On the enumeration of Archimedean polyhedra in the Lobachevsky space

We describe the class of Archimedean polyhedra in the three-dimensional Lobachevsky space, which technically reduces to studying Archimedean tilings of the Lobachevsky plane. We analyze the possibility of obtaining Archimedean tilings by methods that are usually applied on the sphere and in the Euclidean plane. It is pointed out that such tilings can be constructed by using certain types of Fedorov groups in the Lobachevsky plane. We propose a general approach to the problem of classifying Archimedean tilings of the Lobachevsky plane.