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
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.