I will discuss a famous problem posed by S. P. Novikov in 1982 in connection with the conductivity theory of monocrystals: what one can say about different types of asymptotic behavior of plane sections of a triply periodic surface? Apparently the most efficient approach to this problem is closely related to the techniques used in Teichmuller dynamics. I will try to explain historical perspectives and recent developments as well as challenging questions that still remain open.