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So it's many (more than I want to admit :-) years since my Euclidean and Non-Euclidian Geometry class, but isn't this a cross-cap, not a Klein bottle?

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This one is a Klein Bottle but we just released a cross cap one yesterday: https://n-e-r-v-o-u-s.com/blog/?p=8068

No cross-caps here. The first puzzle is a torus, the second a Klein bottle. Informally: the Klein bottle has on pair of edges is glued with a twist and one without; with the cross-cap, both pairs of edges are twisted then glued.

Thanks for the reminder.

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