How important is minimizing surface area? Is it a problem that great minds have worked on? Why?
Does minimizing surface area have anything to do with maximizing volume?
The formulas won’t do much for me, but some sort of histry will help.
I know, I may sound like a broken record at this point, but I’m trying to put my research project together, and I’ve got the “art” down, and I plan on speaking to some math professors on Thursday, and asking them some questions about the math area. I’ve also been looking through some webpages, but I would like a chance to have some discussion on the subject with some of the smartest people I know before I go and talk to my professor. I don’t want to waste his time.
Well, the process of minimizing a given quantity is very frequently used in applied math. From a theoretical standpoint, it’s less interesting.
A question that was asked is, for a given volume, what shape minimizes/maximizes the surface area? That’s something that you might want to look into. The theory used to answer questions like that is the calculus of variations, and while the details might be a bit much for you, you would probably do well to at least mention it.
Presumably you’re talking about “minimizing surface area subject to some constraints”–e.g. mimimizing surface area for a given amount of volume. Otherwise, minimizing surface area is a pretty trivial problem.
I’d bet an engineer could think of some good real-life applications. Alas, I’m not an engineer.
As far as actual applications–very frequently chemical engineers want to maximize the surface area of a shape with a given volume, so as to make a reaction run faster. There’s an organelle in the human cell that has evolved according to this principle.
[On preview I see others have covered some of the same points]
Not to give you too much work to do, but I can only think of somewhat broader subjects. Your particular question is one sort of application – I’d think of it as one math problem, whose solution can be worked on through the general techniques. Also note that to put it in more strict mathematical terms, you have to say “minimizing area subject to certain constraints” (even if they are only aesthetic constraints, or more clearly defined, as in “minimize area & maximize volume”).
The broad subject of optimizing functions is ‘calculus of variations’ (technically, the calculus of variations is only concerned with finding particular points like minima or maxima, but the applications apply to optimizing). The father of this branch is Euler, so that’s one place to start.
Not that optimization problems haven’t always been looked at. There’s the story of Dido (in the Aeneid). She was given the task of getting as much land as could be covered by the hide of an ox. She cut the ox-hide into strips, and tied them together into one long connected strip. The optimization problem then becomes enclosing the most land area with a restricted length curve. (The answer is a circle.)
Minimizing is very important in problem solving. Consider a simple equation such as 2x-3=0. You can solve it with simple algebra [2x=3; x=3/2]. You can also solve it graphically. Plot a line y=2x-3. Now look on the graph and pick the minimum y, with the constraint that x is not negative. By minimumizing y, you can solve the equation.
Solutions to complex problems that can be expressed mathmatically, can be found even if you can’t solve them with simple or complex algebra. Basically you set a computer to work to find minimums rather than solving the equations directly.
You can model chemical reactions and use minimizing techniques to find minimums in such things as free energy. The absolute minimums will be stable states. Local minimums will be metastable and tend to be useful for chemical processing.
It is the practical applications of minimizing techniques that make the theroetical studies valuable.
This really isn’t at the level of sophistication you’re looking for, but minimal surface area and maximum volume tells you why bubbles are round and why water likes to bead into little spheres. I could give more examples, but not much more brainy ones :).
Yes, but only if that’s the only boundary you have to deal with. If, for instance, you’re on the (presumed straight) coast, and the ocean counts as a “free” boundary, then you want a semicircle.
The story is told of a farmer, an engineer, a physicist, and a mathematician. The farmer had 400 feet of fence, and wanted to build the largest possible enclosure for his animals, so he asked the other fellows for advice. The engineer tried several different sorts of rectangles, measured the area of each, and finally settled on a square 100 feet on a side. The physicist took the fencing, and arranged it in a circle of circumference 400 feet. The mathematician took a small segment of fencing, wrapped it as tightly around himself as he could, and said “I declare myself to be on the outside”.
There are loads of times I need to minimise / maximise surface area in my job (I’m an engineer):
Minimise surface area of a tank, subject to constraints, to minimise heat loss.
Maximise surface area in a heat exchanger, again subject to constraints, to achieve good heat transfer.
Maximise design surface area of a catalyst pellet, subject to structural constraints and a consideration of the ability of liquids to get into the pores, to give a good surface for a chemical reaction.
Historically, probably the most celebrated challenge in the field of minimising surfaces was the Plateau problem. Plateau (1801-83) was a Belgian physicist who pioneered the study of the shape of soap films. Fairly formally, the problem was Prove that for a given closed curve in space there is a surface of minimal area having this curve as its boundary. This is intuitively obvious to the likes of physicists, but turned out to be very difficult to prove rigorously. Despite some early major contributions from Riemann and Weierstrass, it only fell in 1930-1 with papers by Tibor Rado and Jesse Douglas. The latter won one of the first Fields Medals in 1936 as a result.
There’s a brief introduction to the history in The Hilbert Challenge (Oxford, 2000) by Jeremy J. Gray.
No one has addressed this yet: In the real world, a common concern is that you have to pay for that surface that comprises the surface area. If you are selling containers, and your factory gets an order for a million, every square inch of surface you can omit is less material used, therefore money that can be spent on your salary, or maybe just on a golden parachute for your CEO.
Another real-world example: Your water heater is a shape which is designed to have the minimum surface area for its volume (to avoid heat loss), with some practical constraints. It would be a sphere, except that it would be harder to mount securely, and it would take up an awkward space (wasting more floor space and using up less airspace above the floorspace it takes up, which you can’t use for much else). Probably some sort of ellipsoid shape would be better than a cylinder, but they’ve also got the constraint that it be easy to manufacture. A cylinder does the job pretty well.