there definitely are some things non-intuitive about high-dimensional objects, in general.
For example, if you peel away the outer 1% of a n-dimensional sphere or cube, you’re left with 0,99^n of its volume. As n goes to infinity, that goes down to zero.
See also the curse of dimensionality (https://en.wikipedia.org/wiki/Curse_of_dimensionality), or https://www.math.ucdavis.edu/~strohmer/courses/180BigData/18..., which, for example, compares the n-dimensional sphere with radius 1 to n-dimensional hypercubes with side length 1. In two and three dimensions, the cube easily fits in the sphere; in four, it just fits. In higher dimensions, you can’t fit that hypercube in the hyper sphere, even though the diameter of the sphere is twice that of a side of the cube.
I have a HN request where someone out there will know the answer as relates to your comment.
Somewhere, in the past, there's a "popular science" level essay on modern string theory that casually discusses the volume of radius ball and similar geometric implications as regards modern string theory in 10 or 11 dimensions as opposed to bosonic 26 dimensional theory and of course normal human scale 3-D. For example, if, hypothetically, in a very thought experiment manner, you wanted to squirt ping pong ball shaped "things" between packed protons in 3D, the gap would max out at X length whereas in 11D you could squoosh a somewhat larger ping pong ball between the packed gaps or was it actually a denser pack, whatever. Yeah I'm well aware the physics doesn't work that way but it was more of an imaginative geometrical essay.
Sounds like the kind of thing you'd find in an 80s or older "mathematical recreations" column in Scientific American magazine, but its not, or I didn't find it in an index. Which doesn't necessarily mean my inability to find it proves its not there, LOL. But that comparison does accurately relate the general level of casualness and length of the essay and being of general interest to the general educated scientific public.
It was just a fun read, a decade or three ago, and I would merely enjoy reading it again if its free on the net.
Its a typical search problem, searching for terms like "string theory essay" is going to return too much whereas overly specific searches return nothing. So there's some magic middle level of search term complexity that would find it; no idea what those search terms would be, mildly curious HOW anyone finds the article as much as I'm curious about re-reading it.
If nothing else this would be an entertaining 2020's physics blog poster topic or maybe a future xkcd comic LOL.
For example, if you peel away the outer 1% of a n-dimensional sphere or cube, you’re left with 0,99^n of its volume. As n goes to infinity, that goes down to zero.
See also the curse of dimensionality (https://en.wikipedia.org/wiki/Curse_of_dimensionality), or https://www.math.ucdavis.edu/~strohmer/courses/180BigData/18..., which, for example, compares the n-dimensional sphere with radius 1 to n-dimensional hypercubes with side length 1. In two and three dimensions, the cube easily fits in the sphere; in four, it just fits. In higher dimensions, you can’t fit that hypercube in the hyper sphere, even though the diameter of the sphere is twice that of a side of the cube.