3D Honeycomb

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3D Honeycomb

Postby lopkiol on November 5th, 2009, 8:30 pm 

What's the equivalent of the honeycomb in three-dimensions?
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Re: 3D Honeycomb

Postby linford86 on November 6th, 2009, 12:05 am 

Could you explain what you mean? I'm not entirely sure what you mean.
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Re: 3D Honeycomb

Postby kudayta on November 6th, 2009, 2:10 am 

Image

One of these?
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Re: 3D Honeycomb

Postby lopkiol on November 7th, 2009, 9:05 pm 

linford86 wrote:Could you explain what you mean? I'm not entirely sure what you mean.

What is the partition of space in cells of equal volume having the least surface area? Is it really the truncated octahedron?
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Re: 3D Honeycomb

Postby xcthulhu on November 8th, 2009, 6:38 am 

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...
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Re: 3D Honeycomb

Postby lopkiol on November 8th, 2009, 8:46 am 

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...


That's true, the conjecture was proved recently but it was formulated more than two millennia ago. Is there at least a conjecture in 3D?

Interesting experiment that of the baloons. I suppose there will be friction between each other. Would that be a problem? And it makes sense to me for the 2D going to a honeycomb, but are you sure about the 3D going to what Kudayta showed?
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Re: 3D Honeycomb

Postby xcthulhu on November 8th, 2009, 10:42 am 

Thanks to google I am now certain that this is an open question.

This problem is known as Kelvin's Conjecture (named after the English physicist Lord Kelvin). Lord Kelvin came to this conjecture by thinking of the lattices that foam cells form, which is fairly similar to my silly idea about balloons. He conjectured that slightly curved truncated octahedrons were optimal...

But Weaire-Phelan structures are apparently better (the number you are concerned about is known as the isoperemetric quotient, and their lattice has a higher quotient than Kelvin's):
http://en.wikipedia.org/wiki/Weaire%E2% ... _structure

Image

I'm not sure what would happen if you did this experiment with balloons. But I bet to get over the problem fiction might cause you could just lube them. There are plenty of lubes that work well with latex (hehe). I think a greater concern would be that air bubbles would probably get caught and cause the balloons to pack into an irregular lattice. But maybe if one agitated the balloons a lot, the air bubbles would be evacuated and they'd form a Kelvin lattice as predicted. This is the basic principle behind (artificial) annealing, cement mixing, and this paper that recently appeared in the journal Nature that I thought was pretty neat.
Last edited by xcthulhu on November 8th, 2009, 10:58 am, edited 1 time in total.
Reason: Upon closure investigation, I decided that a Kelvin foam lattice is probably not optimal... so I changed this post a bit
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Re: 3D Honeycomb

Postby lopkiol on November 8th, 2009, 11:06 am 

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?
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Re: 3D Honeycomb

Postby xcthulhu on November 8th, 2009, 11:47 am 

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?


Oh, so when I originally wrote this I was pretty sure that I was right, since I read it was corroborated by Kelvin, who I presumed to be much smarter than me. I was so convinced, I was pretty sure that if this experiment was run it would be a Kelvin lattice.

But then I started reading about how it was big news in the 90s that Kelvin was wrong, and that Weaire-Phelan structures have a better volume to surface area ratio. So I edited my post... looks like I forgot a little bit. I wouldn't know what to predict for this experiment; but I'd sort of suspect now that the structure would probably be irregular.
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Re: 3D Honeycomb

Postby lopkiol on November 8th, 2009, 12:34 pm 

Your annealing idea has been already performed numerically. See this paper if you're interested.
http://www.aquafoam.com/papers/Kraynik.pdf

What they found - see pictures - is random structures. But for some reason their isoperimetric quotients are way below even that of Kelvin's lattice. FIG. 5 shows the 'energy', which is the reciprocal of the isoperimetric quotient, of all the known structures, plotted against the average number of faces per bubble.
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Re: 3D Honeycomb

Postby lopkiol on January 27th, 2010, 7:40 am 

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.

What do you mean by irregular? What is irregular? And what is regular? Made of the same cell? Then Weaire-Phelan is irregular. Or did you mean periodic? But then also Kraynik's foams are periodic. Please explain.
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Re: 3D Honeycomb

Postby Giacomo on January 27th, 2010, 4:16 pm 

It is still an open problem for n = 3 so not much has been done for n > 3 as far as I know.

Kelvin’s solution to the problem was a honeycomb of truncated octahedrons - shapes with 6 square faces and eight hexagonal faces. A better solution was devised by Weaire and Phelan.

As you know, the Weaire-Phelan structure is composed of two different shapes: an irregular pentagonal dodecahedron (12-faced polyhedron) and a polyhedron with 14 faces.

Ruggero Gabbrielli has developed a new technique for mathematically modelling the structure of foam

His structure is instead composed of 4 different shapes that fit together.

His new structure is not superior to the Weaire-Phelan structure in terms of packing efficiency ... but could lead to a better solution to the Kelvin problem.

Ruggero Gabbrielli's method uses a partial differential equation.

You can get more info from Ruggero Gabbrielli Homepage

And, read more at the following site: The Honeycomb Theorem, Kelvin's problem & counterexamples to Kelvin's conjecture
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Re: 3D Honeycomb

Postby lopkiol on January 29th, 2010, 6:53 pm 

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.
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Re: 3D Honeycomb

Postby Giacomo on January 30th, 2010, 3:42 pm 

lopkiol wrote:I didn't ask for my homepage to be posted but it's ok, I'm glad you did that. :)


So, that's your website! Welcome to the math forum.

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


I'm reading Thomas C. Hales' proof ... It is the 2D version of the 3D Kelvin problem ... Basically, Hales proved that a hexagonal grid represents the best way to divide a surface into regions of equal area with the least total perimeter

Hales took its hexagonal cross section, since it matters most ... Have you tried it, too?

And, later on, I"ll read carefully Fejes Toth's proof (Hungarian mathematician) ... Toth asked, "What if cells were allowed to have curved sides?"

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 ...
Last edited by Giacomo on January 30th, 2010, 4:02 pm, edited 1 time in total.
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Re: 3D Honeycomb

Postby lopkiol on January 30th, 2010, 6:29 pm 

Thanks.
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

Exactly. Before that proof we didn't even know whether a solution existed. Looking for a solution to a problem that doesn't have one would have been a waste of time, I suppose. That's why Morgan's work is fundamental. One option that had never been discared by many scientists, was that the problem could not have had any solution, but only partitions as close as desired to the limit inferior of the cost.
Giacomo wrote:Hales took its hexagonal cross section, since it matters most ... Have you tried it, too?

What do you mean? Tried to do what?
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?"

Do you have a reference for this? I am not sure about what you mean here. As far as I know Toth's problem is related to the surface that separates two sets of different honeycombs coming from opposite directions in a hive.
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 ...

No. I'm referring to randomness in general. Is random non-periodic? No, because aperiodic order would be random. What is then?
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