Geometrical interpretation for the sum of factorial numbers
I am in need of a way to represent the sum
$1! + 2! + 3! + 4! = 1 + 2 + 6 + 24 = 33$
in a geometrical way. What I mean by this is that for example, the sum
$1^2 + 2^2 + 3^2 + 4^2 = 1 + 4 + 9 + 16 = 30$
can be represented geometrically as a pyramid with layers consisting of 1, 4, 9 and 16 pieces respectively a regular manner. Image from Wikipedia to illustrate the geometrical construction of the square numbers.
I have tried to find such a regular pattern to construct a geometrical shape from the factorial numbers, but to no avail. How could this be done?
Also, a follow up question: Is there a way to represent the factorial numbers up to an arbitrary number $n!$, instead of ending at $4!$ as stated in this question above? (less important, but interesting nonetheless)
Thanks in advance!
EDIT: The probably most important part is that the 1, 2, 6 and 24 are discrete and somewhat separated from each other, kind of like the different layers in the comparison between te sum of squares (see linked image above).
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$\begingroup$The only idea I could come up with was counting the number of vertices in a tree graph that had branching ratios of $2, 3, 4, \ldots$, so that the number of vertices on each level were $1!, 2!, 3!, 4!, \ldots$.
If you need more layers ($n=6$), you might want a different layout (thanks to @HenrikSchumacher):
Radial embedding is particularly elegant and helpful too ($n=6$):
Perhaps such a three-dimensional representation would be appropriate:
The answer to your question "Also, a follow up question: Is there a way to represent the factorial numbers up to an arbitrary number n!, instead of ending at 4! as stated in this question above?" is:
$$\sum\limits_{n=1}^k n! = (-1)^{k+1} \Gamma (k+2) \text{Subfactorial}[-k-2]-\text{Subfactorial}[-1]-1$$
$\endgroup$ 7 $\begingroup$Polynomial functions of degree $d$ can be represented in a space of $d$ dimensions, using segments, squares, cubes then hypercubes. For example, the square pyramidal numbers can be sketeched in 3D as a stack of squares.
This does not generalize to factorials as they are of "unbounded degree" and would require an unbounded number of dimensions.
Even if you find a trick to limit the number of dimensions, the value of the numbers quickly becomes unmanageable ($10!=3628800$).
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