Often in basic mathematics, we can visualize things very easily, which I believe helps understanding (instead of just working out a number theoretical proof). For example: $$(n+1)^2 - n^2 = (n+1) +n$$

can be visualized by squares. Remove a square with sided $n$ from a square with sides $n+1$ leaves the top row ($n+1$) and the right row without the top ($n$) (done here with diamonds and bullets for $n = 4$).

$$ \diamond \diamond \diamond \diamond \diamond \\ \bullet \bullet \bullet \bullet \diamond \\ \bullet \bullet \bullet \bullet \diamond \\ \bullet \bullet \bullet \bullet \diamond \\ \bullet \bullet \bullet \bullet \diamond $$

Another example is proving that $$\sum_{i = 1}^n 2\cdot i = n^2 + n$$ which can be done in the following way (for $n = 4$):

$$ \diamond \diamond \diamond \diamond \\ \diamond \diamond \diamond \bullet \\ \diamond \diamond \bullet \bullet \\ \diamond \bullet \bullet \bullet \\ \bullet \bullet \bullet \bullet $$

Here, we see two triangles, the one with diamonds with row lengths from $1$ to $n$ and the one with bullets going from $1$ to $n$, which represents the sum. We also see a $(n+1) \times n$ rectangle, which represents the right hand side. This proves the theorem.

I was working through same basic number theory proofs and induction proof because I like to visualize these. It is easy enough to visuale $n^a$ as an $a$-dimensional cube with sides $n$. The problem is that I have often difficulty to visualize the factorial: $n!$

Does anybody know of a nice way to visualise the factorial?

The best I could come up with is the following:

See $2!$ as just two dots $\bullet \bullet$.

See $3!$ as a triangle with the sides made with $2!$, e.g. $$ \cdot \\ \bullet \quad \bullet \\ \bullet \quad \quad \bullet \\ \cdot \space \space \bullet \bullet \space \space \cdot $$

Now see $n!$ as an $n$-gon with the sides made of the $(n-1)$-gon. (So $4!$ would be a square with a $3!$-triangle on its sides.)

This visualization is not very easy to work with when you want to visualize proofs. Are there better ways to visualize $n!$?

EDIT: I should emphasis that I would like to visualize $n!$ using dots or lines or so, not so much with concepts ( it is definitely easier to understand the factorial using permutations, just as it is easier to prove some statements using algebra, however the point is that I am trying to prove these things using these very concrete and real visualizations.)