To start: what are geodesics? Well, they’re the shortest paths between two points in curved space-time (or any other type of geometry). In a nutshell, theyre like the straight lines we know from flat spaces but with an added twist they follow the curvature of the surface.

Now that you have a basic understanding of geodesics Let’s begin exploring with our topic: triangles made up of these shortest paths in different types of space-time. In positive curved spaces, like the inside of a sphere or the universe as we know it, geodesic triangles are called spherical triangles. Theyre pretty straightforward just draw three lines connecting points on the surface and you’ve got yourself a triangle!

In negative curved space-time (like in black holes), things get a bit more interesting. Geodesics become timelike, meaning they can only be traversed by objects with mass or energy. This leads to some pretty mind-bending phenomena like time dilation and gravitational lensing. In this type of geometry, geodesic triangles are called hyperbolic triangles think of them as the opposite of spherical triangles in a way. Instead of being enclosed by curves that converge at their endpoints (like on a sphere), they’re made up of lines that diverge from each other.

Finally, we have zero curvature spaces like flat planes or Minkowski space-time (which is what we use to describe the universe in special relativity). In these types of geometries, geodesics are just straight lines and geodesic triangles are called Euclidean triangles. They’re pretty much identical to their counterparts in flat spaces except they follow the curvature of the surface instead of being completely flat.

Geodesic triangles in positive, negative, and zero curved space-time. It might sound like a mouthful but trust us, it’s worth your time to learn about these fascinating concepts. Who knows maybe one day well be able to use them to unlock the secrets of our universe or even travel through time itself!

Until next time, keep exploring and stay curious!