NavList:

Thanks to AdrianF who has contacted me privately suggesting a parallel and probably more easiliy understood way of looking at paragraphs 9 &10 of my proposed explanation in my earlier post.

A plane through the centres of the three spheres would also contain three pairs of external tangents, each part of one of the three cones.  These pairs of tangents would also intersect at the three points along the fold.  Moreover, a view at 90 degrees to this third plane would be a copy of the 2-D diagram we started with, and the distances along the fold would be true distances.

I think we now have a complete solution.  DaveP