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?