Just think of the high dimension n-cube like a spiky sea urchin. It has 2^n spikes, and the spheres live in those spikes near the ends. The central sphere is large because it extends out to those spheres, extending outside the sea urchin's "body".
But ... it's not. It's not concave anywhere. If you draw a line from any point of the n-cube to any other point, it never passes outside the body of the cube. Perhaps your model gives better intuition in "curse of dimensionality" cases like this one, but it's clearly worse in other ways, right? It's simply not at all an accurate description of the shape.
It depends on how you define "concaveness". A cube is concaver than a square in a sense. The travel from the center of a unit square to its side takes 0.5, and to its vertex ~0.7. For a cube, 0.5 and ~0.87. For a 100d cube, 0.5 and 5. (For completeness, for 1d cube it's 0.5, 0.5). Of course that is not a true concaveness, but it gives a nice sense of inside distances, especially with spheres, which are defined as equidistant.
We are just used to project a cube in a way that prevents to see its linearly-spatial configuration (a projection messes up lengths), but if you preserve these lengths and "flatten" them instead, a cube will flatten out to a sort of a shuriken.
> It depends on how you define "concaveness". A cube is concaver than a square in a sense.
Yup. Even a regular 3D cube (and 2D square) has concave faces if you're viewing it from a polar perspective. As I stand in the center of the cube, measuring distances from me to the surface, I'll see that measurement follow a concave pattern.
Yeah, I know that's not the definition of concavity or whatever, but when relating a sphere to a cube, and trying to get an intuition of higher-dimensional spaces, I think it helps to look at it from the sphere's perspective rather than a Cartesian one.
Thank you! This comment solidified for me that the increasing ratio of distances from origin to vertex/side together with the fact that the hypercube surface is still not curved in any way whatsoever is the major thing my spatial intuition (unsurprisingly) struggles with.
There’s a certain way in which a cube “feels sharper” or “feels spikier” than a square. Trying to formalize that, you can compare the edge of a 3d box where two faces meet to the point where three faces meet. I’d rather step on the two-face edge than the three-face corner, and there’s definitely a sense in which the cube is spikier.
It seems reasonable to extend that same intuition to n-D sharpness/spikiness in an accurate way. Adding an extra “face” just chops more off making those vertices sharper and sharper, at least relative to the high dimensional space around it.
I think that's a great insight; I especially like comparing the sharpness of an two-face edge to a three-face corner; we could expect stepping on a seven-face corner to be slightly worse than stepping on a six-face. I think that "sharpness" idea surely must be related to this phenomenon. However, one should be careful not to let that "increasing sharpness" idea lead to the mental image of a concave shape, especially not a sea urchin. That would be a false resolution to this "paradox" and that image of a N-cube would lead to all sorts of other incorrect ideas, e.g. regarding where the volume of the shape is.
Maybe the right intuition is a sea urchin that is, due to high dimensional unintuitive properties, still convex. It’s almost entirely extremely pointy corners, and yet they’re “magically” connected to each other in an entirely convex manner.
It works if you aren't rigid about your mental representations. Pull an inverse wittgenstein on the intuitions. Instead of stool, chair, recliner all being instances of the general category "chair" -- sea urchin, cube, dodecahedron are all partially accurate descriptions of the specific 10D cube.
Maybe? I'm not claiming there's no way to have a good intuition about 4D space -- in fact articles like this make me want to figure out how to achieve such a thing. But it seems likely to me that even if your brain is somehow capable of visualizing 4D things, it would be just as weird to move to 5D as it is for normal people to move to 4D. Did Stott have any special intuition about 5+D? And we're talking about making that cognitive jump five more times to get to 10D.
However, it's clear the "starfish" intuition is simply not accurate. That's not what N-cubes look like. The point of this post is that we should have cognitive dissonance when we try to think about 10-cubes, because it's weird that (A) the "inner sphere" pokes out of a shape that is (B) convex everywhere. You can resolve the cognitive dissonance easily by simply ignoring or rejecting B -- sure, it's not weird that such a sphere would poke out of a starfish. But you are wrong. It's not a starfish! It's convex everywhere! So you can't say "why do y'all have cognitive dissonance about this?"
It is accurate, just not completely accurate. You only get cognitive dissonance if you try to resolve it all the way.. stack multiple imperfect intuitions to approximate the real thing.
