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.
Answer
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 4… 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 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 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 R which allow division (they’re called division algebras) but there is a unique 4 dimensional one. (Look up the Frobenius theorem
Attribution
Source : Link , Question Author : jojafett , Answer Author : rschwieb
