I have found interesting problem in Gilbert's Strang book, ,,Introduction to Linear Algebra'' (3rd edition):
How many corners does a cube have in 4 dimensions? How many faces? How many edges? A typical corner is $(0,0,1,0)$
I have found the answer for corners:
We know, that corner is $(x_1,x_2,x_3,x_4)$. For every $x_i$ we can use either $1$ or $0$. We can do this in $2 \cdot 2 \cdot 2 \cdot 2 = 2^4 = 16$ ways.
The same method can be used for general problem of cube in $n$ dimensions (I suppose):
Let's say, we have $n$-dimensional cube (I assume, that length of edge is $1$, but it can be some $a$, where $a \in \mathbb{R}$ [1]). Here, corner of this cube looks like this: $(x_1,x_2, \ldots , x_n)$. For every $x_i$ there are $2$ possibilities: $x_i = 0$ or $x_i = 1$ ($x_i = a$ in general). So, this cube has $2^n$ corners.
It was pretty simple, I think. But now, there are also faces and edges. To be honest, I do not know, how to find the answer in this cases.
I know, that solution for this problem is:
A four-dimensional cube has $2^4 = 16$ corners and $2 \cdot 4 = 8$ three-dimensional sides and $24$ two-dimensional faces and $32$ on-dimensional edges.
Could You somehow explain me, how to figure out this solution? I have found solution for corners by myself, using Linear Algebra methods & language. Could You show me, how to find the number of edges and faces, using Linear Algebra methods?
Is there other method to find these numbers? (I suppose, that answer for this question is positive)
I am also interested in articles/textbooks/etc. about space dimensions, if You know some interesting positions about that, share with me (and community).
As I wrote: I am interested in mathematical explanations (in particular using Linear Algebra methods/language but other methods may be also interesting) and some intuitions (how to find solution using imagination etc. [2]).
Thank You for help.
[1] I am not sure of this assumption, because:
(a) I am not sure, how edges (and faces) behave in $n$ dimensions
(b) I am not sure, how should I think about the distance in $n$ dimensions. I mean, I know, that my intuition may play tricks here
[2] I am not asking, how to imagine $4$ dimensional cube, but I think, that there is a way to find the solution, using reasoning, not only Linear Algebra.
Addition
My definition of face (there was a comment about that) is the same as definition here: http://en.wikipedia.org/wiki/Face_(geometry), especially:
In geometry, a face of a polyhedron is any of the polygons that make up its boundaries.