David Gale writes re the Teabag Problem: << It surprised me until I found an origami example on the internet. There's whole lot of stuff on this problem, e.g. http://www.ics.uci.edu/~eppstein/junkyard/teabag.html but I couldn't find anything on the natural question, what if instead of a square you have a disc? Can there be a positive volume?
Here are some references for the corresponding problem using two unit disks identified along their boundaries: The Mylar Balloon Revisited Ivaïlo M. Mladenov; John Oprea The American Mathematical Monthly, Vol. 110, No. 9. (Nov., 2003), pp. 761-784. What Is the Shape of a Mylar Balloon? William H. Paulsen The American Mathematical Monthly, Vol. 101, No. 10. (Dec., 1994), pp. 953-958. Both papers use some hairy integrals and differential geometry. But, the first paper states that the goal is to determine the *convex hull* of the wrinkly shape that the Mylar balloon would actually assume. It's not clear to me what assumptions the second paper is making. I doubt either paper rigorously determines the maximum volume the metric space formed by a circular teabag can assume when embedded isometrically in 3-space. --Dan