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One way to very concretely describe piece movement on a given board is to
use a (combinatorial) graph:  each vertex is a location available, and
there are several types of edges between these vertices.  Each piece is
allowed to move from vertex to vertex, provided that there is an edge of
the appropriate type between them.  This is good for simple pieces, but
becomes a little complicated even if we just want to allow sliders.

So the question is how do our traditional notions of 'topological' boards
translate into actual game mechanics, i.e. graph play.  The octagon-square
tiling that Joe has presented brings up some interesting questions.  That
type of tiling allows us to choose different sizes for the sides of the
octagon, so we can make the squares larger or smaller.  It seems most
natural to have all edges the same length, but do different side length
promote (in our mind, looking at the board) different movements?