This is in the ‘so simple it seems to have been overlooked’ category. Or maybe it was just too obvious to mention. In any case I haven’t seen it before.
It is just an ordinary fan from a rectangle, with one edge glued to the other to make a full circle. Only, instead of the corrugations meeting the long edge at 90 degrees, they meet it at slightly less. That small change is enough to allow the whole thing to collapse, as a regular circular fan cannot.
This is another instance of the guideline--I hesitate to call it a "law"--that in collapsible origami, skewed angles work better than perpendiculars. Perpendicularity is all about equilibrium, among multiple options and stresses; skewedness is all about disequilibrium--and decisiveness.
If you make this, don’t forget that for a fan to form a circle, the long edge of the rectangle must be at least 2pi times the short one. A ratio of 6.5 to 1 is a reasonable approximation.
How it came about: I’d been studying fan disk shapes made with curves instead of straight lines, some of which are like those which Nishimura and Christine Edison have also been making. These disks collapse to some extent---that is one of the first things one notices about them. Always on the lookout for “special properties of curved folds”, for a while I thought the collapsibility was a function of the curviness. Alas: closer inspection reveals that it is because curves are, by definition, not everywhere perpendicular to the long edges. Simple curves either hit the bottom edge at an angle, the top edge at an angle, or both. It is that angle, and not the curviness as such, that is doing the work of allowing the collapse. And an angled straight fold does a better & purer job of it than a curved one.
It is hard for me to believe that the familiar paper fan, one of the oldest of forms in origami, can give up ANY new tricks at this point. But you never know. If anyone has seen this before, please drop me a note. --Until I hear otherwise I’ll claim this one as my own.
invented December 3, 2008
P.S. Ray Schamp just pointed me to the beautiful images in a paper by Taketoshi Nojima on collapsible forms. This shape is not in it, but others using a very similar principle (from a flat disk instead of a rectangle) are.