What are some examples of a mathematical result being counterintuitive? As I procrastinate studying for my Maths Exams, I want to know what are some cool examples of where math counters intuition.
My first and favorite experience of this is Gabriel's Horn that you see in intro Calc course, where the figure has finite volume but infinite surface area (I later learned of Koch's snowflake which is a 1d analog). I just remember doing out the integrals for it and thinking that it was unreal. I later heard the remark that you can fill it with paint, but you can't paint it, which blew my mind.
Also, philosophically/psychologically speaking, why does this happen? It seems that our intuition often guides us and is often correct for "finite" things, but when things become "infinite" our intuition flat-out fails.

Another elementary example: Connelly spheres, also known as flexible polyhedra.  These are non-convex polyhedra, homeomorphic to a sphere, with triangular faces; the polyhedra can be deformed continuously, while the faces remain rigid.  It took about 211 years to find a
counterexample to Euler's conjecture that embedded polyhedra are rigid.   See e.g. http://www.reocities.com/jshum_1999/polyhedra/introduction.htm
