Analysis of the rules underlying possible configurations of pieces upon a chess board are secondary to the axioms of the rules upon which the possibility and ranges or limits of motion for pieces are defined. Preliminary (and perpendicular) to this definition of limits and ranges are the boundary conditions and container coordinate system defined by the board itself.

Considering possible variations-on-a-theme of boards and coordinate systems; definition of relationships of coordinates (squares, triangles, hexagons, other ?) or the topological boundary conditions of the board itself (square, rectangular, infinite, spherical, cylindrical, multi-layered, other ?) indicates one extra direction of systems analysis available. While real-world chess boards are limited to 8×8 square domains, the analysis or at least the potential for other forms of organisation and coordinate system potentially allow for insight into novel approaches to that limited or orthodox board space and coordinate system.

It is a mathematical and logical fact that there is always at least one way to extend a sufficiently sophisticated axiomatic system into new (and potentially useful) reconfigurations. It is probable that creatively re-imagining boundaries and coordinate systems at a more sophisticated layer allows for the cultivation of useful intuitions and strategies to approach rules-based systems and other games.

While altering the rules of movement and strategy on the board can be interesting and lead to unexpected insights and creative novelty, altering the properties of the coordinate system and notional space within which action occurs has a dramatic effect on multiplying possibilities. This is not without it’s own problems in regard to identifying or recognising useful patterns and possibilities or insights, for e.g.:

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