Consider a lattice of squares in a rectangular universe. The top edge connects to the bottom edge, except offset to the right by one square. That is, one can visit every square by simply going up (or down) repeatedly.
Similarly the left and right edges. Similar to a toroidal universe except for the offset. Does something weird happen at the corners?
Originally inspired by the idea of playing go 囲碁 on such a board. Interesting things could happen with ladders, perhaps with a different offset.
Also consider an offset grid of diamonds.
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