How to intuitively understand the $n$-dimensional cube as the dimension grows large

Question

So I read^* that for the convex body, i.e. the cube $[-1,1]^n$ in $\mathbb{R}^n$, the smallest ball containing it has radius $\sqrt{n}$, while the largest ball inside the cube has radius $1$.

Also,

"...as the dimension grows, the cube resembles a ball less and less."

How do I visualize these things when $n\geq 4$? I just can't see it!

It'd be great if I could get some help with the intuition involved here. Thanks!

^*See page 2 of

Keith Ball, "An elementary introduction to modern convex geometry" in Flavors of Geometry, Silvio Levy ed., Cambridge 1997.

Edit: While the suggested answers are very good, I don't think they address the particular geometric structure I am concerned with in my question.

Possible duplicate of : https://math.stackexchange.com/questions/2286180/visualizing-the-4th-dimension/2286207 or https://math.stackexchange.com/questions/2125036/how-can-i-visualize-a-four-dimensional-point-inside-a-schlegel-diagram-of-a-tess — Martin Hansen, Dec 20 '20 at 07:46
Does this answer your question? Visualizing the 4th dimension. — Martin Hansen, Dec 20 '20 at 07:48
Here's one I always thought was fun: take a square of side 2 and divide it into four squares of side 1. inscribe a circle in each of these four squares. Now draw the circle with the same center as the square of side 2 that is tangent to the four other circles. In the analogous 9-dimensional construction the sphere at the center is tangent to the (8-dimensional) sides of the 9-dimensional cube of side 2; in dimensions 10 and higher, the sphere at the center lies partly outside the cube of side 2. In two dimensions, the sphere at the center has about 3% the area of the cube... — Will Orrick, Dec 22 '20 at 12:53
... of side 2. This fraction shrinks until dimension 264, after which it increases, exceeding 1 for the first time in dimension 1206. — Will Orrick, Dec 22 '20 at 12:54
The most closely related questions I've been able to find so far are Clarification on higher dimensional unit sphere/unit cube properties and Where is the mass of a hypercube?. Neither, in my opinion, is a duplicate of your question. Nor are the two questions previously linked. — Will Orrick, Dec 24 '20 at 18:15
What's new in higher dimensions? is also worth looking at. One of the answers there mentions the construction in one of my earlier comments. — Will Orrick, Dec 24 '20 at 18:33

score 2 · Accepted Answer · answered Dec 20 '20 at 09:15

What makes you think that we can visualize higher cubes and spheres? For $n=4$ you might play games like using some sort of time slider to draw the intersection of your object with the $xyz$-hyperplane, but for $n>4$ those kind of hacks will become unavailable very quickly.

The intuition behind facts like the ones you quote, isn’t intuition but computation. In some sense mathematics builds around our intuition for 2, 3 or maybe even 4-dimensional space, by which I mean that most definitions mimick something in this low dimensional worlds. Yet the definitions are vastly more general in the way that the dimension is inessential, so we might as well try to find out what they do in higher dimensions (thinking of manifolds). It is a pity for sure that we can’t see what is happening there, because sure enough things start to break down. Manifolds become unsmoothable or have multiple distinct smooth structures, classification results are impossible to obtain and spheres become pointy and computationally start to look and behave rather alien. To state one example: The Poincare-conjecture was one of the millenium problems (ie was on the same level of difficulty than the Riemann hypothesis or $P$ vs $NP$) and was about $3$-spheres. Higher geometry is hard.

On the other hand this is the whole fun about abstract mathematics. Intuitive definitions derived from a small collection of examples soon enough turn out to have more exotic but interesting instances, which makes the definition even more interesting and worthy of studying.

How to intuitively understand the $n$-dimensional cube as the dimension grows large

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