Regulres Parkettierungsproblem

The regular parqueting problem is considered as the problem of finding all forms of bounded domains by means of which the Euclidean plane can be covered in at least one manner, simply, without holes, by adjacent congruent samples, so that every sample is surrounded in an identical manner by the totality thereof, or, what is equivalent, that one of the seventeen properly discontinuous groups of congruent mappings of the plane on itself, operates on the covering in a manner which is transitive on the domains. In these mappings, both direct and indirect congruence are allowed. Furthermore, it is required that the boundary of a domain be piecewise smooth. The solution consists of a system of 28 basic types of tessellations. Any basic type includes infinitely many forms of domains; among them are infinitely many belonging only to that one basic type. Every basic type is characterized by a rule by which all domains of this type are constructed. The element of these constructions for all basic types is the random line.