deep thought wrote on Tue, Jul 27, 2021 05:17 PM UTC in reply to half sick of shadows from Tue Jul 8 2008 05:20 PM:
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.)
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.)