Sphere packaging

Geometry Level 3

What is the highest average density that can be achieved by a regular lattice arrangement of congruent spheres?

84% 74% 54% 95%

1 solution

Krishna Sharma
Sep 20, 2014

I did by chemistry In solid state Face Centered cubical lattice has maximum packing fraction of 74%

How do you know that is the maximum (other than someone claiming that it is)?

Calvin Lin Staff - 6 years, 8 months ago

Kepler's Conjecture states that it is. It wasn't until 1998 when Thomas Hales developed a proof of it. Because his proof was so difficult to verify, in 2003 he started the Flyspeck Project, which proposed to transform his proof into one that could be systematically checked by computer. In August 10, 2014, it was announced that this task was complete, and that his proof "has been verified"---by computers. It's been estimated that it will take 20 man years for mathematicians to confirm it "by hand".

There have been other famous problems that were ultimately solved by computers, and still waiting for "humans to confirm by hand", such as the 4-Color Theorem. It's probably has become a fact of life that there will be more such conjectures in mathematics that will ultimately be proven by computer, and not by hand. For example, we have the digits of pi calculated by computers to "a trillion places", but who's going to check that by hand?

Michael Mendrin - 6 years, 8 months ago

Note that this is a different question from Kepler's conjecture. It specifically restricts our attention to "regular lattice arrangement", as opposed to Kepler's conjecture which allows for arbitrary arrangements like the "dirty dozen".

The problem of close packing of spheres was first mathematically analyzed by Thomas Harriot around 1587, after he was posed a question on piling cannonballs on ships. It was solved by Gauss in 1831.

Calvin Lin Staff - 6 years, 8 months ago

@Calvin Lin – Ah, yeah, you're right. Nemo did say "regular lattice". Well, it was interesting how the Kepler Conjecture had made recent news.

Michael Mendrin - 6 years, 8 months ago

@Michael Mendrin – Indeed, and I made a post about it in What does stacking spheres have to do with writing proofs ,

Calvin Lin Staff - 6 years, 8 months ago

@Calvin Lin – Oh, I KNEW I had read that somewhere. So, it was you!

Well, this is giving me an idea about a new problem, which I'll post in a while, if there is a solution to it.

Michael Mendrin - 6 years, 8 months ago

@Michael Mendrin – Please correct me if I am wrong sir, but I thought the Kepler Conjuncture suggested that it was true for both regular and irregular arrangements and that cubic and hexagonal close packing were two ways in which they could be arranged. Maybe I, misunderstood it. Please tell me if I need to rephrase the question. I had only read about it recently somewhere, and maybe I might not have understood it right.

nemo maximus - 6 years, 8 months ago

@Nemo Maximus – Calvin has explained this well. The Kepler Conjecture states that there is no packing method, regular or irregular, that can have a density higher than 74%. In the 19th century, Gauss proved this for regular lattice packing. Bales more recently proved this for any packing, regular or irregular. This is one of out of the 23 famous "Hilbert's Problems", which includes the Riemann hypothesis. Many are still unsolved, but this one just has been solved.

Michael Mendrin - 6 years, 8 months ago

@Michael Mendrin – Mr Mendrin, I think I saw your new problem, is this the one?

nemo maximus - 6 years, 8 months ago

@Nemo Maximus – Yeah, I found a solution last night. I wasn't sure I'd find a realistic one, i.e., something less than "millions of balls".

Michael Mendrin - 6 years, 8 months ago

Sphere packaging

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