  Four rigid panels connected by hinges that meet at a point form a 4-vertex,
the fundamental building block of origami metamaterials. Here we show how the
geometry of 4-vertices, given by the sector angles of each plate, affects their
folding behavior. For generic vertices, we distinguish three vertex types and
two subtypes. We establish relationships based on the relative sizes of the
sector angles to determine which folds can fully close and the possible
mountain-valley assignments. Next, we consider what occurs when sector angles
or sums thereof are set equal, which results in 16 special vertex types. One of
these, flat-foldable vertices, has been studied extensively, but we show that a
wide variety of qualitatively different folding motions exist for the other 15
special and 3 generic types. Our work establishes a straightforward set of
rules for understanding the folding motion of both generic and special
4-vertices and serves as a roadmap for designing origami metamaterials.
