TITLE: Spheres cause contradictions in dimensions $10$ and more?
QUESTION [15 upvotes]: According to this Numberphile video, if you tightly pack hyper-spheres into a hyper-box and then find the radius of the largest hyper-sphere that could possibly fit in the remaining space, the resulting hyper-sphere would somehow exceed the confines of the box that contained all of the hyper-spheres (where the number of dimensions are greater or equal to 10).
Isn't a logical contradiction generally considered a disproof of something?
Wouldn't this disprove the generalised formula being used to find the radius of the resulting sphere on n dimensions?
Is it possible that mathematicians simply do not understand extra dimensional geometry and its inherent rules?

REPLY [4 votes]: Just take an $n$-dimensional hypercube of edge size 2 centered at the origin. Take furthermore hyperballs with unit radius centered at each of its vertices. Then those evidently will be touching by construction.
By Pythagoras the distance from the origin, i.e. from the body center of that hypercube, towards either of its vertices happens to be
$$\sqrt{1^2+1^2+...+1^2}=\sqrt{n}$$
Accordingly an hyperball, which is centered at the origin and is touching those corner-centered unit hyperballs, should have a radius of
$$\sqrt{n}-1$$
But that number clearly gets larger than $1$, which in turn clearly is the radius of the in-hyperball of that hypercube, as soon as $n>4$. - That is all of that counterintuitive magic. 
--- rk