Chiral Polyhedra in Ordinary Space, I

Chiral polyhedra in ordinary euclidean space E³ are nearly regular polyhedra; theirgeometric symmetry groups have two orbits on the flags, such that adjacent flags are in distinctorbits. This paper completely enumerates the discrete infinite chiral polyhedra in E³ withfinite skew faces and finite skew vertex-figures. There are several families of suchpolyhedra of types {4,6}, {6,4} and {6,6}. Their geometry and combinatorics arediscussed 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. Allchiral polyhedra not described in Part I have infinite, helical faces and again occur infamilies. So, in effect, Part I enumerates all chiral polyhedra in E³ with finite faces.