Chiral Polyhedra in Ordinary Space, I

Chiral polyhedra in ordinary euclidean space E3 are nearly regular polyhedra; their geometric symmetry groups have two orbits on the flags, such that adjacent flags are in distinct orbits. This paper completely enumerates the discrete infinite chiral polyhedra in E3 with finite skew faces and finite skew vertex-figures. There are several families of such polyhedra of types {4, 6}, {6, 4} and {6, 6}. Their geometry and combinatorics are discussed in detail. It is also proved that a chiral polyhedron in E3 cannot be finite. Part II of the paper will complete the classification of all chiral polyhedra in E3. All chiral polyhedra not described in Part I have infinite, helical faces and again occur in families. So, in effect, Part I enumerates all chiral polyhedra in E3 with finite faces.