Maximum Surface Area of a Flat Surface

[omalleyt]

What is the pattern that maximizes the ratio of surface area of a flat surface to the volume of material used in this surface? The surface has to be flat on a macroscopic scale, but on the microscopic scale could have dimples or other imperfections that increase the SA/Volume ratio. I am wondering what the pattern of microscopic imperfections that maximizes this ratio would be. I'm visualizing something like a flat surface with the texture of a golf ball.

 

[mathman]

I don't believe such a maximum exists. In one dimension, it is possible to construct a line (L) of infinite length on an interval (I) between the 0 and 1 points on a straight line, where the maximum deviation between L and I is as small you want. Generate plane figures from L and I and you will have an example to show what you want is impossible.

Construction: nth iterate - interval 1/n, curve oscillate between 1/√n and -1/√n. The total length is then ~ √n.

This post should be in the mathematics forum.

 

[Andy Resnick]

There should be a maximum, because the OP constrained the surface to be 'flat on a macroscopic scale'- whatever that means.

 

[Drakkith]

What is the pattern that maximizes the ratio of surface area of a flat surface to the volume of material used in this surface? The surface has to be flat on a macroscopic scale, but on the microscopic scale could have dimples or other imperfections that increase the SA/Volume ratio. I am wondering what the pattern of microscopic imperfections that maximizes this ratio would be. I'm visualizing something like a flat surface with the texture of a golf ball.

I would guess that it would be microscopic pits or bumps on the surface of the materiel. As to a specific pattern, that would depend on how flat/smooth you wanted the materiel to be on a macroscopic scale. I can't give you a specif ratio though.

 

[mathman]

There should be a maximum, because the OP constrained the surface to be 'flat on a macroscopic scale'- whatever that means.

There is no maximum! If you looked at my post, the nth iterate has a maximum deviation from flat of 1/√n which ->0 as n->∞. This is certainly flat on a macroscopic scale.

Meanwhile the total length ~√n which ->∞ with n.

To get a surface, simply project the line along the y direction (assuming the above interval is in the x direction).

 

[Drakkith]

There is no maximum! If you looked at my post, the nth iterate has a maximum deviation from flat of 1/√n which ->0 as n->∞. This is certainly flat on a macroscopic scale.

Meanwhile the total length ~√n which ->∞ with n.

To get a surface, simply project the line along the y direction (assuming the above interval is in the x direction).

Wouldn't the thickness of the material set the maximum?

 

[Andy Resnick]

There is no maximum! If you looked at my post, the nth iterate has a maximum deviation from flat of 1/√n which ->0 as n->∞. This is certainly flat on a macroscopic scale.

Meanwhile the total length ~√n which ->∞ with n.

To get a surface, simply project the line along the y direction (assuming the above interval is in the x direction).

I couldn't follow your construction- I didn't get the same result (I'll take your word for it, tho).

In any case, one of the essential differences between math and real objects is that n can't go to ∞.

 

[kcdodd]

it could go to the number of atoms on the surface though, assuming something like 10^15 atoms/meter^2. sqrt(10^15) ~ 10^7, whatever that is.

 

[DaveC426913]

1] It seems to me the answer depends strongly on the rather arbitrary deliniation between microscopic and macroscopic.


2] It also seems to me that it would be theoretically possible to have a surface analagous to the alveoli in the lungs, which have been perfecting maximum area for hundreds of millions of years.

 

[Andy Resnick]

2] It also seems to me that it would be theoretically possible to have a surface analagous to the alveoli in the lungs, which have been perfecting maximum area for hundreds of millions of years.

I like the way you think, but alveoli are (semi)spherical, to use the fewest number of cells.

Random comment: whale kidneys are anatomically identical to lungs.

 

[DaveC426913]

I like the way you think, but alveoli are (semi)spherical, to use the fewest number of cells.

You sure about that? Lungs' performance (and thus the critter's metabolism) is rate-limited by the surface area it can expose. Surface area is a high priority.


Anyway, the point is, you could have a 2D surface whose fractal dimension arbitrarily approaches 3 by having the surface so convoluted as to virtually entirely fill a volume, dependent only on material thickness.

