Jim writes: << I propose that someone build a tetrahedron with vertices at . . . (Thanks to Fred Helenius for identifying this "land tetrahedron". . .) . . .

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Hmmm, how about the connecting "beams" being (ends of) cylinders, upon each of which is a TV camera and inside of which is a TV screen showing what's at the other end? Plus, there may be nice arrangements of hundreds of such tetrahedra, so that everyone's "near" everyone. --Dan