Chiral Polyhedra in 3-Dimensional Geometries and from a Petrie–Coxeter Construction

We study chiral polyhedra in 3-dimensional geometries (Euclidean, hyperbolic, and projective) in a unified manner. This extends to hyperbolic and projective spaces some structural results in the classification of chiral polyhedra in Euclidean 3-space given in 2005 by Schulte. Then, we describe a way to produce examples with helical faces based on a classic Petrie–Coxeter construction, yielding a new family in $$\mathbb {S}^3$$ which is described exhaustively.