I'm a biology teacher attempting to resolve a discussion with one of my colleagues.

The ratio of surface area to volume is of extreme importance to the efficiency of cells, and I understand that this ratio decreases whenever you proportionally enlarge any three dimensional object. I can easily demonstrate this for a particular shape, but is there a mathematical proof that can show it is true for ANY 3D geometric shape?

Someone was disagreeing with me when I said that this can be universally applied, and I don't have the mathematical expertise to do more than take one shape at a time and do the calculations to demonstrate it. I was hoping for a more universal mathematical proof if it can be done, or possibly to be set straight if there are any odd exceptions that I have not considered...

Many thanks!

Latimeria wrote:The ratio of surface area to volume is of extreme importance to the efficiency of cells, and I understand that this ratio decreases whenever you proportionally enlarge any three dimensional object. I can easily demonstrate this for a particular shape, but is there a mathematical proof that can show it is true for ANY 3D geometric shape?

Someone was disagreeing with me when I said that this can be universally applied. . . . .

I suspect that it doesn't always hold unless the object is convex. But otherwise, the surface varies as a square of the inradius, but the volume varies as a cube of the inradius, and cubes grow more quickly than squares.

I rather think it does apply generally, whatever the shape.

...proportionally enlarge any three dimensional object...

- that is a linear transformation represented by a diagonal 3 x 3 matrix with the same number (the scaling factor) at all three points on the diagonal. So the volume of any shape on which that linear transformation operates will be increased by the cube of that scaling factor.

Not sure whether there is an equally straightforward way of gauging the increase in surface area, but intuitively it seems obvious that it increases by the square of the scaling factor.

Thanks, that makes sense.

Except one thing. Ughaibu, you seem to imply that concavities in an object could create exceptions to this. Is that accurate? If so, could you give an example? I'm trying to figure out how that might work. Thanks.

Latimeria wrote:Ughaibu, you seem to imply that concavities in an object could create exceptions to this. Is that accurate?

I dont know.

Latimeria wrote:could you give an example?

There are mathematical objects with infinitely large surfaces and zero or finite volume. I was thinking of a sea urchin type object with spikes approaching Gabriel's horn. I dont know if this can be done with finite objects, you might find something if you look up Besicovitch sets.

For a shape defined as

{(x,y,z) in ℝ3 s.t. F(x,y,z) = 0} where F is a smooth function, I think the answer is yes, the surface area to volume ratio scales as 1/𝜆 where 𝜆 is the scaling factor applied to x, y and z.

The volume of the shape is just the integral of the volume three-form dx^dy^dz and so scales as 𝜆3

To get the area you'd have to parametrise the surface of the shape in local coordinates u,v

x=f(u, v)
y=g(u, v)
z=h(u,v)

If I write the tangent vectors (directional derivatives in directions u and v) to the surface as

Tu = (∂f/∂u, ∂g/∂u, ∂h/∂u)
and
Tv = (∂f/∂v, ∂g/∂v, ∂h/∂v)

then the area of the surface is

∬ sqrt(|Tu|2|Tv|2-(Tu.Tv)2)dudv

Now as x->𝜆x, y->𝜆y, z->𝜆z, then clearly f->𝜆f, g->𝜆g, h->𝜆h
and so Tu ->𝜆Tu, Tv ->𝜆Tv

hence square root in the double integral scales as 𝜆2

I think these results hold for any smooth manifold in ℝ3, regardless of convexity and concavity. I'm sure that if you relax the smoothness, there will be pathological examples where it doesn't hold though.

Very thorough, thank you twistor.

Ughaibu, when you said sea urchin, I initially thought of reducing it to a spheroid object with very long cone-shapes emanating from it (both of which would clearly obey the SA:V decrease), but your mention of Gabriel's horn and Besicovitch sets are new to me. Very interesting at a glance, but I'll have to sit down for a while looking at the math before I feel as though I fully understand the implications. Thanks!

It seems to me that if the ratio decreases for a cube, probably the simplest shape to calculate, do the tetrahedron if you want it as the simplest. If also it decreases for the sphere as the ultimate shape in the progressive increase in the number of faces and the ratio still decreases then it seems safe to assume that it is generally true for all in between. The ratio is important for the size of warm blooded animals since they need a surface area commensurate to their volume to dissipate their internally generated heat. This limits the size warm blooded animals can acheive. At least that is how I always understood it.

I have seen many interesting answers to the question concerning cells. When I think of cells, I think of one large cell, such as a humming bird egg or much larger birds. The only time the cell exceeds its shell is when the egg is cracked. But if the eggs get larger then the shell wall gets thicker. It seems to me the birds and egg layers have a natural formula that is in practice.