The $n$-dimensional analogue of a square or cube is a hypercube. A side-$2R$ hypercube centered at the origin is

$$[-R,R]^n.$$

It has

$$2^n$$

vertices, one for each choice of signs in

$$(\pm R,\ldots,\pm R).$$

Distance to a corner

Every vertex is at distance

$$R\sqrt n$$

from the center because

$$\sqrt{R^2+\cdots+R^2}=R\sqrt n.$$

The cube therefore extends increasingly far from its center along diagonal directions as dimension grows, even though each coordinate remains between $-R$ and $R$.

A counterintuitive packing example

Place equal balls at the vertices of a side-$2$ hypercube and ask how large a centered ball can be before it touches them. The distance from the center to a corner is $\sqrt n$, so the gap after subtracting a unit corner-ball radius grows like

$$\sqrt n-1.$$

In sufficiently high dimension, a centered ball can therefore become surprisingly large relative to the cube's side length.

Where the volume is

High-dimensional cubes have exponentially many corner regions. Much of their geometry is unlike our three-dimensional intuition because adding dimensions continually introduces new mutually perpendicular directions.