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Topic: honeycombs
Replies: 9   Last Post: Jan 24, 2013 5:40 PM

 Messages: [ Previous | Next ]
 Martin Brown Posts: 259 Registered: 12/13/04
Re: honeycombs
Posted: Jan 24, 2013 3:43 AM

On 23/01/2013 22:22, RichD wrote:
> On Jan 21, Martin Brown <|||newspam...@nezumi.demon.co.uk> wrote:
>>> I saw a news item about a new technique to draw CO2 from
>>> the atmosphere. It's a chemical process, using amines,
>>> which binds with the molecule, coated on a large structure,
>>> in the shape of a honeycomb.

>>
>>> According to the story, this maximizes surface area.
>>> ok, mathematicians, which function gets optimized by a
>>> honeycomb? What are the constraints and assumptions?

>>
>> 2D problem to *minimise* the surface area to occupy a given volume. Bees
>> use it to make honeycomb with the least amount of wax.
>>
>> It is not difficult to show that the angle between sides must be 120
>> degrees and that equal lengths minimise total length/area occupied.
>>
>> They have the structure just about as wrong as it is possible to be
>> unless the stuff they are making it out of is extremely precious.

>
> ?

Their claim is incorrect as is obvious by inspection of a few other
simple shapes length to area ratios.

>> The 3D problem to occupy volume with a foam of minimum surface area is
>> far more interesting and gives rise to Plateau's laws of soap films. The
>> Kelvin foam structure was optimal until fairly recently when
>> Weare-Phelan discovered a 3% better solution using a pair of shapes. A
>> whole new family has been found but as yet a proof of optimality eludes.

>
> The claim is that honeycomb maximizes surface area
> (of what?). This is new to me, so I'm looking for a
> rigorous statement of the problem, and proof of the solution.

You won't find one since it is trivial to demonstrate that a hexagonal
2D honeycomb does not maximise surface area inside a given volume.

This is because it is the global *MINIMUM* of length to volume in a
configuration that is required to tile the plane.

Equilateral triangle, rectangle or regular hexagon will all tile in 2D.

Consider a single hexagon with equal unit sides
_
/ \
\_/
L = 6 A= 3sqrt(3) L/A = 2/sqrt(3) ~ 2.31

A square
_
|_|
L =4 A=1 L/A = 4

A triangle
/\
L=3 A= sqrt(3)/4 L/A = 4*sqrt(3) ~ 6.93

See

http://en.wikipedia.org/wiki/Regular_polygon

This is trivially obvious to anyone but a chemical engineer since you
can clearly pack six extra unit lengths into a hexagon by joining
together diametrically opposite vertices to make equilateral triangles.

The best that you can do in a given finite volume is to take a square
and convert it into a rectangle with one side as short as possible
Keeping L=4

Regular square is L=4 A=1 L/A = 4

Make the sides of a rectangle 2-e, e and as e -> 0
L=4 A=e(2-e) and L/A = 4/e(2-e)

A ratio that can be made arbitarily large within engineering
constraints. Rectangular grids are also easier to manufacture.

http://en.wikipedia.org/wiki/Variable_capacitor

In fact most capacitors are internally but rolled up as a cylinder.

Absolute maximum surface area contained in a given volume is either a
fractal like activated charcoal gas mask filters at 500m^2/g or if that
isn't allowed then parallel plates like in capacitors.

If the rest of the project is as well researched then don't invest!

--
Regards,
Martin Brown

Date Subject Author
1/22/13 Rich Delaney
1/22/13 Martin Brown
1/22/13 Repeating Rifle
1/23/13 Rich Delaney
1/24/13 Repeating Rifle
1/23/13 Rich Delaney
1/23/13 glen herrmannsfeldt
1/24/13 Martin Brown
1/24/13 Martin Brown
1/24/13 Brian Q. Hutchings