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

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Martin Brown

Posts: 259
Registered: 12/13/04
Re: honeycombs
Posted: Jan 24, 2013 3:43 AM
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On 23/01/2013 22:22, RichD wrote:
> On Jan 21, Martin Brown <|||> 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


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.
Traditional variable capacitors are made this way eg.

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!

Martin Brown

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