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.
I bet there exists a non-Euclidean surface such that all these lines are straight.
A cone should do the trick, no?
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.