The unit “1” in terms of hypervolume is a n-hypercube with side length 1, so the question is really, how many of those hypercubes fit into an n-ball of a given radius.
Now, the thing I realized is that a unit hypercube in n dimensions is an unimaginably weird object. Its side length is, by definition, 1. But as n goes to infinity, the distance between the center and the corners of this unit hypercube goes to infinity — this follows directly from the Pythagorean Theorem. The center of the hyperfaces, on the other hand, is always only 0.5 distance units away from the center of the cube. An inscribed ball touches those hyperfaces, so 0.5 is also the radius of that inscribed ball. This makes the hypercube in some sense very “spiky”, with parts of the boundary close to the center (0.5) and other parts infinitely far away. Also, there are 2^n of those corners.
Indeed, and that’s just another way of stating the submission headline, most of the volume of the hypercube is near those spiky corners. Stated another way, as n goes to infinity, none of the volume of a hypercube is at a bounded distance from the center. And note that we’re still talking about a unit hypercube, so the side length in all of this always remains 1.
So, if you accept the math, the question is: Is the quoted result really something unintuitive about high-dimensional balls, which are perfectly symmetrical objects? Or is the “weirdness”, in terms of human intuition, better blamed on the hypercubes we implicitly compare them to when we talk about volume?
> This makes the hypercube in some sense very “spiky”, with parts of the boundary close to the center (0.5) and other parts infinitely far away.
One way to make the weirdness of the "other parts infinitely far away" concrete is to recall that, say, the Empire State Building can fit entirely inside of an n-cube with sides of 1 cm, for sufficiently large n. The length of the diagonal of such a cube grows without bound, and so does any constant-dimensional cross-section.
Extremely high-dimensional spaces are extremely spacious. (People experience this with ML and statistics, where random points in very high-dimensional data sets are almost certain to be extremely far apart from one another in a Euclidean metric.)
> So, if you accept the math, the question is: Is the quoted result really something unintuitive about high-dimensional balls, which are perfectly symmetrical objects? Or is the “weirdness”, in terms of human intuition, better blamed on the hypercubes we implicitly compare them to when we talk about volume?
That's a terrific way to put it. We can think of the n-balls as bizarrely small (and/or bizarrely round), or we can think of the n-cubes as bizarrely huge (and/or bizarrely spiky). Then we could say that the latter eventually contain absurdly much space, while the former continue to contain only a moderate, reasonable amount of space!
> the Empire State Building can fit entirely inside of an n-cube with sides of 1 cm, for sufficiently large n.
My sole attempt at writing an essay for my local SF fan club, when I was in high school, was to apply this observation to explain ... err .. handwave .. how the TARDIS can be bigger on the inside, eg, as seen in The Invasion of Time.
Of course, back then the TARDIS was only 68,000 tons - about the size of an aircraft carrier. (In Castrovalva, 1/4 of the TARDIS - 17,000 tons - was ejected.) Figuring 400 meters length in a (2m)^n box gives at least n = 40,000 dimensions.
Nowadays it has enough mass to fracture a planet, contains a collapsing star, etc. Life was simpler in the 1980s. :)
> One way to make the weirdness of the "other parts infinitely far away" concrete is to recall that, say, the Empire State Building can fit entirely inside of an n-cube with sides of 1 cm, for sufficiently large n.
That’s a striking illustration. Even the entire Earth fits into a 1 cm n-cube, but — and that’s important — only diagonally.
> the Empire State Building can fit entirely inside of an n-cube with sides of 1 cm, for sufficiently large n. The length of the diagonal of such a cube grows without bound, and so does any constant-dimensional cross-section.
I still find this analogy confusing, because the units don’t match. The Empire State Building has some volume in cm^3, but the hypercube volume is cm^n. 1 cm^5 is not 100 times more volume than 1 cm^3, right?
Am I thinking about it wrong? I guess you can think about filling up a 3D cube with quasi-2D slices, is it like that?
Ah, thinking about the other comments a bit, you mean the diagonal of the hypercube can be larger than the biggest dimension of the building?
It’s not about area, but about actually physically placing an object in a higher dimensional object, like placing a long stick (1D) in a room (3D): the stick can be longer than the longest side of the room, but not by that much. In higher dimensions, this effect is more extreme.
