A dual graph does NOT necessarily mean a self-dual. 2nd half of your statement is true. DUAL(DUAL(X)) = X.
Nice drawing! You almost got it. In the dual from the graph of “19 lines by 19 lines”, you have 18x18+1 vertices, 684 edges, and 361 faces, which is exactly the dual of the original graph of 361 vertices, 684 edge, and 18x18+1 faces.
Of course the dual of the 19x19 graph (either you see it as lines, or equivalently, squares) has a different shape (not a self-dual), but there exists a dual graph for it which switches vertices and faces. The dual doesn’t mean anything with regard to the game. I just wanted to have fun.
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