Couldn't accomplish this in a comment to [Rahul's](https://mathoverflow.net/a/29695) #1, but here is a physical realization of my drawing with colored pencils. Compare:

![Line drawing][1]
![Physical realisation with pencils][2]

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….


  [1]: https://i.sstatic.net/mXuKC.jpg
  [2]: https://i.sstatic.net/6ceaA.jpg