Here's a seven sided ring: it's fairly easy to build but
a small child destroyed mine before I could verify whether
it could be made to stand upright.
Any takers to rebuild and find out?
Each side is slightly curved, so you must use four panels
per side for the corners to work out. The corners themselves are
simply octahedra.
There are some fun calculations to do to make this all work out,
but I'll post them to the Math forum.
Here's the flickr set: http://www.flickr.com/photos/karlhorton/sets/72157612698334867