nice. isn't this the path on the edges of a tetrahedron I mentioned?

I'd certainly think that would have the maximum for a 4-segment closed polygon: 4 consecutive edges of a regular tetrahedron that returns to where it started.

--no. The 4-edge closed polygonal path (all edge lengths 1/4) I mentioned has hinge angle 90 (note 90 is trivially seen best). The thing is made by folding a rhombus with angles 70.522 and 109.478 along its short diagonal at 90 degrees. This is not an edge path on a regular tetrahedron, but some regtet angles worm in there somehow.