The folds of the brain do the same thing.

 

[Andy Resnick]

You sure about that? Lungs' performance (and thus the critter's metabolism) is rate-limited by the surface area it can expose. Surface area is a high priority.

The folds of the brain do the same thing.

I am sure of almost nothing, but aveoli look like tiny little grapes:

http://www.asthmahelpline.com/photos%20site/ALVEOLI.jpg
http://nhscience.lonestar.edu/biol/respiratory/alveoli.htm
http://worldphotocollections.blogspot.com/2009/07/beautiful-microscopic-images-from.html

It's also true that our body is essentially composed of nothing but functional surfaces: the brain, lung, kidney, intestine, blood vessels, liver, pancreas, skin... can all be modeled as a surface.

 

[Dr Lots-o'watts]

What is the pattern that maximizes the ratio of surface area of a flat surface to the volume of material used in this surface? The surface has to be flat on a macroscopic scale, but on the microscopic scale could have dimples or other imperfections that increase the SA/Volume ratio. I am wondering what the pattern of microscopic imperfections that maximizes this ratio would be. I'm visualizing something like a flat surface with the texture of a golf ball.

Mathematically, I agree with mathman.

Physically, I am thinking "spiked hair". As narrow as a column can possibly be (atom diameter, or whichever chemical bonding allows the nearest structure), and as close to each other as allowed (again limited by allowed bonding).

Standing (carbon) nanotubes may be the closest thing technologically feasable.

At the moment, it's not obvious to me that a sponge-like, or alveolar structure, has necessarily more surface area -on the "macroscopic surface"- than a hair-like structure.

 

[mishrashubham]

Alveoli are a nice comparision. So just like alveoli try to maximize surface area to the nearest cell which is the fundamental unit of life, the thing that omalleyt is asking for will have to be small projections of small groups of atoms. But you can't go further than that because in maths you don't have a point that is next is to any given point i.e there are infinitely many points between any two given points but in this case we do have a "next atom".

 

[DaveC426913]

I am sure of almost nothing, but aveoli look like tiny little grapes:

I know what they look like, it's your take on what that maximizes/minimizes that I'm dubious of.

 

[DaveC426913]

Alveoli are a nice comparision. So just like alveoli try to maximize surface area to the nearest cell which is the fundamental unit of life, the thing that omalleyt is asking for will have to be small projections of small groups of atoms. But you can't go further than that because in maths you don't have a point that is next is to any given point i.e there are infinitely many points between any two given points but in this case we do have a "next atom".

No. what would be even more efficient (max surface area) is tiny trees, with limbs, branches and leaves.

Here's the start of the idea:

[URL]http://local.wasp.uwa.edu.au/~pbourke/fractals/fracintro/fracintro17.gif[/URL]

 

[mikeph]

Do you mean a smoothed version will be flat but if you zoom in, it is bumpy?

The above one is a nice example, also I think the continuous random walk (Wiener process) can have an average value of 0 (mean) but have infinite length over a finite Cartesian space.

The delta function is also flat "macroscopically" but has a pretty big surface area if you stretch the definition of "surface" for the generalised function. I think the question is difficult to answer because it does not contain strict guidelines of what constitutes a "macroscopically flat" surface: your own "golf ball" example is not what I think of as macroscopically flat.

 

[mathman]

/\/\/\/\/\/\/\/\/\/\...

Make width (horizontal) of each line 1/n. Make height (slant) of each line 1/√n. Total length (horizontal) of n lines = 1. Total length of n segments = √n.

There is no max, since n can be as large as you want. The physical limit is atomic.

 

[Andy Resnick]

Construction: nth iterate - interval 1/n, curve oscillate between 1/√n and -1/√n. The total length is then ~ √n.

Make width (horizontal) of each line 1/n. Make height (slant) of each line 1/√n. Total length (horizontal) of n lines = 1. Total length of n segments = √n.

Ok- now I get the same result.

I know what they look like, it's your take on what that maximizes/minimizes that I'm dubious of.

Well, the surface area could increase by making the cell membrane rougher, for example.

