As we know, the tangent function repeats every $\pi$, so it would make sense for $f(x)=\sum_{n=0}^{\lfloor x\rfloor}\tan n$ to be wildly erratic, since no two $\tan\lfloor x\rfloor$ for different $\lfloor x\rfloor$ will be the same, and it's fairly erratic what $\lfloor\pi \lfloor n\rfloor\rfloor$ is(yes I know it's fairly simple, but based on all the values from $x={1,2,...,n}$, it's hard to tell what $x=n+1$ is). But what is seen is tiny inverted humps making small humps making much larger inverted humps that increase in a way that makes me think it'll make another hump, but it would be massive so I'm not sure, since Desmos refuses to calculate this after x=25,000, which is over a trillion operations and the free version of WolframAlpha stopped at x=1,300.
Why does this happen? (The fractal pattern where the larger shape is made up of smaller inverted shapes.) And does this pattern continue?
I'm not sure, but it seems that the smaller shapes in the fractal often differ from each other from each other quite a bit, while the larger the shape, the less off it is, which leads me to believe that this won't show any underlying properties of $\pi$ and the tangent function and their relation to the integers..
Here are some clues.
Notice that, since $22$ is very close to $7\pi$, $\tan(x+22)$ is quite close to $\tan(x)$ for many $x$. This is the cause of the most fundamental near-periodic nature in the plots.
Added: Also adding to this is the fact that $\tan(22k+x)$ is approximately $-\tan(22k-x)$ for integer $k$, so we've got lots of near cancellation happening. This is why the smallest little U-shapes start and end at around the same $y$ values.
Similarly, using continued fractions to approximate $\pi$, we find $355$ is very close to $113\pi$, and so we have a larger scale near-periodic feature in the graph, too.
Added: Also, we can notice that $52174$ is very close to $16607.5\pi$, and that $\tan 52174$ has the extraordinarily large value of $-181570.29570$, resulting in the massive downward jump in the thid plot.
