<i>In a 3d world one can escape from the Earth's gravitational field. On the flat-world equivalent there is no escape.</i> You can checkout but never leave. There ought to be a science fiction story based on this premise.<p>Things get interesting in 2d flatland.<p>There the field or force has to decay as 1/r because the circumference of the boundary scales as O(r). But that's the <i>field</i>, to get to the potential you need to integrate and then you get a function that is logarithmic with distance.<p>A consequence of that is the potential does not drop to zero as you go further away. Unlike what is the case for the 3d case.<p>If you have heard that about a random-walking bird that returns infinitely often but a random-flying bird one that doesn't, that's sorta related.<p>Totally unrelated, was quite thrilled to see this map of the sphere<p><a href="https://substackcdn.com/image/fetch/$s_!8JEP!,f_auto,q_auto:good,fl_progressive:steep/https%3A%2F%2Fsubstack-post-media.s3.amazonaws.com%2Fpublic%2Fimages%2F49ffbc16-5e0e-4bce-9a37-e5c7b09019c7_1600x900.png" rel="nofollow">https://substackcdn.com/image/fetch/$s_!8JEP!,f_auto,q_auto:...</a><p>because I had been entertaining myself by making toy globes out of paper and it seems this polyconic map was the one I had used (an interrupted version of this).<p>The more conventional way is to use gores using interrupted sinusoidal that look like a string of lobes connected at their common equatorial hip.<p><a href="https://www.wolframcloud.com/obj/resourcesystem/published/DemonstrationRepository/deployments/InterruptedSinusoidalMapProjection/img/InterruptedSinusoidalMapProjection_Snapshot-2.png" rel="nofollow">https://www.wolframcloud.com/obj/resourcesystem/published/De...</a><p>What I was working with were more like flowers, one for each hemisphere, with the pole at the center.
> A consequence of that is the potential does not drop to zero as you go further away. Unlike what is the case for the 3d case.<p>The integral of const/r^2 doesn't drop to zero as r gets larger, but approaches some constant. Unlike const/r, which is unbounded as r gets larger. So potential is bounded in R3, but not in R2.