What's maximized is the rate of mass transfer in the lung; that's clear. It makes no sense to make the surface rougher than some characteristic diffusion length- there won't be an increase of mass transfer, but extra material must be used. Lung capacity is increased by increasing the number of aveoli (lung size), not by increasing the surface area of the lung.

 

[mishrashubham]

No. what would be even more efficient (max surface area) is tiny trees, with limbs, branches and leaves.

Here's the start of the idea:

[URL]http://local.wasp.uwa.edu.au/~pbourke/fractals/fracintro/fracintro17.gif

[/URL]

I get what you are trying to tell. Those are pictures of fractals that you are trying to show, but the trees and branches that you are talking about have to be made of something, and that something would be atoms. But nothing smaller than that. So, while dividing them into smaller and smaller branches you would reach a point where one of those branches would be an atom. But you can't make a smaller branch out of that atom. So that would be the limit to how long you can go on making branches. So just like you have long strands of fibres jutting out of a carpet, you would have small branches of atoms coming out of a surface that would look macroscopically flat but would maximize the actual surface area. (Hmm...I wonder why we don't have such branching systems on the surface of our small intestine. Dave if you could answet that I would be glad)

 

[Andy Resnick]

(Hmm...I wonder why we don't have such branching systems on the surface of our small intestine.

We do- there are gross features, the villi, and the brush border (microvilli) of individual epithelial cells.

http://www.edoctoronline.com/medical-atlas.asp?c=4&m=3&p=31&cid=1108&s=

 

[mishrashubham]

Yes I am aware of villi and microvilli. Its just that they don't have the extreme divisions into extremely small parts. But may be it can't go smaller than that (Projections of the membrane may be the furthest it can go). That might be sufficient for our body I guess. Anyways thanks for the reply.

 

[DaveC426913]

I get what you are trying to tell. Those are pictures of fractals that you are trying to show, but the trees and branches that you are talking about have to be made of something, and that something would be atoms. But nothing smaller than that. So, while dividing them into smaller and smaller branches you would reach a point where one of those branches would be an atom. But you can't make a smaller branch out of that atom. So that would be the limit to how long you can go on making branches. So just like you have long strands of fibres jutting out of a carpet, you would have small branches of atoms coming out of a surface that would look macroscopically flat but would maximize the actual surface area. (Hmm...I wonder why we don't have such branching systems on the surface of our small intestine. Dave if you could answet that I would be glad)

Yes, the limit would be twice the thickness of the material that comprises the surface layer.

 

[mishrashubham]

Yes, the limit would be twice the thickness of the material that comprises the surface layer.

I didn't get it. Please elaborate.

 

[DaveC426913]

I didn't get it. Please elaborate.

The surface of your physical layer is comprised of atoms, true, but it also serves some function, which probably requires more than just a single layer of atoms to operate. The functional surface can't be any thinner than that. And since it must loop up then down, there are two sides to every protruberance. Thus, any protruberance will be a mininum width of twice the function thickness of the surface layer.

Ideally, your functional surface could be a single atom thick, making for a minimum width of 2 atoms, but practically, it will be thicker.

 

[mishrashubham]

Yes. Right. Though it doesn't always have to be one loop (going up then down). A long molecule like the ones consisting of carbon chains that we generally see in hydrocarbons can be long enough to count as a protuberance(theoretically). But for practical purposes yes this cannot serve any particular function.

 

[DaveC426913]

(I've added an illustration to my earlier post.)

Yes. Right. Though it doesn't always have to be one loop (going up then down). A long molecule like the ones consisting of carbon chains that we generally see in hydrocarbons can be long enough to count as a protuberance(theoretically). But for practical purposes yes this cannot serve any particular function.

Well, it depends on how you define the surface and for what purpose. My assumption is that the surface is homogenous in function. i.e. any given area of the surface can be chosen and serves exactly the same function as any other area.

With your idea of molecular protruberances, only certain areas of the outcropping will perform the function (because only those areas will expose the right atoms for bonding or whatever). If the entire surface is the same (say hydrogen bonds) then you're back to the layer concept that I'm talking about. You've got a backbone molecule, but your "surface" is in fact a one-layer thickness of hydrogen atoms.

 

[mishrashubham]

Hmm.. You are right.