Couldn't accomplish this in a comment to Rahul's #1, but here is a physical realization of my drawing with colored pencils. Compare:
alt text http://cs.smith.edu/%7Eorourke/MathOverflow/weaving4.jpg alt text http://cs.smith.edu/%7Eorourke/MathOverflow/pencils4.jpg
The pencils are in contact, but the lead inside could serve as lines in space without contacts. I don't know how to reconcile this with your quote from Whiteley...