The aim of this work is to establish a systematic methodology for generating automatically different tessellations and double-layer grids (DLGs) following a defined and specific nomenclature proposed originally for such a task. This particular nomenclature defines the notation of mosaics and DLGs in a synthesized and unique manner, with the advantage that it shows how to generate and design them after the parameters expressed on their own names. As a result, by means of an algorithm and some computational codes, it is possible to recreate in 3D any of those grids directly from their own names. Current nomenclature for tessellations is also analyzed, finding severe disadvantages, such as the excessive length of their notations or their non-uniqueness character. A new nomenclature is proposed in order to define and generate consistently and unequivocally n-uniform mosaics in a consistent manner with the current nomenclature used for the Archimedean (regular and semiregular) tessellations.