Couldn't accomplish this in a comment to Rahul's #1, but my drawing is based on 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...