  Here are two problems. First, understand the dynamics of a tiling billiard in
a cyclic quadrilateral periodic tiling. Second, describe the topology of
connected components of plane sections of a centrally symmetric subsurface $S
\subset \mathbb{T}^3$ of genus $3$. In this note we show that these two
problems are related via a helicoidal construction proposed recently by Ivan
Dynnikov. The second problem is a particular case of a classical question
formulated by Sergei Novikov. The exploration of the relationship between a
large class of tiling billiards (periodic locally foldable tiling billiards)
and Novikov's problem in higher genus seems promising, as we show in the end of
this note.
