Two different Math-Fun threads have cross-fertilized in my brain, leading to the following rumination. Thanks to Jim Propp, and indirectly to my wife, for the origami theme; and thanks to Tom Karzes for his Gaussian-integer puzzle (to which I think the answer is mostly "Gaussian GCD"). I propose two operations, folding and unfolding, on any set of Gaussian integers. For both operations you start by selecting a crease-line, which must be one of the reflection lines of the Gaussian integers -- that is, a horizontal or vertical line with integer or half-integer intercept, or a 45-degree diagonal with integer intercepts. To "fold" a set, you remove all the points on one side of the crease, while adding their mirror images to the other side. (The constraint on the crease line ensures that the mirror images will also be Gaussian integers). To "unfold" a set, just add all mirror images, with no erasing. The central question of the "Theory of Gaussian origami" is, what sets are achievable from what starting sets? In particular, what sets are achievable starting from a single point? I know that by repeated folding I can get a single point from any finite starting set. But, for example, starting with {0}, can I get {0, 2+i} by any number of folding and unfolding operations?