When considering tilings of the plane by the unit cube. That is, a tiling by the unit square $I^2=[0,1]^2$ which covers $\mathbb{R}^2$. Usually one says that the unit cube tiles the plane using translations. But can the same be said for reflections?
That is, does there exist reflection lines in such a tiling of the plane?
I would say yes, almost obviously so. However, can one have an infinite amount of reflection lines, since it tiles a plane which is infinite in all directions, and thus one can have a reflection line at every cube? If that makes sense. Or does one fix the notion of the reflections lines to be centred around origo, as to get "half and half on both sides". In which the cube tiling of the plane would have exactly as many reflection lines as the square, which is four.
We say that the square tiles the plane with translations because you can start with one square and translate it to all the other positions necessary to create the tiling.
After you've done that, the whole tiling will have many symmetries. Some are the translations you used to move the first square. Some are reflections about lines (parallel to the edges of the squares or to the diagonals of the squares). There are rotational symmetries too.
To tile the plane by equilateral triangles you must both translate and rotate or reflect the original triangle. Then you can think about the symmetries of the whole tiling.