Number of faces visible in different dimensions [duplicate]

Question

In 3D, if I look at a cube, at most I can see three faces at one time from any given perspective, e.g. by looking at a corner dead on.

In 2D, if I look at a square and my perspective is in the plane determined by the square, at most I can see two "faces" (lines).

In 1D, if I look at a segment (which is I suppose the notion of a square/cube for 1D), at most I can see one "face" (the segment endpoint).

My question is, how does this concept generalize to an $n$ dimensional hypercube? Does it grow linearly as the pattern suggests?

Answer 1

In $n$ dimensions the hypercube will have $2n$ faces with normals in the directions

$$(\pm 1,0,0,0, \ldots ,0),$$ $$(0,\pm 1,0,0, \ldots ,0),$$ $$(0,0,\pm 1,0, \ldots ,0),$$ $$\vdots$$ $$(0,0,0,0, \ldots ,\pm 1)$$

You can only 'see' one of each of the +/- pairs as if one is directed towards the observer the one will be facing away (unless you are inside the hypercube), so cannot see more than half the faces.

But an observer at point $(x,x,x,x, \ldots, x)$, with $x > 1/2$, will always be able to see all $n$ of the positive directed faces of a unit hypercube.

So, yes - the number of faces grows linearly, and you can see half of them from the right direction.