Pretty 3D Polygons: Exploration and Proofs

We explore ‘pretty’ 3D (skew) polygons and prove their existence. These generalize the well-known regular 2D polygons. In 3D, an additional regularity condition is imposed: all edge torsion angles must be equal in absolute value. The torsion angle of an edge is the dihedral angle between the planes spanned by the edge and each of its two adjacent edges. We define an infinite family of pretty 3D polygons with both rotation and reflection symmetries. This resolves an open problem about the existence of certain pretty 3D polygons. Moreover we present some ad hoc specimens, including two trefoil knots, that do not have reflection symmetry. Finally, we present some pretty 3D polygons that can be morphed while preserving their prettiness.