Roots of Bernoulli polynomials - a pattern Contemplating a question on math.SE, I have stumbled on this:



Here, the point labeled $n$ is that root of the $n$th Bernoulli polynomial which has smallest positive imaginary part.
Does anyone know an explanation of this pattern? Specifically, some periodicity modulo 5 must be involved somehow.
 A: It is known that $B_n(x)$ has $\asymp \frac2{\pi e}n$ real roots (see Veselov, A. & Ward, J. On the real roots of the Bernoulli polynomials and the Hurwitz zeta-function) Probably it means that the first "positive" complex root becomes real after each $\frac{\pi e}2=4.27\ldots$ steps.
A: For the limiting curve formed by the complex zeros of the Bernoulli polynomials, see Goh, Boyer, On the Zero Attractor of the Euler Polynomials (Adv. Appl. Math. 2007, doi:10.1016/j.aam.2005.05.008). Actually, the authors treat the closely related Euler polynomials and say that Bernoulli polynomials "are easily handled with the techniques in this paper." The connection is made explicit on page 21 of http://www.math.drexel.edu/~rboyer/talks/MIT_FINAL.pdf ; the limiting curve for Bernoulli polynomials is half that of Euler polynomials. Perhaps the answer to your question can be obtained from this curve.
A: (too long for a comment)
I think there is nothing happening modulo 5, at least not in a number-theoretical sense. It is first of all the visual impression because of the fact that most of the segments linking a point $P_n$ to $P_{n+5}$ have a slope close to 0 (about $-.06$), so they are geometrically closer than, say, $P_n$ and $P_{n+6}$, meaning that we easily perceive those patterns as "almost horizontal" lines (depending on the scaling), slightly descending.
IMO what is more striking is the fact that the real parts show a "very close to linear" progression of about 1/17 from $P_n$ to $P_{n+1}$, and that every 5 of them (from time to time 6), the imaginary part, usually increasing, "jumps" back to the bottom. The top points (11, 32, 53, ...) are 21 apart (from time to time 26), with the points  116, 184, 231..., which "should" be on top but cause a distance 26 instead of 21, jumping even lower. This reminds me a lot of what happens when projecting a sinusoidal curve wrapped around a cylinder (Lissajous curves): If you choose points on the sinus curve with linearly progressing x-coordinates, their projections will behave very similarly as the $P_n$'s here.
There should be some constant $\approx 5.2$ for the average period of the jumps, and so the main question would be how to find this constant, which does not seem to be related to $\frac{\pi e}2$.
