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I've finally realised what kind of topology Quadruple Besiege boards have. It's a torus with a twist. If you join the array shown end to end to form a cyclinder, you then curve the cylinder further as if to make a normal torus, but twist one end by 180° just before actually making the join. Not a Moebius strip, not a Klein bottle, but a torus with a twisted tesselation.