Interpolations

To begin, I am not entirely sure how best to describe this page's intention and what exactly I am doing! However, the merest of glances will show much better than words. In short, I here show the underlying structure of the parquet deformations, as exemplified on my dedicated page. Simply stated, these ‘interpolations’ (see below) serve as a framework, to which lines are then added. How best to describe this is unclear. ‘Grids’? Not strictly as such, as grids imply an overlapping of lines. ‘Open Lattice’? Again, it is not really a lattice as such. Evolving Motifs? Deforming Devices? Nothing seems ideal. However, for the sake of some description, I have chosen ‘Interpolations’, but not without a great deal of reservation. This has echoes with Craig Kaplan’s descriptions of a true interpolation in his work, but my instance is not as exact. Pending a better description, ‘Interpolations’ will have to serve for now. If any reader has a better description, do let me know!

Identifying/referencing each interpolation is by a two-fold process, first by a simple numerical sequence, for convenience, and then by a country's name, such as Australia, Brazil etc., as according to my own, original identifying system and which will likely grate on any mathematician reading this! To this end, I retain an old idiosyncratic system I used from my earliest days. This refers to the finished parquet deformation, where I title the interpolation after a country, with the ‘filling-in’ of towns/cities thereof. For example Canada (interpolation) - Ottawa (filling-in); Canada - Banff etc. Although not strictly mathematical, it makes referring to any specific parquet deformation and its underlying grid much easier than trying to remember a strictly mathematical procedure e.g. instance 64 (interpolation) and what 64a, 64b, 64c (variations) represents, or, even more involved, with second generation instances, e.g. 64a(i)! Of course, I could have used any other ‘two-part’ system; there is nothing unique about the ‘country and town’ system.

A variety of devices can be seen, some of which are ‘open’ e.g. Argentina, and continuous of line e.g. Austria. These typically expand, rotate or increase in angularity in some way, typically, but not necessarily, from dots on a square lattice. A striking feature of the interpolations is the typical sheer simplicity, even straight lines can lead to aesthetic parquet deformations! The overriding premise of this page, and others, is that you too, dear reader, no matter what your level of mathematical ability, can do this as well!