Angle measure in non-Euclidean geometry

Question

I'm reading Donal O'Shea's The Poincare Conjecture, a nontechnical book for mainstream audiences. It's reminded me of a question I've long had -- which this book hasn't answered.

In non-Euclidean space, triangles don't necessarily measure $180^\circ$. When we look at triangles on spheres or saddles, they are >$180^\circ$ or <$180^\circ$, respectively. But is that only for us considering those angular measurements (in those non-Euclidean spaces) from our Euclidean perspective?

Here's my thinking: If I measure a triangle at my desk, it's $180^\circ$. 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 $180^\circ$. The angles might sum to <$180^\circ$ from a Euclidean perspective, but shouldn't my hyperbolically embedded protractor continue to measure $180^\circ$?

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. Analogously, couldn't hyperbolic triangles that measure <$180^\circ$ to us Euclideans measure as $180^\circ$ to Hyperboleans?

(The book says that "Gaussian curvature" can be determined based on measurements taken only from the surface -- no need to see off the surface. Elsewhere, the book also mentions isometries and preservation of distances. Do those issues relate to this issue of measuring angles?)

Answer 1

That the sum of the angles of a triangle in the hyperbolic plane is less than 180° is a fact that depends only on the axioms chosen and is completely independent from the model we use to visualize the said plane in the euclidean plane.

For instance, the Poincaré disk model does not distort the angles: the euclidean perspective (in this model) is not different from the hyprlerbolic perspective. However, in this model the straights look like parts of euclidean circles. Or in the Klein model the hyperbolic straights look like euklidean segments and the triangles are like as if the sum of their angles were 180°.

What we see in an euclidean model does not have much to do with what we would see if we -- boom! -- turned to be hyperbolic. Would we see straights? It depends on the physical thing that we would cosider straight. The path of a light ray there? Then yes, if the light rays would be considered straight, and if they behaved like straights (in the hyperbolic sense) then we would consider everithing straight that would behave like light rays.

Answer 2

A Hyperbolean can immediately disprove the notion of a constant angle sum simply by constructing equilateral triangles with distinct side-lengths. The triangle with larger sides will have smaller angles. (Indeed, a triangle with infinitely-long sides will have measure-zero angles.) Consequently, the larger equilateral will have a smaller angle sum.

A slightly-more-inquisitive Hyperbolean might construct three copies of any particular (not-necessarily-equilateral) triangle $\triangle ABC$, fanning them out, with the $A$ angle of one next to the $B$ and $C$ angles of the others. This Hyperbolean would then observe that the total is never quite "two right angles" (that is, the outer edges of the fan don't align with a line), although smaller and smaller triangles get closer and closer to that total.

In these scenarios, "isometries" and "preservation of distances" definitely play a role. For instance, in the latter, it is assumed that the mere act of moving copies of $\triangle ABC$ around, so as to make the fan, will not change the shapes of those copies. Basically, assuming isometric uniformity is declaring that a segment of length $5$ here will stay length $5$ even if I (and my ruler) move over there. If lengths are preserved, then so are angles, and vice-versa, since they're related by the Laws of Cosines, but at that point, things start getting technical.

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Incidentally, in an ancient MathOverflow post, I asked "What are trig classes like within a universe that's noticeably hyperbolic?". Elsewhere, I've phrased this as "What could Hyperbolean Pythagoras been thinking?" ... because the Hyperbolic Pythagorean Theorem ---$\cosh a \cosh b = \cosh c$--- seems far more difficult to intuit within the Hyperbolic Experience than $a^2 + b^2 = c^2$ is for us Euclideans. (But, of course, I'm biased.) I still don't have a satisfactory answer, or even a truly satisfactory notion of what a satisfactory answer would be, but it seemed worth mentioning here.

Answer 3

The other answers are pretty detailed already, so I'll just add that this statement from your question may be the source of a misconception:

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 [degrees for the sum of the angles of a triangle].

Hyperbolic geometry is intrinsically a different geometry than Euclidean geometry; you can't "morph" one to the other while preserving all angles, as you seem to suggest.