In this post we consider as reference the article Arithmetic billiards from Wikipedia. We consider the arithmetic billiard that is explained in the article.
I've wondered if we can compute some simple particular values for the sum of divisors function $\sigma(n)=\sum_{1\leq d\mid n}d^a$ with $a=1$. I think that it could be interesting for other post compute values for the Euler's totient function $\varphi(n)$ (I'm interested also for subsequent posts do computations with refractions and polarization of light rays).
Question. I would like to know if you can to help me to compute more particular values of the sum of divisors function using the definition of billiards and their arithmetic. Below I've added two examples that I think that are easy to get. Many thanks
Example 1. We consider $p\geq 5$ a prime number and our billiard has the form of a capital letter L, these are the column $1\times\varphi(p)$ squared units plus the square unit $\square$. We place the left bottom corner at $(0,0)$ of the plane. Then the corner that is the starting point for our ball is placed at the point $(1,p-1)$. The value $\sigma(p)$ equals to $$3+\operatorname{number}(\text{diagonal trajectories})-2,\tag{1}$$ where one can to motivate that $3$ counts certain regions, and $-2$ is to discount two corner pockets of our billiard. Notice that the ball moves inside the billiard.
Example 2. We consider $p$ and $q$ distinct odd prime numbers and our billiard is a rextangle $q\times p$, our billiard is placed at the corner $(0,0)$ of the plane that is the starting point for our ball at $45º$ angles, again. Then the value $\sigma(pq)=\sigma(p)\sigma(q)$ equals to $$4+2\cdot\operatorname{number}(\text{bounces})+2\cdot\frac{\varphi(p)\varphi(q)}{2},\tag{2}$$ where one can to motivate that $4$ counts certain regions, and $\frac{\varphi(p)\varphi(q)}{2}$ can be interpreted as the number of delimited squares that were drawed by the trajectories of our ball in this billiard (a full turn, from the corner pocket that is the starting point for our ball to the exit corner pocket).
Additionally I tried to compute $\sigma(pqr)$ for pairwise distinct odd prime numbers $p,q$ and $r$, I can to get the closed-form of a term (in terms of the number of certain triangles of a billiard with the shape of a rectangle $p\times qr$, but I cann't get a closed-form similar than $(1)$ and $2$ for $\sigma(qpr)=\sigma(p)\sigma(qr)$.
Remarks. 1) As was said I think that the approach to get particular values for $\sigma(n)$ (I didn't try get values as $\sigma(N^2)$ seems very limited for me, this is my motivation to continue studying billiards with refractions. 2) I don't know if my computations are known from the literature, if this kind of billiard are in the literature to compute the sum of divisors function please add a comment or the reference in your answer.