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I am curious, is the pythagorean theorem or some version of it applicable to spherical trig (triangles on the surfaces of spheres and other shapes)?

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If a Möbius strip is going to be included, shouldn't a Klein bottle be too?

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'Topology' is about mapping one shape onto another(and hopefully back), same applies to the triangle onto sphere, any shape onto another, there's more than one way around the barn though I think the branches (algebraic or point set) are converging (if not considered already so). Relatively simple questions like this can be astonishingly difficult to prove - as was the recently solved Poincare Conjecture. It's important to consider exactly which space(flat, curved, other) you are mapping to and from and backwards upside down. It's in many ways important to consider if asking for all potential sizes or small from one to another. Small triangle onto large Klein bottle, Moebius stip, sphere... or large onto small... helps if creating a Klein bottle 'opener'

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