Comment by aleph_minus_one

> In higher dimensions, are the spheres just a visual metaphor based on the 3-dimensional problem, or are mathematicians really visualising spheres with physical space between them?

For such discrete geometry problems, high-dimensional spaces often behave "weirdly" - your geometric intuition from R^3 will often barely help you.

You thus typically rather rely on ideas such as symmetry, or calculations whether "there is still space inbetween that you can fill", or sometimes stochastic/averaging arguments to show the existence of some configuration.