Click on an image to see what happens.
The idea here is
(1) the opened flower should have much more than twice the surface area of the closed one; it should not just fan out but expand in all directions as it opens.
(2) It must close back reliably the same way.
(3) Should be pretty, and look like a flower.
(4) If possible, it should change dramatically through different stages as it is opened.
(5) The flower must be origami—from a square, no cuts, and glued only to the background card.
The only mechanisms I know of so far (but I’m pretty ignorant) that do any of this reliably are the preliminary fold (opens to 4X; & with a variation, to 9X), the Miura fold (opens to an indefinite multiple of X), and twist folds. All of the flowers shown combine these principles to one degree or another.