xcthulhu wrote:I'm pretty sure this is an open question. The honeycomb conjecture was only proved by Thomas Hales in the last decade; he tells me it relies on Almgren's work on geometric measure theory and that it's fairly challenging.
But I suspect you can figure this out experimentally. Just get a bunch of little, transparent balloons and put them in a bigger transparent balloon. If you contract the sides of the bigger balloon, it will compress the little balloons and I bet they'll fall into a regular lattice configuration, and the shape of the balloons will contort to the truncated octohedron like Kudayta showed. I know in the planar version of this experiment that the balloons contract to a honeycomb...
lopkiol wrote:xcthulhu wrote:But maybe if one agitated the balloons a lot, the air bubbles would be evacuated and they'd form a Kelvin lattice as predicted.
Why a Kelvin lattice?
xcthulhu wrote:I wouldn't know what to predict for this experiment; but I'd sort of suspect now that the structure would probably be irregular.
lopkiol wrote:I didn't ask for my homepage to be posted but it's ok, I'm glad you did that. :)
I only wanted to draw the attention on the fact that even though we might not know it, a solution to the problem exists (http://arxiv.org/pdf/0711.4228) but nothing can be said on its internal order yet.
To be honest, I would like to know how people define randomness here.
Giacomo wrote:I only wanted to draw the attention on the fact that even though we might not know it, a solution to the problem exists (http://arxiv.org/pdf/0711.4228) but nothing can be said on its internal order yet.
The arxiv document simply proves an existence theorem ... it does not demonstrate the objects in the partition. Not what one would call the "solution" ... The Honeycomb problem consists of finding the min "perimeter" partition not showing one exist
Giacomo wrote:Hales took its hexagonal cross section, since it matters most ... Have you tried it, too?
Giacomo wrote:And, later on, I"ll read carefully Fejes Toth's proof (Hungarian mathematician) ... Toth asked, "What if cells were allowed to have curved sides?"
Giacomo wrote:To be honest, I would like to know how people define randomness here.
Are you referring to random close packing, ie a quilt of random polygons or cells with curved rather than straight sides?
So, instead of having straight sides, we would have convex and concave shapes ...
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