Suppose I have n orthogonal unit vectors (in Euclidean space). These unit vectors may be used to describe an n-dimensional unit hypercube. A subset of these unit vectors may be combined to make lower dimensional faces on the n-dimensional unit hypercube. What I mean is that one or more of these unit vectors may be used to define $a_1$ 1-D edges, $a_2$ 2-D faces, $a_3$ 3-D cells, etc. (up to $a_{n-1}$ $(n-1)$-dimensional faces). For example, in 3D, you may take the x,y,z unit vectors to form three 1-D edges (along the three basis vectors) and three 2-D faces (in the xy, xz, and yz plane).
My question is: is there a rule that allows me to make an n-dimensional lattice using these lower dimensional faces (the $a_1$ 1-faces, the $a_2$ 2-faces, etc)?
By inspection, I may draw a 6x6 lattice in two dimensions as a collection of 36 letter L's (one 0-D corner + two 1-D edges) and one big rotated letter 'L' to close the figure. Similarly, I may make a $k^3$ cubic lattice as a combination of $k^3\times$ (one 0-D corner $+$ three 1-D edges $+$ three 2-D faces) plus one large figure similar to the others (3 faces) rotated to close the lattice.
I think the solution is that I can build a $k^n$ (n-dimensional) lattice from the $a_1$ 1-D edges (the n orthogonal unit vectors), the $a_2$ 2-D faces that these unit vectors can describe a basis for, the $a_3$ 3-D cells that these unit vectors can describe a basis for, etc., and than have one identical (but large) figure with the same number of 1-faces, 2-faces, etc. rotated to close the lattice. I figured this out, but would like to hear from people who may have worked on this problem more than I (and I haven' found a proof this is even a solution one way or the other yet).
If this line of reasoning is correct, that I may describe a $k \times k \times k \times k$ (4-D) lattice as a collection of $k^4 \times$ (one 0-D face $+$ four 1-D faces $+$ six 2-D faces $+$ four 3-D faces) and one large (one 0-D face $+$ four 1-D faces $+$ six 2-D faces $+$ four 3-D faces) rotated to close the figure.