define the 6-sided { Tetrahedron } .

• The method of construction is as follows :

• Take 3 sticks of any arbitrary lengths and orient these in { 3-D space } so that :

• the 3 sticks bisect one-another .

The { point-of-intersection } for the { 3 sticks } is then the { centroid } of the { Tetrahedron } , and :

• the 6 end-points on the { 3-sticks } then match exactly onto the 6 mid-points on the { 6 sides } .

This then fully defines { Tetrahedron } .

• Hopefully , this will lead to a better understanding of the { Four Body Problem } , and further .

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Section ISetting up the Reference Frames
Section IIThe 12 Degrees-Of-Freedom for the Tetrahedron
Section IIIThe 5 Categories of Tetrahedra
Section IVConstructing the { General Tetrahedron }
Section VThe { Flattened Tetrahedron }
Section VIBrief Summary on the Tetrahedron
Section VIISetting up for Angular Momentum Issues
Section VIIICross-Product Considerations
Section IXConcluding Remarks
Appendix AA Numerical Example for the Cross-Product of 2 Tetrahedra