What's the difference between 2 and 3? Here are two classical results which depend on whether a parameter is 2 or 3:


*

*It is possible to bisect an arbitrary angle with ruler and compass, but impossible to trisect it. 

*While there are infinitely many Pythagorean triples, i.e. integer solutions to $x^2+y^2=z^2$, there are no non-trivial integer solutions to $x^3+y^3=z^3$. 
There are several other instances where the dividing line seems to be between 2 and 3:


*

*A 2-regular tree is countable, a 3-regular tree is uncountable.

*2SAT is solvable in polynomial time, 3SAT is NP-complete.

*A random walk on $\mathbf Z^2$ is recurrent, while a random walk on $\mathbf Z^3$ is transient. 
What other examples can you think of? 
 A: There are infinitely many regular polytopes in $\mathbb R^2$ and only five in $\mathbb R^3$.
A: $SL_2(\mathbb{Z})$ is an amalgam whereas $SL_3(\mathbb{Z})$ is not.
A: Autonomous systems of ODEs produce simple dynamics in two dimensions, but complex dynamics in three or more. This is directly related to the fact that curves in three or more dimensions can pass each other without crossing.
A: The permutation group of two elements is abelian, the permutation group of three elements is not. There are thus non-galoisian number fields of degree $3$. 
A: In 2-dimensional Euclidean space every two lines intersect (maybe in the infinite), in 3-dimensional Euclidean space there are skew (?) lines.
A: More examples are given as answers to a similar question about problems NP-hard in $\mathbb R^3$ but not in $\mathbb R^2$:


*

*Set-cover by half-spaces.

*Finding a shortest path between two points among polygonal obstacles.

*Determining whether a non-convex polygon/polyhedron can be triangulated without Steiner points.

*Realizability problem for $d$-dimensional polytopes is a candidate ($d \leq 3$ vs $d \geq 4$).

A: $\mathbb R^3$ is much more rigid than $\mathbb R^2$ when considering conformality:
Conformal transformations of $\mathbb R^2$ do not form a finite-dimensional Lie-group.
A: Elements of order $2$ in a group are the only non-trivial elements which their own inverse.
