We prove that any given well-behaved folded state of a piece of paper can be reached via a continuous folding process starting from the unfolded paper and ending with the folded state. The argument is an extension of that originally presented in (DM01). 1 Introduction. In defining an "origami" or "folding" of a piece of paper, there is a distinction between specifying the geometry of the final folded state (a single folding, e.g., an origami crane) and specifying a continuous folding motion from the unfolded sheet to the final folded state (an entire animation of foldings). It is conceivable that some folded state exists, but that the piece of paper could not reach that state via a continuous folding process, e.g., the state could only be reached by passing portions of the paper through itself, or by cutting and regluing. Our main result is that in fact every well-behaved folded state of a simple polygonal piece of paper can be reached by a continuous folding motion, and so the entire config uration space of all well-behaved folded states of a piece of paper is connected. As a consequence, other results that define fold- ings with specific properties need not distinguish between folded states and continuous folding motions, and can use the more convenient specification of a single folded state. The same result as ours was established in (DM01) for the special case of a rectangular piece of paper and a folded state having a flat patch. Here we extend the result to an arbitrary simple polygonal piece of paper and to any well- behaved, possibly entirely curved, folded state, in addition to formalizing definitions and adding detail to the proof. 2 Definitions. We believe that research in mathematical origami has been somewhat hampered by lack of clear, for- mal foundation, so we devote a relatively lengthy section to this topic before turning to the proofs. At a high level, our definitions generalize Justin' s definition of flat folded states (Jus94).