I'm excited to have discovered a near miss!
I'll do a proper writeup shortly, but want to post so that others can try it: it's pretty simple.
Nine octahedra arranged point to point in a circle can be close internally by joining the remaining corners.
ok, so its a teeny weeny bit off, and one rod always jumps away, but if you can forgive that its a trip to have such a compact near-miss.
Let alpha = arcsin(1/sqrt(3)) = 35.264 degrees, be the semi-apex angle of an equilateral square-based prism.
Notice that there are three pyramids touching on their faces at each joint. Applying simple trig leads to the angle subtended by each octahedron being: (180 -4*alpha) degrees = 38.94 degrees.
So, nine octahedra make: 351 degrees - and the ten degree difference can (nearly) be absorbed.
--Karl Horton 20:38, 3 June 2007 (UTC)