In this question I try to colour infinite grid paper.

There are $k$ colours and $N$ patterns (pattern is a $2\times 2$ square that coloured some way).
The colouring $C$ is called the "correct" if every $2\times 2$ square in it is a pattern.

Suppose that there is correct coloring on the infinite grid plane. It seems that in this case there is a periodic correct colouring (i.e. there are $u,v$ such that for any $x,y$ cells $(x,y), (x+u,y), (x, y+v) $ are coloured same way), but I failed to prove that.

Is it true?