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.

A famous example in mathematical physics is the Ising model.  This was invented by Wilhelm Lenz, who gave it as a thesis topic to his student Ernst Ising.  Ising solved the problem in one dimension (which is rather easy), finding that there is no phase transition, and incorrectly concluded from this that there is no phase transition in three dimensions.  In fact the model does have a phase transition in two or more dimensions, which is what makes it interesting.