Thanks, I see what you mean now. I think I was keying off the word “entirely” and trying to make it work for all three dimensions of the building at once. It seems like the building will fit along each coordinate inside the diagonals, but the edges of the hypercube would still cut through the building, right? That’s where I was getting confused
> the building will fit along each coordinate inside the diagonals, but the edges of the hypercube would still cut through the building, right?
No, you can fit the whole thing inside the 1mm hypercube, as long as you have enough dimensions. Once you have fit the height of the building along one diagonal, you can add new dimensions and rotate the building along that diagonal so that the width and the height will fit along the diagonals of the other dimensions.
And in that case n is about 1,964,262,400. According to Wikipedia the overall height of the Empire State Building is 443.2m. That's 44320cm. So we need a cube that -- corner-to-opposite-corner -- is that many cm long. If the cube is 1 cm on a side, its corner-to-opposite-corner distance is the square root of n, so the number of dimensions of the cube needs to be about 1.96 billion for [a line the same length as] the Empire State Building to fit inside.
and thought about all of those circular craters, then realized that the circularity of the craters is a consequence of symmetry (and also of conservation of energy). The sphericity of Mars itself is also a consequence of physical principles like that.
So in terms of physical laws and physical phenomena, n-balls show up much more readily than n-cubes, and in some senses have a much simpler description.
If you weren't already on a street grid in a city and using that grid to navigate, a parent might say to a child "don't wander more than one kilometer from home" and would naturally generate a circle rather than a square as the locus of points that are close enough.
Nature and people care a lot about proximity, which is to say distance, which then generates n-balls. So maybe they are more fundamental and more intuitive for certain purposes. But the curse of dimensionality is going to make either balls or cubes weird in enough dimensions, and counterintuitive to intuitions formed with just three dimensions. The balls are going to be surprisingly tiny ("there's very little nearby here") and the cubes are going to be surprisingly vast ("space is so spacious").
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.
Ball packing (no, it's not some new thing Bay Area VCs are doing--I mean, it probably is also that but it's not what I'm referring to here; rather, it's the n-dimensional analogue of sphere packing) gets stranger in n-dimensions as well.
If you have an n-dimensional integer lattice with unit n-spheres at each even-numbered point, you get inscribed n-spheres. For n = 2, those circles are tiny: about 29% (1-1/sqrt(2)) of the radius of the bigger (r = 1) ones. At n = 3, they're somewhat bigger but still small. At n = 4, they're the same size. At n = 5+, the inscribed n-spheres are actually larger than the unit spheres, which means they extend farther despite being "inscribed". This is probably impossible to visualize, because it's never that way in our 3-dimensional world.
High dimensional spaces also make it difficult to come up with good distance metrics. In 1000+ dimensions, nearly all the volume is on the boundary, and neighborhoods in the classical sense (e.g. visible clusters) don't exist. It's trivial to come up with metrics that "work" but it often requires domain knowledge to come up with ones that are useful.
> Is the quoted result really something unintuitive about high-dimensional balls, which are perfectly symmetrical objects?
That reference to symmetry made me thing of another way to look at this: the overlap between two random unit n-hypercubes goes to zero as n goes to infinity (or does it?).
_with_poles_HiRes.jpg){kind=link}
The unit “1” in terms of hypervolume is a n-hypercube with side length 1, so the question is really, how many of those hypercubes fit into an n-ball of a given radius.
Now, the thing I realized is that a unit hypercube in n dimensions is an unimaginably weird object. Its side length is, by definition, 1. But as n goes to infinity, the distance between the center and the corners of this unit hypercube goes to infinity — this follows directly from the Pythagorean Theorem. The center of the hyperfaces, on the other hand, is always only 0.5 distance units away from the center of the cube. An inscribed ball touches those hyperfaces, so 0.5 is also the radius of that inscribed ball. This makes the hypercube in some sense very “spiky”, with parts of the boundary close to the center (0.5) and other parts infinitely far away. Also, there are 2^n of those corners.
Indeed, and that’s just another way of stating the submission headline, most of the volume of the hypercube is near those spiky corners. Stated another way, as n goes to infinity, none of the volume of a hypercube is at a bounded distance from the center. And note that we’re still talking about a unit hypercube, so the side length in all of this always remains 1.
So, if you accept the math, the question is: Is the quoted result really something unintuitive about high-dimensional balls, which are perfectly symmetrical objects? Or is the “weirdness”, in terms of human intuition, better blamed on the hypercubes we implicitly compare them to when we talk about volume?