24. The Tesseract and beyond

Let $C_n$ be an $n$-dimensional cube (hypercube) of side length $a$.

Determine, in terms of $n$ and $a$, all of the fundamental geometric and combinatorial properties of $C_n$.

Your solution should determine and justify:

1. The number of vertices, edges, and $k$-dimensional faces for every $0\leq k\leq n$.

2. The number of $(n-1)$-dimensional facets and their geometric structure.

3. The number of diagonals of each possible type, the length of each type, and the number of diagonals of each type.

4. The total number of diagonals.

5. The number of $k$-dimensional faces incident to each vertex, edge, or higher-dimensional face, where applicable.

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$Optional$: Determine any other natural geometric or combinatorial properties of $C_n$ that admit a meaningful generalization to arbitrary dimension. This part is not included in the marks and may be used to earn a starred solution.