on/to the other side of (a line): across and over
Across and over can both be used to mean ‚on or to the other side of a line, river, road, bridge, etc’.
- His village is just across/over the border.
- See if you can jump across/over the stream.
high things: over preferred
We prefer over to say „on/to the other side of something high“.
- Why are you climbing over the wall? (not across the wall?)
flat areas: across preferred
We usually prefer across to say ‚on/to the other side of a flat area or surface’.
- He walked right across the desert.
- It took them six hours to row across the lake.
the adverb over (to)
Note that the adverb over has a wider meaning than the preposition over.
We often use over (to) for short journeys.
I’m going over to Jack’s. Shall we drive over and see your mother?
across and through
The difference between across and through is like the difference between on and in. Through, unlike across, is used for a movement in a three-dimensional space, with things on all sides.
- We walked across the ice. (We were on the ice.)
- I walked through the wood. (I was in the wood.)
- We drove across the desert.
- We drove through several towns.