Let $a/b$ be a rational in $(0,1)$ expressed in lowest terms. The height of $a/b$ is $\max \{a,b\}$. For each rational in $(0,1)$ of height $\le h$, form the sum $a+b$. So, for example, here are the rationals of height $\le 6$ and their sums: \begin{array}{ccccccccccc} \frac{1}{6} & \frac{1}{5} & \frac{1}{4} & \frac{1}{3} & \frac{2}{5} & \frac{1}{2} & \frac{3}{5} & \frac{2}{3} & \frac{3}{4} & \frac{4}{5} & \frac{5}{6} \\ 7 & 6 & 5 & 4 & 7 & 3 & 8 & 5 & 7 & 9 & 11 \\ \end{array} Now form a histogram of these sums. So above, the bin for $3$ gets $1$ count because only $\frac{1}{2}$ leads to $3$, but the bin for $7$ gets a count of $3$ because each of $\{ \frac{1}{6}, \frac{2}{5}, \frac{3}{4} \}$ sum to $7$.
Here is the histogram for $h \le 24$:
You can see the bin for $3$ still has a count of $1$, the bin for $47$ has a count of $1$ for $\frac{23}{24}$, and the bin for $46$ is empty. The tallest bin is for $23$, whose sum is achieved by $11$ fractions: $\{ \frac{1}{22}, \frac{2}{21}, \ldots, \frac{11}{12} \}$.
Here is the histogram for $h \le 256$:
My question is: What explains the features of this plot.
I would appreciate explanations of some of the structure that seems to be emerging in this plot, from the left-right symmetry to more subtle features. Perhaps I am hallucinating structure where there is none, but it seems one can discern a series of nested triangles that roughly demarcate regions of differing density, something like this:
Effectively, you have a histogram of $a + b$ for all $a < b \leq n$ such that $\gcd(a, b) = 1$.
For $a = 1$, we get all $b > 1$.
For $a = 2$, we get roughly half of all $b > 2$.
What does that look like when we add them together?
We continue like this, for $a = 3$, roughly one third, etc.
For $a \leq 5$, we get this:
Does this illuminate the pattern?