## ABSTRACT

This chapter analyzes some of the conditions that make a rigidly foldable origami tessellation. The fold angle coloring establishes matching rules on fold angle that must apply for rigid foldability to persist. The flat-foldable vertex inequality establishes a qualitative relationship between the major and minor creases of a degree-4 vertex. The relationship between adjacent fold angles, coupled with the fact that opposite fold angles are pairwise equal, implies that there is a simple proportionality between the half-angle tangents for any pair among all four angles around a flat-foldable degree-4 vertex. There is also a relationship between adjacent fold angles at a degree-4 vertex. Using distinct colors for different fold angle magnitudes can help keep track of matching fold angles. The mathematics makes substantial use of spherical trigonometry, the trigonometry of angular relationships in 3D, as opposed to the planar trigonometry that is often the sole focus of current-day high school mathematics.