How could we estimate the surface of a tree, a sponge, a caulliflower and things like that?

Question

Please help, I really need to know.

Answer 1

Why do you really need to know? Is some gangster threatening to rub you out if you don't figure out exactly how much paint it will take to cover every last square millimeter of a tree in his front yard for White Tree Day?


I don't have any great ideas, but if you're desperate..., well these might help someone get desperately close.

Cauliflower:
Find some moderately viscous liquid. Pour enough in a container to dip the cauliflower in. Record the original volume of the liquid. Dip the cauliflower in the liquid. Pull it out. Let the excess drain off of it. Then measure the volume of the liquid that's not on the cauliflower. The difference in volume is how much is on the cauliflower. "Estimate" the average thickness of the liquid that's on the cauliflower (this is the desperate part). The viscosity of the liquid is important here, because you want something thick enough to measure. It'll still be small, but a millimeter is easier to measure than a micron. Anyway, suppose that every part of the cauliflower were covered in the same thickness and that a rectangular prism could be made of the same thickness. The area of one of the big faces of the prism could be calculated by dividing the volume of the liquid on the vegetable by the thickness.

I have no idea how accurate this would be.
A better question would be, why in the world would this be useful?


Tree:
Estimate the surface area of a leaf. Estimate the number of leaves on a tree.
Assume that each branch (or section of a branch) is a truncated cone. Calculate the surface area of each truncated cone.
Decide if you were being generous or stingy in your calculations and add or subtract some.


Sponge:
Weigh the sponge. Dissolve a bunch of salt in water. (Measure the salt first. Or, weight the water before adding salt. Then weight it afterward.) Dip the sponge in the water. Let the excess drain out. Make sure there's not enough liquid on the sponge to drip off. Measure the weight/volume of the water left over. Let the sponge dry completely. Weigh it again. (Mathemagical part:) Estimate the thickness of the dried salt on the sponge. A microscope would be good for this. Figure out what volume of salt is still on the sponge, and do the same trick as with the cauliflower.
This type of thing might work on the cauliflower as well, but you'd have to choose a solute and a solvent that won't get absorbed by the vegetable. Perhaps paint would work. You'd have to do some experimenting to see how thick the paint dries and its dry weight.


If you had some really expensive modeling equipment, you could have some laser-guided modeling equipment map out the surface of the object and arrive at a really great estimate of the surface area.
If anyone says that this solution is not practical, asks him what practical value there is to estimating the surface area of a tree.

Answer 2

Some kinds of fractals are abstractly considered to be somewhere between a line and a plane and thus are often described as having 1.3 dimensions or so on. (I think Serpinski's Gasket is like that.)
Let's consider a 10" cube of sponge. Its volume is 10^3; the external surface area of one face is 10^2. Perhaps its "surface area" in three dimension may be approximated in terms of 10^2.5 or so.

Answer 3

This is actually an interesting, non-trivial question with important applications. Here's a couple of links dealing with this subject matter. Short of extraordinarily difficult methods such as freezing your object, then vapor coating them, and measuring the increase in mass and the thickness of the coating, probably the best way is through fractal analysis. There are computer programs that allow you to create a variety of 3D fractal textures, in which you can find one that matches your specimen the best, and then you can mathematically estimate the surface area of a given volume of the specimen.

Answer 4

Wow, toughie. Assuming the object is not too big (in other words, rule out trees for this), I say dip it in chocolate. The chocolate should coat the object evenly. Then, if you know how much chocoate is needed to coat a given square inch of the object, the amount of chocolate before and after you coat the object should be a measurement of the surface area.

Answer 5

This is an interesting field of mathematical research.

This site, for example, extensively explores

the fractal structure of a cauliflower :

http://icpr.snu.ac.kr/resource/wop.pdf/J01/2005/046/R02/J012005046R020474.pdf

Answer 6

try surface area.

Answer 7

break it up into either polygons or 3-d shapes