TITLE: Are there mathematical concepts that exist in dimension $4$, but not in dimension $3$?
QUESTION [58 upvotes]: 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.

REPLY [5 votes]: Things that are completely new to four dimensions, that is, they appear first in four dimensions, but not in one/two/three dimensions, and which are also not straight forwards generalizations of concepts in one/two/three dimensions:

Double rotation (a way to rotate in two independent way simultaneously)
duo-prisms and duo-cylinders (geometric objects that "visualize" the concept of double-rotation).
A centrally symmetric self-dual polytope other than a polygon (24-cell, which is even regular),
More than one way to develope differential geometry on a manifold (e.g. multiple differentiable structures on $\Bbb R^4$ and maybe the 4-sphere)
Polytopes that cannot have rational vertex-coordinates, or polytopes which have strange realization spaces (not connected or not simply-connected, ...). For example, there is a (combinatial type of) polytope that can be realized in two ways, but these two ways cannot be continuously deformed into each other without changing the combinatorial structure of the polytope at some point.
Polytopes in which there is an edge between any pair of vertices, other than a simplex (so-called neighborly polytopes).
A knotted surface, e.g. a sphere embedded into 4-space, in such a way that it is free of self-intersections, but cannot be continuously transformed into the "standard embedding" without creating self-intersections in the process (okay, this is kind of a straight foward generalization of knots from 3D).