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deep thought wrote on 2021-07-27 UTC

The reason people are calling it a 'Klein bottle' rather than a sphere is because of the weird situation around the poles.

If you draw the eight spaces surrounding each pole as triangles, then the rook shouldn't pass -- it only crosses edges. If you draw them as squares and try to connect opposites, each pole becomes a 'cross-cap' rather than a point. A sphere with two cross-caps is topologically equivalent to a Klein bottle.

(Alternatively each pole could be a double-handle, then you have an orientable surface of genus 4.)

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