# Are there mathematical concepts that exist in dimension 44, but not in dimension 33?

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 $$33$$ dimensions, and they all have analogues in $$44$$ dimensions, but there is another regular polytope in $$44$$ dimensions: the 24 cell.

The kicker is that in dimensions higher than $$44$$… there are only three regular polytopes!

Another thing that can happen in $$44$$ dimensional space but not $$33$$ is that you can have two planes which only intersect at the origin (and nowhere else.) In $$33$$ dimensions you’d get at least a line in the intersection.

I don’t know if this also counts, but linear transformations in $$33$$-dimensions always scale 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 $$44$$ 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 $$33$$ dimensional asociative algebras over $$R\mathbb R$$ which allow division (they’re called division algebras) but there is a unique $$44$$ dimensional one. (Look up the Frobenius theorem