Paradigm shifts from $2\to 3$ Recently, I've been thinking about a common theme that I've seen all over mathematics. One often finds, that when the number of dimensions/degrees of freedom in a given scenario/problem changes from $2$ to $3$, that some fundamental shifts in the solution or resulting behavior occur. I'll list five examples of what I'm talking about, three of which have to do with differential equations. (What can I say? Read my bio.) Before I do so, my questions are

*

*What are some other examples of this theme?

*Is there a universal reason why this happens? Does it have to do with $3$ being the first odd prime? Or perhaps something else?

1: Poincaré-Bendixson Theorem
For those that don't know, the Poincare-Bendixson theorem is a deep result in ODEs/dynamical systems. Consider the autonomous ODE
$$\dot{\mathrm{x}}=\mathrm{u}(\mathrm{x})\tag{1}$$
Where $\mathrm{x}:\mathbb{R}\to\mathbb{R}^n~;~\mathrm{x}:t\mapsto \mathrm{x}(t)$, and $\mathrm{u}:\mathbb{R}^n\to\mathbb{R}^n$. When $n=1$ the behavior of the solution is typically very easily to analyze heuristically, and in particular, it is rather obvious that only monotonic solutions exist. When $n=2$, things obviously get a lot more complicated but there is in fact the powerful P-B theorem:

Let $\Omega\subset \mathbb{R}^2$ be closed and bounded. If $u$ is $C^1$ in $\Omega$, $\Omega$ contains no fixed points, and $\exists x_0\in\Omega$ such that the solution of the IVP
$$\dot {\mathrm{x}}=\mathrm{u}(\mathrm{x})~~;~~\mathrm{x}(0)=\mathrm{x}_0$$
Is entirely contained in $\Omega$, then there is at least one closed orbit in $\Omega$.

Quite a remarkable theorem if you ask me. The consequence of this theorem is that there is no chaotic behavior in two dimensions. In the plane, any solution of $1$ is either

*

*unbounded

*bounded and approaching a periodic limit cycle

*periodic

So solutions that are bounded, but do not approach a stable limit cycle, i.e strange attractors, are impossible in the plane.
However, when we jump from dimension two to dimension three, and any subsequent dimension, no similar result exists. Solutions of autonomous ODEs in $n>2$ dimensions can be as "strange" as you like.
2: Fermat's Last Theorem
Consider the simple equation
$$a^n+b^n=c^n\tag{2}$$
Where $a,b,c\in\mathbb{Z}\setminus \{0\}$ and $n\in\mathbb{N}$. The question is, given $n$, how many solutions $(a,b,c)$ exist to $\boldsymbol{(2)}$? When $n=1$ it is obvious that there are infinitely many solutions - the sum of any two integers is an integer. When $n=2$, proving that infinitely many solutions exist is still rather easy, and known thousands of years ago to the Greeks. We can simply let $r,k\in\mathbb{N}$ with $k>r$ and observe that
$$(k^2-r^2)^2+(2rk)^2=(k^2+r^2)^2$$
Since there are infinitely many pairs of positive integers $(k,r)$ with $k>r$ there are infinitely many solutions. However, as I am sure you are all aware, the general answer was not known until 1995, when Andrew Wiles published his complete and peer-reviewed proof of the problem, $358$(!) years after the problem's conception by Fermat. His result was that

For $n>2$, no solutions to $\boldsymbol{(2)}$ exist.

