Fixed point theorem

The fixed point theorem by Brouwer is something I learnt in graduate school. Formally,

The Brouwer Fixed Point Theorem states that a continuous mapping, f, of a convex, closed bounded set in Rⁿ into itself necessarily has a point, x₀, where f(x₀) = x₀.


Put simply, it says that there is a point (say on a surface) that will remain unaffected by some transformation that affects all other points around it. Imagine a point on a beach that remains untouched by a tsunami. That is a fixed point. Or a point in a cup of coffee that remains unaffected by the stirring of a spoon. 


For a better explanation, please click here.


The coffee example (attributed to Prof William Z) came to mind some weeks ago when I was holding my water bottle during a run. I noticed that I could minimize the stirring and swishing of the water inside the bottle if I held it in  a particular way and ran with as little sideways arm motion. In other words, I believe that there can be a fixed point inside of the water in my water bottle. 


Doing that made me run with a slightly better form. Or so I believe.