Xjyuk

Are there mathematical concepts that exist in the fourth dimension, but not in the third dimension? Of course, mathematical concepts include geometrical concepts, but I don't mean to say geometrical concept exclusively. I am not a mathematician and I am more of a layman, so it would be appreciated if you could tell what the concepts are in your answer so that a layman can understand.

The one that sticks out for me the most is that there are five regular polytopes (called Platonic solids) in $3$ dimensions, and they all have analogues in $4$ dimensions, but there is another regular polytope in $4$ dimensions: the 24 cell.

The kicker is that in dimensions higher than $5$... there are only three regular polytopes!

Another thing that can happen in $4$ dimensional space but not $3$ is that you can have two planes which only intersect at the origin (and nowhere else.) In $3$ dimensions you'd get at least a line in the intersection.

I don't know if this also counts, but linear transformations in $3$-dimensions always stretch one direction (that is, they have a real eigenvector). This means that in all cases, a line in one direction must either stay put or be reversed to lie upon itself. In $4$ dimensions, it's possible to have transformations (even nonsingular ones) that don't have any real eigenvectors, so all lines get shifted.

Also not sure if this counts, but there are no $3$ dimensional asociative algebras over $mathbb R$ which allow division (they're called division algebras) but there is a unique $4$ dimensional one. (Look up the Frobenius theorem

The fact that $mathbbR^4$ can be given a multiplication such that with vector's addition, it has a structure of (non commutative) field, i.e. quaternions.

Whereas one can prove that it is impossible for $mathbbR^3$ to have a vector multiplication such that with usual addition of vectors, $mathbbR^3$ is a field (think for example to an unsuccessfull candidate, cross product, which is not even associative).

I just saw now that at the end of his answer @rschwieb has mentionned "division algebras" ; but the concept of "field" is more powerful.

The fourth dimension is, in some ways, very peculiar.

The $3$-dimensional Euclidean space has a unique differentiable structure. In fact, the same is true in any dimension not equal to $4$. By contrast, the $4$-dimensional space has a continuum of incompatible differentiable structures. Intuitively, this means that something that looks like the familiar $4$-dimensional space can have a large variety of rather odd geometries.

An at most $3$-dimensional sphere admits a unique differentiable structure (there are no exotic spheres).

It is not known whether there are exotic $4$-dimensional spheres, or whether there are finitely many of them (in contrast, in any other dimension, there are only finitely many differentiable structures on the sphere --- e.g. $28$ in $7$ dimensions).