How many points are needed to uniquely specify a box? In $\mathbb{R}$, to specify a line segment, we need two points. In $\mathbb{R}^2$, to specify a given rectangle, all we need are three points. Then the fourth point is determined, and there is only one rectangle with those three points as vertices. In $\mathbb{R}^3$, to specify a cuboid, we can do it with just $4$ vertices (e.g. pick any vertex and its three adjacent vertices). Then there is only one cuboid that can have these $4$ vertices.
In general, what is the minimum number of points/vertices needed to uniquely specify a given box in $\mathbb{R}^n$? Is it $n + 1$?
 A: In general, a parallelotope (the generalization of parallelograms to many dimensions) can be specified by $n+1$ points: a vertex $u$ and all the vertices $v_1,\ldots,v_n$ adjacent to that vertex - and, as long as long as there is no hyperplane on which all vertices lie, any collection of $n+1$ vertices really does extend to a parallelotope (since, essentially, one can just take the Minkowski sum of the edge vectors from $u$ to each $v_i$ to get a volume). 
It seems like a box, for you, is just a parallelotope in which the faces are perpendicular to one another - they can be specified the same way, except they additionally require that the edge vectors from $u$ to each $v_i$ be pairwise perpendicular - which doesn't really bring down the number of vertices required, but it means that, if we specify $u$ then $v_1$, we would know that $v_2$ lies on some specific hyperplane, then that $v_3$ lives on a codimension two subspace, and so on - bringing down the degrees of freedom. In general, you have $n$ dimensions of choice for $u$ and $v_1$, then $n-1$ for $v_2$ and $n-2$ for $v_3$ and so on, until you only have a single dimension of choice for $v_n$.
All told, you get $n(n+1)$ degrees of freedom in choosing a parallelotope in $n$ dimensions, but only $\frac{n(n+3)}2=n+(1+2+3+\ldots+n)$ degrees of freedom in choosing a box - which is probably a more meaningful measurement than "number of points required."
A: Specifying a parallelotope can always be done with 3 points in ℝ3 or higher.
A box in ℝ3 has size (width, length, height), position (x,y,z), and rotation (alpha, beta), all of which can be represented with three points.
In ℝ4, there is one additional size value, one additional position value, and one additional rotation value; again this fits in three points, because an ℝ4 point has the extra dimension.
Growth continues proportionally as dimensions increase.  Argument is also valid for ℝ2 and ℝ, but not as interesting.
QED.
An alternate way to specify an ℝ3 box with three points: Start, End, and Size.  Start and End specify one vertex of the box.  The Size point is defined to be on the plane orthogonal to the vertex containing End.  Thereby constrained, the dimensions, position, and orientation of the box are defined.
