You can create a real 3D paper cone, with preprinted geodesics, by printing out a suitable template and then cutting and gluing. This provides “hands on” demonstrations of several interesting concepts about life in a non-Euclidean space, such as the fact that geodesics that start out parallel do not remain parallel, and may eventually cross, even though they are absolutely straight.
The cone is a valuable pedagogical stepping stone, because even though it is a non-Euclidean space, it is as nearly Euclidean as it could possibly be. The cone has zero intrinsic curvature everywhere except at the apex. The only way for denizens of the cone-world to discover that their space is not completely flat is to do an experiment that somehow circumnavigates the apex.
I provide PDF files with templates you can print out, to make it easy to create your own models. Actually I'm not sure that "model" is the right word; it's an actual /instance/ of a non-Euclidean space, not merely an analogy. (You can of course use it as a model for other curved spaces.) If you have access to a glue stick, scissors, and a printer, you're all set.
It is one thing to write down the equation of geodesic deviation, but it is something else to look at actual geodesics on an actual cone, and watch how they wrap around.
The question arises, what's the appropriate grade level? I don't really know. A fifth-grader can assemble a cone in about five minutes, and can learn some valuable lessons by staring at it. Meanwhile, physics grad students can also learn some valuable lessons. To fully understand what the model is trying to tell you takes a lot longer than five minutes.
It is important to learn the correct lessons from all this:
*) Fact: We have two straight lines (A and B) that start out parallel but do not remain parallel. They may eventually cross. By the same token, lines that start out non-parallel may cross each other twice.
*) Fact: This happens even though the space is absolutely flat everywhere in the neighborhood of line A, and also absolutely flat everywhere in the neighborhood of line B.
*) Do not let this undermine your notion of straightness. The lines really are straight. Also do not let it undermine your notion of flatness. The space really does have zero intrinsic curvature in the neighborhood of each of the lines of interest.
*) The notion of parallel that you learned in high-school geometry needs to be updated. The idea that parallel lines are everywhere parallel only works in a Euclidean space. You can formulate a reliable local definition of parallelism using a Schild’s Ladder construction. This requires constructing straight lines and measuring distances, as we now discuss.
*) The concept of distance is nontrivial. You cannot expect some magical oracle to tell you the distance between point a (on line A) and point b (on line B). If you want to know the distance, you have to run a measuring tape from a to b, or perform some similar experiment that crosses the space between the two points. The operational definition of distance therefore depends on curvature in regions between the two lines, even if the lines themselves never actually touch the high-curvature regions.
*) The cone is not a good model of gravitation. The fact that geodesics partially wrap themselves around the cone is not a good model of orbits that wrap themselves around a planet. A vastly better model is presented in section 3.
The fact that geodesics are sensitive to curvature elsewhere has direct application to astronomy and cosmology: You can measure the mass of a planet without ever setting foot on the surface. Just put a satellite in orbit around it, and apply Kepler’s laws. You can even measure the mass of a black hole this way, which is significant because the hole doesn’t have a surface that you could even imagine setting foot on.
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This is fairly new and probably infested with typos or worse.
Let me know if you have questions or suggestions.