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Thank you that was a great read. I will be so happy if we find the optimal packing for 56 oranges one day
They're going to be so mad when they hear about gift bags...
Seriously though, this is really interesting, thanks for posting. Also, the term "Sausage Catastrophe" is...incredible.
Yeah, arbitrary sphere packing problems turn out to be quite tricky. Thankfully there are some nice simple cases that actually have physical relevance, like 3D packing of infinite regular spheres (FCC/HCP packings win). Edit: actually proving that isn't exactly easy, but the fact that that's the densest packing of atoms we observe in nature was a good hint!
But when you try to generalize things in some way... Well, the article does a pretty good job explaining why it gets hard!