"no matter how you pack spheres, there'll always be gaps between them" Para 1

What this suggests to me is that no matter how many rigid 3D same size (or even different size maybe) spheres you stack together as densely as possible, there will always be space enough for an ant to crawl about absolutely anywhere in the stack. Is that so? Has it already been conjectured and/or proved? (I've already started experimenting)

This doesn't apply to circles in that 3 tangent circles can completely enclose and block off the space between them so that there's no passage into it, at least not in the same plane.