Sphere Packing

[PatrickPowers]

It so happens that sphere packing is hard to deal with mathematically. Here in 3D it's obvious what the best packing arrangement is but this wasn't proven until 2015 and that by brute force. The only N dimensions with known neat proofs are 8 and 24. In 24 each sphere osculates with 196,560 others and it's possible to prove that this is the best possible arrangement.

There have long been results that show that the more dimensions you have, the less efficient the packing. Minkowski, Einstein's teacher, proved that the upper bound for sphere-packing density decreases exponentially as dimension increases. According to Scientific American,

Efficiently stacked oranges can fill about 74% of three-dimensional space. The sphere-packing problem has not been solved yet in four dimensions, but in eight dimensions, Viazovska showed that the densest packing fills about 25% of space, and in 24 dimensions, the best packing fills only 0.1% of space.

I find this very surprising.

[Vector_Graphics]

Hmm, I'm immediately thinking of the packing in 4D where you put a sphere into the gap formed by 16 spheres at the corners of a tesseract. That should be... I think 2 spheres per 16 units of volume, so about 60% of the space is filled. The absolute optimal packing is unsolved, but based on the numbers for the optimal packings in other dimensions this seems... pretty close. Perhaps it is optimal, and it's one of those intuitive things that is annoying to prove mathematically. Perhaps there's a better configuration. Who knows.

[PatrickPowers]

Take a sphere in ND, what is the max number of osculating spheres? Are they then all mutually osculating or are there gaps? A cursory search found no answers.

There's no particular reason to think there are or are not simple answers.

[PatrickPowers]

Take a sphere in ND, what is the max number of osculating spheres? Are they then all mutually osculating or are there gaps? A cursory search found no answers.

There's no particular reason to think there are or are not simple answers.

It's called the kissing problem. It turns out that even in 3D it wasn't until 1953 the answer was proved to be 12. The surrounding spheres do not touch one another (I thought they did) so it's conceivable that a 13th could fit in there. In 4D the answer is a maximum of 24 spheres kissing one. There are two possible such arrangements. The number of possible solutions increases with N.

Known upper bounds for 5,6,7,and 8 are 46,82,140,and 240,

They use something called the Gegenbauer polynomials. Gegenbauer literally means anti-farmer.

https://annals.math.princeton.edu/wp-content/uploads/annals-v168-n1-p01.pdf

[Embi]

It's crazy how in 3D, it's straightforward, but beyond that, it gets really tricky. I was amazed when I first learned about the 8 and 24 dimensions having neat proofs for the best packing arrangements. The exponential decrease in packing efficiency as dimensions increase is mind-boggling.