What is the maximum number of faces of totally convex solid that one can "see" from a point?
…and, more importantly, how can I ask this question better? (I'm a college student with little experience in asking well formed questions, much less answering them.)
By "see" I mean something like this: you point a camera from a point at the solid, and look at the picture. How many of the faces of the solid look like faces and not just lines? Let's assume that the lens is just a point in space (no lenses wider than the solid itself) and that the camera is a finite distance from the solid. I know this is a crude definition… if you have any ideas for a more rigorous definition, this would be awesome, then maybe there's ways to prove the answer to my question mathematically.
For example, in the picture of this cube, you can see 3 faces. This is the maximum you can see for a cube. How can that be proved?
What methods might you use to prove this for a convex solid of any size and shape? Are there ways to do so using only relatively basic math (Multivariable calc, linear algebra, high school geometry)?
Best Answer
You can always find a place from which to see at least half the faces.
To see why, start by considering a polyhedron with central symmetry. Imagine a viewpoint from which you don't see any lines as points or faces as lines (i.e. general position) and far enough away so that you can see all the faces whose normal points into your side of the half plane perpendicular to your line of sight. Then think about what you see from far enough away in the opposite direction. You can see all the faces from one side or the other and no face from both sides, so the symmetry says you see half each time.
Four of the five regular polyhedra have a center of symmetry. The tetrahedron does not: there's no place to put the origin that allows invariance under the map $x \to -x$.
Even without central symmetry, you see all the faces from one side or the other, so you see at least half from at least one side. Pyramids represent an extreme case. You can see all but one face from one direction and just one from the other, as @almagest points out in a comment.
Since the polyhedron has only finitely many faces, "far enough away" in the preceding proof does not have to be at infinity (though it may be pretty far). As @JohhHughes comments, if you put your camera close enough to any face that's the only face you'll see.
Note: the arguments work in all dimensions. They are particularly easy to visualize in the plane. (On the line they're trivial.)