Why the inverse-square law works — and it's not just about distance
The inverse-square law explains why light, sound, and radio signals weaken with the square of distance. But why square, not cube? This article derives the surface area of a sphere from scratch, showing that the area-preserving mapping between a cylinder and an inscribed sphere gives 4πr². A marble analogy then makes the falloff intuitive: in 1D, intensity stays constant; in 2D, it drops with distance; in 3D, with distance squared.
In a two-dimensional world, the falloff in marble-intensity, as witnessed by those living in bottomless hole, decreases with the first power of distance to the origin.
- HarHarVeryFunny
That seems to be milking the explanation for all it's worth!
The short answer being that if something is radiating out 3-dimensinally in all directions, then at a given distance, r, that thing (light/whatever) is being spread out over an area equal to the surface area of a sphere of that radius, which is 4 x pi x r^2, i.e. the intensity is inversely proportional to the square of the distance (r^2).
- doug_durham
Thanks for putting the effort in on this. The author pointed out that no LLMs were used in the production of the blog post. To be honest the first thing I did was put the blog post into an LLM and asked it to simplify. The LLM did a great job of providing a very clear explanation for me.
This is one of the super powers I find for LLMs. I am certain that this particular derivation spoke to the author and seemed like the obvious approach. In my experience math derivations a highly personal. What resonates with me will not resonate with someone else. An LLM allows an infinite number of variations. Thanks again to the author.
- srean
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
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.