3: The Three Body Problem
Take a system of $n$ particles in $\mathbb{R}^3$ with positions $\mathrm{r}_1,\dots ,\mathrm{r}_n$ and masses $m_1,\dots, m_n$ and consider the the coupled vector IVP
$$\ddot{\mathrm{r}}_{i}=\sum _{j\in \{1,\dotsc ,n\} \setminus \{i\}}\frac{-Gm_{i} m_{j}}{\| \mathrm{r}_{i} -\mathrm{r}_{j} \| ^{3}}(\mathrm{r}_{i} -\mathrm{r}_{j})$$
$$\mathrm{r}_i(0)=\mathrm{r}_{i,0}~~,~~\dot{\mathrm{r}}_i(0)=\dot{\mathrm{r}}_{i,0}$$
Where $i\in\{1,\dots ,n\}$. When $n=1$ we just have a single stationary body. When $n=2$ things get a lot more interesting, but still the equations are easy to analyze and their behavior is easy to predict with numerical simulation - it is why we are able to predict solar ecplipses years in advance. In fact, taking the limiting case $m_2 \gg m_1$ some very precise equations for the motion of the bodies have been known for hundreds of years, namely Kepler's laws. However, when $n\geq 3$ the system becomes chaotic, with the particles exhibiting no obvious or predictable behavior. Is this because, unlike two points, one cannot in general find a line that goes through three arbitrary points?
This reminds me a lot of example 1.
4: Commutative Division Algebras over $\mathbb{R}$ (Frobenius's Theorem)
My shortest entry on this list, due to my extreme lack of knowledge about abstract algebra.

*

*There is (trivially) a one dimensional commutative division algrebra over $\mathbb{R}$, namely $\mathbb{R}$ itself.


*There a two dimensional commutative division algebra over $\mathbb{R}$, namely the complex numbers $\mathbb{C}$.
But there is in fact no commutative division algebra over $\mathbb{R}$ when $n>2$. Once again, we see that changing the dimension from $2$ to $3$ completely changes the behavior.
5: The fundamental solution of Laplace's equation
We seek to solve the equation
$$(\boldsymbol{\triangle}u)(\mathrm{x})=\delta(\mathrm{x})$$
Here $u:\mathbb{R}^n\to\mathbb{R}$, $\mathrm{x}\in\mathbb{R}^n$, and $\delta$ is Dirac's delta distribution. It can be shown that, letting $V_n=\frac{\pi^{n/2}}{\Gamma(1+n/2)}$ be the volume (where, by volume I really mean the $n$ dimensional measure) of the unit $n$ ball, the solution is
$$\Phi_n:\mathbb{R}^n\setminus \{0\}\to\mathbb{R}$$
$$\Phi _{n}(\mathrm{x}) =\begin{cases}
\frac{1}{2} |\mathrm{x} | & n=1\\
\frac{1}{2\pi }\log |\mathrm{x} | & n=2\\
\frac{-1}{n( n-2) V_{n}} \ \frac{1}{|\mathrm{x} |^{n-2}} & n\geq 3
\end{cases}$$
This is actually more interesting - going from $n=1,2,3$ we start with a power law in $|\mathrm{x}|$, then a logarithm, and then again a power law. However, once again we see a stark change in behavior when $n$ goes from $2\to 3$.
If you made it this far, thanks for reading. Consider leaving an answer giving other examples or perhaps a hand-wavy explanation.
 A: To explain the comment from @Peter Kagey, a random walk on a 1- or 2-dimensional grid returns to the origin with probability one, but in three dimensions or higher the probability of eventual return is strictly less than one.
A: Two results in combinatorics on words to add to your list.
Square-free words.
There exist infinite square-free words on a three-letter alphabet, but none on a two-letter alphabet.
Post correspondence problem. Given a finite set of pairs of words $\{(r_i, v_i) \mid 1 \leqslant i \leqslant n\}$, determine whether or not there exists a nonempty sequence $i_1, \ldots, i_k \in \{1, \ldots, n\}$ such that $r_{i_1}r_{i_2} \dotsm r_{i_k} = v_{i_1}v_{i_2} \dotsm v_{i_k}$. The Post correspondence problem is decidable for $n = 2$, undecidable for $n \geqslant 5$ and conjectured to be undecidable for $n = 3$ and $n = 4$.
A: Regular polytopes
Infinitely many in $2d$ but only finitely many in higher dimensions.
https://en.m.wikipedia.org/wiki/Regular_polytope
Orbits
Stable closed orbits exist in the $3d$ two body problem with the usual inverse square central force.  In the $4d$ analogy with an inverse cube law, only circular orbits are possible and they are unstable.  Similarly for even higher dimensions.
