the tetrahedron, with 6 pencils (edges) and 4 elastics (vertices),
Here's an 'imploded' version of the tetrahedron; the elastics have been moved much closer to the middle of the pencils.
I tried to make a cube with hypotenuses as I mentioned. This has a few, but the pencil thickness starts becoming a large enough factor to cause distortion by itself. Also, the hypotenuse is longer than the edge, so I had to move the elastics closer to the middle of the pencils.
Lastly, I made some DNA. This is something I try with any modeling set I come across, e. g. Legos or Zome. The joint I used to attach here was a little ridiculous, but was the most stable thing I could come up with. It works perfectly fine if the DNA is infinitely long, otherwise you need to pull on it.
Overall this was very fun, and the first two structures were easy. Grab some pencils and elastics and see what you come up with.