# If the world had four spatial dimensions, then area would be a tensor?

In three dimensions area is a vector because two dimensions have a direction relative to the third. If the world had four spatial dimensions then area would be a tensor?

And what form then the laws of physics which imply the concept of area, as electromagnetism or the definition of pressure, would have?

• It's not the concept of area that changes, it's the relation between vectors and pseudovectors that changes. Mar 9, 2021 at 11:10
• I didn't want to put it in answer since it does not address your more important question of "how the laws of physics change" Mar 9, 2021 at 13:18
• Mar 9, 2021 at 16:39
• Regarding physics, you would need to distinguish laws that use 2 dimensions (e.g. based on rotations) and laws that use n-1 dimensions (e.g. based on enclosing).
– M.S.
Mar 11, 2021 at 16:44
• I'd say that a more accurate heuristic is that area is a rank-2 tensor in any number of dimensions, but in three dimensions (only) it can also be throught of as a vector. Mar 13, 2021 at 4:02