4 ms·

> 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 mak

by schoen 4y ago

> 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!

infogulch 4y ago
Would unit-balls be a better concept for describing volume since it doesn't exhibit these unintuitive blow-up features?
- schoen 4y ago
  Sorry for the late reply, but I just saw this featured picture on Wikipedia https://en.wikipedia.org/wiki/Main_Page#/media/File:Mars_topography_(MOLA_dataset)_with_poles_HiRes.jpg https://en.wikipedia.org/wiki/Main_Page#/media/File:Mars_top... 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").
dalke 4y ago
> 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. :)
codeflo 4y ago
> 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.
rsfern 4y ago
> 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?
- codeflo 4y ago
  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.
  - rsfern 4y ago
    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
    - enriquto 4y ago
      > 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.
dreamcompiler 4y ago
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.

by schoen 4y ago

infogulch 4y ago

Would unit-balls be a better concept for describing volume since it doesn't exhibit these unintuitive blow-up features?

schoen 4y ago
Sorry for the late reply, but I just saw this featured picture on Wikipedia https://en.wikipedia.org/wiki/Main_Page#/media/File:Mars_topography_(MOLA_dataset)_with_poles_HiRes.jpg https://en.wikipedia.org/wiki/Main_Page#/media/File:Mars_top... 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").

dalke 4y ago

> 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. :)

codeflo 4y ago

> 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.

rsfern 4y ago

> 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?

codeflo 4y ago
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.
- rsfern 4y ago
  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
  - enriquto 4y ago
    > 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.

dreamcompiler 4y ago

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.