Oh -- and the metric used on GxG is just the cartesian product of that on G with itself, where G is a regular octagonal curve.
OK. Allow me to suggest that that's less natural than the different definition I proposed, because it seems freaky to consider {a1,c1,e1,g1} a square. But it's your game, so you get to choose the rules :-).
If you mean (0,0),(0,2),(0,4),(0,6),
(Yup.)
that's *not* a square -- all angles are straight, not right. A regular octagonal curve has the same metric as a circle (with 8 equally spaced points on it).
Ohhh, I see. I entirely misunderstood. I thought you meant the metric induced by the obvious embedding of G in R^2. With the clarification you've now provided, I agree that that's a fine definition, not least because it's equivalent to the one I prefer :-). -- g