From line-systems to sphere-systems - Schlafli's double six, Lie's line-sphere transformation, and Grace's theorem

摘要

If each four spheres in a set of five unit spheres in R 3 have nonempty intersection, then all five spheres have nonempty intersection. This result is proved Using Grace's theorem: the circumsphere of a tetrahedron encloses none of its escribed spheres. This paper provides self-contained proofs of these results: including Schlafli's double six theorem and modified version of Lie's line-sphere transformation. Some related problems are also posed.