this post was submitted on 06 Oct 2024
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Spheres and hyperbolas. EDIT: oh they mean this specific shape not just any non-Euclidean square, so those won't work.
I haven't thought about a cone much but I think no.
The definition of a square is a polygon with four equal sides and four equal angles (nothing about 90 degrees and nothing about internal vs external angles since people love to get hung up on that)
A cone is half a hyperbola but I think without the symmetry of a hyperbola you can only get two angles equal at a time or two sides equal at a time.
You have the section of the smaller circle wrapping around the back of a narrower section of the cone, a right angle and then a straight line running down the side of the cone, another right angle in the other direction, then the section of the larger circle, this time going around the front of the cone, another right angle, and then the straight line completing the shape.
But you are right, the circle sections are not geodesics on that manifold, so not 'straight' by the most reasonable extension of that word to non -euclidian geometries. They'd be more like lines of latitude in that they seem straight from the outside, but in the context of that manifold are curved.
A cylinder would allow all geodesics, but then it would still have two 90 degree angles and two 270 degree angles so still not a square. I think it would be a trapezoid/trapezium, and might be a parallelogram depending on what definition you use.
There might be some crazy custom shape that makes the angles on the more complete circle segment actually 90 degrees but I don't think there's a common easy-to-conceive shape that works.
But you just said internal vs. external angles was a distraction that people shouldn't get hung up on.
To be clear, apart from that one bit, I agree with you completely. I don't even necessarily disagree with what you said back there per se. I just don't think it was useful to bring it up because even if it isn't explicitly in the definition of a square, it is an implicit assumption that when talking about the angles of a shape, you're always talking about all internal angles (or, equivalently, all external angles, just no mixing), so getting "hung up on" internal vs. external angles is not a bad thing.