![]() Do those issues relate to this issue of measuring angles?) Elsewhere, the book also mentions isometries and preservation of distances. (The book says that "Gaussian curvature" can be determined based on measurements taken only from the surface - no need to see off the surface. Analogously, couldn't hyperbolic triangles that measure <$180^\circ$ to us Euclideans measure as $180^\circ$ to Hyperboleans? I'd expect angle measurements to be affected by one's "space" just as much as the concept of "straightness." So if we look at a line in hyperbolic space, it may look curved to us Euclideans, but it looks straight to Hyperboleans. The angles might sum to <$180^\circ$ from a Euclidean perspective, but shouldn't my hyperbolically embedded protractor continue to measure $180^\circ$? If - Poof!- our universe morphs into some hyperbolic geometry, I'd assume my protractor would also morph just as much, so I'd still measure $180^\circ$. Here's my thinking: If I measure a triangle at my desk, it's $180^\circ$. But is that only for us considering those angular measurements (in those non-Euclidean spaces) from our Euclidean perspective? When we look at triangles on spheres or saddles, they are >$180^\circ$ or <$180^\circ$, respectively. ![]() In non-Euclidean space, triangles don't necessarily measure $180^\circ$. It's reminded me of a question I've long had - which this book hasn't answered. ![]() I'm reading Donal O'Shea's The Poincare Conjecture, a nontechnical book for mainstream audiences. ![]()
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