ABSTRACT
Summary Students use spherical trigonometry to discover strong relationships between the dihedral angles of a four-valent flat vertex fold as it opens and closes rigidly (that is, each region of paper between the creases remains rigid). These results can then be used to prove that certain flat-foldable crease patterns cannot be folded rigidly.
Content The spherical law of cosines is extensively used, as well as Kawasaki’s Theorem (the four-valent case) from the Exploring Flat Vertex Folds activity. This is meant to follow the Rigid Folds 1 activity, although it does not make use of Gaussian curvature. However the results about non-rigid folds fit nicely with the previous non-rigid results. To fully appreciate the results on the square twist’s rigidity, students will need to have seen the Folding a Square Twist activity previously.
