Examples of results failing in higher dimensions

A number of economists do not appreciate rigor in their usage of mathematics and I find it very discouraging.

One of the examples of rigor-lacking approach are proofs done via graphs or pictures without formalizing the reasoning. I would like thus to come up with a few examples of theorems (or other important results) which may be true in low dimensions (and are pretty intuitive graphically) but fail in higher dimensions.

By the way, these examples are directed towards people who do not have a strong mathematical background (some linear algebra and calculus), so avoiding technical statements would be appreciated.

Jordan-Schoenflies theorem could be such an example (though most economists are unfamiliar with the notion of a homeomorphism). Could you point me to any others?

Thanks.

Simple symmetric random walks in 1 and 2 dimensions return to the origin infinitely many times, but not in 3 and higher dimensions.

That is, if you are on a number line (or coordinate plane) and repeatedly flip a coin to determine whether to take one step in the positive direction or one step in the negative direction (or do something to randomly choose 1 step in the positive or negative x or y direction), the probability that you'll get back to the origin is 1. If you do the same in 3 dimensions (or higher), where you're choosing randomly between 6 (or more) directions for taking 1 step, the probability of returning to the origin is less than 1.

edit: The probability $p(d)$ of a $d$-dimensional simple symmetric random walk returning to the origin is apparently called Pólya's Random Walk Constant. $p(1)=p(2)=1$, but $p(3)\approx 0.34$, $p(4)\approx 0.19$, $p(5)\approx 0.14$, $p(6)\approx 0.10$, $p(7)\approx 0.09$, and $p(8)\approx 0.07$ (from the linked MathWorld article).

Chaos cannot exist in one- or two-dimensional continuous dynamical systems. Among other things, this means that in one or two dimensions if you supply similar inputs you will get similar outputs.

However, in 3 or more dimensions the dynamics can be chaotic, meaning that similar inputs do not necessarily lead to similar outputs (you can witness this in the Lorenz equations, a crude model for atmospheric dynamics).