Only homothetic motion is possible for four masses fixed at the vertices of a regular tetrahedron; no rotating, shape-preserving solution exists, but the proof offered here is incomplete.
and Lass, H., A note on relative motion in the g eneral three-body problem, Celestial mechanics, 8, 5-10 (1973)
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On the existence of regular tetrahedral non-homothetic homographic solution
Only homothetic motion is possible for four masses fixed at the vertices of a regular tetrahedron; no rotating, shape-preserving solution exists, but the proof offered here is incomplete.