Visualizing the factorial

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.)

One way is the total number of leaves of a (single) rooted tree in which each leaf is minimally linked to the root by exactly $n-1$ edges, and which has the following property: the root has $2$ children, each child of the root has $3$ children, each child of each child of the root has $4$ children, and so on until the leaves are reached. A natural term for this is factorial tree, but I don't know if this phrase is in general use for this notion.

For example, for $n = 4$:

Here's a geometric visualization in higher dimensions. You can take a hyper-cube in dimension $d$ (basically the Cartesian product of $n$ copies of the interval $[0,c]$ for any $c > 0$ that you want), and then you triangulate (i.e. partition) into equal volume simplices (a simplex in $d$ dimensions is a full dimensional convex hulls of $d+1$ points, i.e. higher dimensional analog of triangles for $d = 2$) by first drawing the edge from the origin to the opposite corner of the hypercube (so the opposite corners are vertices included in each simplex), and then move along one edge of the cube incident to the origin to get your next vertex, then move closer to the opposite corner by taking one edge incident to that vertex to get the next vertex, and so on until you reach the opposite corner. You can traverse the dimension-aligned edges in any order you want to get $d+1$ vertices of a distinct simplex, and the interiors of the simplexes are disjoint, and the number of congruent simplexes you get in this partition is equal to the number of ways you can order the dimensions, which is $d!$. Thus, if $c = 1$, then each simplex in this partition has volume $1/d!$ and they are all congruent.

A related construction is to consider the volume of the simplex whose vertices are the origin along with the endpoints of $d$ linearly independent vectors $v_i$ extending from the origin. This solid has the description $\{ \sum_i c_i v_i \, | \, \sum_i c_i \leq 1, c_i \geq 0$ } where the $v_i$ are your vectors. The parallelepiped (analog of hypercube) spanned by these vectors $v_i$ on the other hand has the description $\{ \sum_i c_i v_i \, | \, 0 \leq c_i \leq 1 \}$. It is a geometric fact that the volume of the parallelepiped is $d!$ times the volume of the simplex, and the volume of the parallelepiped is $|\det V|$ where $V$ is the matrix of the vectors that span the parallelepiped.

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...

This might not be what you're looking for, but I visualize a factorial as a process. $5!$ is how many ways you can arrange 5 things. I visualize arranging 5 things. Not a representation as dots but personally it's powerful.