ABSTRACT
The way the vertex traces mate with each other on the Gaussian sphere also explains why the pattern curls ever tighter with increasing fold angle. As the fold angles increase, the arc lengths in the Gauss map increase, and the sizes of the individual bow-tie traces increase. The Gauss map is completely independent of any distances in the folded form: it is affected only by fold angles and orientations. The trace for a developable degree-4 vertex will always be some variant of a bow-tie, but it can be rather distorted, depending on the sector angles, the degree of folding, and how far it is wrapped around the Gaussian sphere. For flat-foldable vertices, the relationship between adjacent fold angles was surprisingly simple and elegant, but that made use of the considerable simplifications afforded by the condition of flat-foldability.
