  Rigid origami, with applications ranging from nano-robots to unfolding solar
sails in space, describes when a material is folded along straight crease line
segments while keeping the regions between the creases planar. Prior work has
found explicit equations for the folding angles of a flat-foldable degree-4
origami vertex and some cases of degree-6 vertices. We extend this work to
generalized symmetries of the degree-6 vertex where all sector angles equal
$60^\circ$. We enumerate the different viable rigid folding modes of these
degree-6 crease patterns and then use $2^{nd}$-order Taylor expansions and
prior rigid folding techniques to find algebraic folding angle relationships
between the creases. This allows us to explicitly compute the configuration
space of these degree-6 vertices, and in the process we uncover new
explanations for the effectiveness of Weierstrass substitutions in modeling
rigid origami. These results expand the toolbox of rigid origami mechanisms
that engineers and materials scientists may use in origami-inspired designs.
