TITLE: Visualizing the factorial
QUESTION [27 upvotes]: 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.)

REPLY [4 votes]: The way I see $n!$ is a hybrid of avid19's and Dave L. Renfro's visualizations: I imagine $n$ people lining up one by one. I think it really helps to imagine people or animals or fruits or something, rather than boring symbols: that's the way it's done in Burns and Weston's Math For Smarty Pants, and it seems to have made quite an impression on me. My keyboard has no fruits on it, unfortunately, so maybe try to imagine the digits below pinned to some hockey players.

The first person doesn't have any choice about where they join the line, since there is no line yet.

1

The second person can join in two places: the front or the back.

21 12

The third person can join in three places: the front, the middle, or the back.

321 231 213
312 132 123

The fourth person can join in four places.

4321 3421 3241 3214
4231 2431 2341 2314
4213 2413 2143 2134
4312 3412 3142 3124
4132 1432 1342 1324
4123 1423 1243 1234

The fifth person can join in five places...