In a 3d world one can escape from the Earth's gravitational field. On the flat-world equivalent there is no escape. You can checkout but never leave. There ought to be a science fiction story based on this premise.
Things get interesting in 2d flatland.
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 field, to get to the potential you need to integrate and then you get a function that is logarithmic with distance.
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.
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.
Totally unrelated, was quite thrilled to see this map of the sphere
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).
The more conventional way is to use gores using interrupted sinusoidal that look like a string of lobes connected at their common equatorial hip.
> 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.
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.
I should have said that it rises to zero, assuming standard sign conventions.
The gravitational potential infinity of a point mass at the origin is defined to be zero.
Could you expand on why you think it approaches a constant, presumably a non-zero constant. In any case absolute values do not matter, only the potential difference matters, but am still interested in your thought.
It's fun to puzzle through why optical reflections, active radar, and things like that are inverse R^4. (It's just two inverse-square laws composed together).
Dipoles, like magnetic fields and planetary tidal forces, decay as an inverse R^3. That's less intuitive.
While we're on this... it is arguably even more befuddling to recognize that the information content potential of space apparently correlates with the surface area rather than volume
https://en.wikipedia.org/wiki/Holographic_principle
Dimensions, and hence surfaces and volumes are part of our “tooling” used to understand the world (epistemic) rather than being intrinsic to nature. So the befuddling is related to our own interpretation.
Things get interesting in 2d flatland.
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 field, to get to the potential you need to integrate and then you get a function that is logarithmic with distance.
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.
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.
Totally unrelated, was quite thrilled to see this map of the sphere
https://substackcdn.com/image/fetch/$s_!8JEP!,f_auto,q_auto:...
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).
The more conventional way is to use gores using interrupted sinusoidal that look like a string of lobes connected at their common equatorial hip.
https://www.wolframcloud.com/obj/resourcesystem/published/De...
What I was working with were more like flowers, one for each hemisphere, with the pole at the center.
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