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.

Answer

Here’s a counterintuitive example from The Cauchy Schwarz Master Class, about what happens to cubes and spheres in high dimensions:

Consider a n-dimensional cube with side length 4, B=[2,2]n, with radius 1 spheres placed inside it at every corner of the smaller cube [1,1]n. Ie, the set of spheres centered at coordinates (±1,±1,,±1) that all just barely touch their neighbor and the wall of the enclosing box. Place another sphere S at the center of the box at 0, large enough so that it just barely touches all of the other spheres in each corner.

Below is a diagram for dimensions n=2 and n=3.

enter image description here

Does the box always contain the central sphere? (Ie, SB?)

Surprisingly, No! The radius of the blue sphere S actually diverges as the dimension increases, as shown by the simple calculation in the following image,

calculation of inner sphere radius

The crossover point is dimension n=9, where the central sphere just barely touches the faces of the red box, as well as each of the 512(!) spheres in the corners. In fact, in high dimensions nearly all of the central sphere’s volume is outside the box.

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