# 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)

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).

• Why not use a triangle? Place a column of $1$s at the right. Starting from the second row, place a column of $2$s to the left of the column of $1$s. Repeat the column construction until you get to $n$ which should be a column of length one. – John Douma Nov 30 '18 at 19:45
• @JohnDouma I am not quite sure how this would produce something connected to factorial sums? Thank you for your comment. Would I have first 4 1:s, then 3 2:s, 2 3:s, 1 4:s? That makes a sum of 20. – Sigfrid Stjärnholm Nov 30 '18 at 20:13

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:

$$\sum\limits_{n=1}^k n! = (-1)^{k+1} \Gamma (k+2) \text{Subfactorial}[-k-2]-\text{Subfactorial}[-1]-1$$

• OP wants to know geometrical interpretation. Is there any geometrical interpretation using this? Please explain :) – tarit goswami Nov 30 '18 at 19:35
• In no way does this answer OP's question. – MPW Nov 30 '18 at 19:39
• @MPW: "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? " – David G. Stork Nov 30 '18 at 19:40
• Thank you for your response! This gives me the sum of the factorial numbers up to a certain point k, and in my case I want k = 4. I see your argument that it would be hard to find such a geometric interpretation of factorial sums, but if we limit the value of k to 4, only considering 4!, is it not possible that for such small values of factorial there would exist a beautiful and simple interpretation as it did for the squares? – Sigfrid Stjärnholm Nov 30 '18 at 20:02
• @SigfridStjärnholm: Well, not precisely a fractal because the branching ratio changes at each level... hence the graph is not self-similar. – David G. Stork Nov 30 '18 at 22:46

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 ($$10!=3628800$$).

• Ah, i see! Would there however be some sort of way to represent just up to 4!, if ignoring the general case? – Sigfrid Stjärnholm Nov 30 '18 at 19:59