A chain of edges can close iff the longest edge is not longer than the sum of the lengths of all the other edges. This is Theorem 8.6.3 (p.326) in [*Computational Geometry in C*][1] and Theorem 5.1.2 (p.61) in [*Geometric Folding Algorithms*][2]. You can easily see the necessity of this condition: If one edge $e$ is too long, the others all together cannot reach from end-to-end of $e$.

This result has nothing to do with $6$; it holds for $n \ge 3$ edges.


  [1]: http://cs.smith.edu/~jorourke/books/compgeom.html
  [2]: http://gfalop.org/