[math-fun] Re: Partitioning 3-space into straight lines, exactly one in each direction

Recently someone asked on sci.math.research whether R^3 could be foliated by straight lines, with exactly one line in each direction.

That was me. Unfortunately, I borrowed the word "foliate" from an earlier posting, unaware that it implied some sort of continuity.

If we drop the continuity requirement for a foliation, the problem becomes this one:

QUESTION: Is R^3 the disjoint union of [bi-infinite] straight lines L_d such that each direction d occurs exactly once?

That's the question I was trying to ask. Thanks to William Thurston for the proof that this follows from the axiom of choice. I wonder if there's a constructive proof. Dean Hickerson dean@math.ucdavis.edu