What is going on, or has gone on, in the math world dealing with the concept of maximizing volume while minimizing mass?
I’m doing some research now (Duck Duck Goose, how have you mastered Google? I mean, I know it’s just a search engine, but it kicks my ass all over the place. Eh, I guess I should read the “help” link…)
Anyway, I have found some interesting links, but then I was thinking, “Hmmm, I know there are some brilliant math types on the board…let’s ask them!” So, is there any information you can give me?
Thanks in advance.
Density is a material property which, for the most part, is a constant for a given material. (Pickers of nits will note that density will change with temperature but I am assuming that the question is regarding a solid at a relatively constant temperature).
So, if you increase the volume (of a solid) the mass will increase linearly.
Ok…so, what you are saying is that has volume increases, mass also has to increase. (Ok, I did know that. I retained something from Calculus last year). So how does one go about minimizing said mass? Is it possible?
I’m asking because I’m doing a research paper on Alexander Calder, and I have to link his work to math conceptually. The website says that his work deals with the volume and mass relationship, specifically max volume min mass.
So uh…what does all of this mean?
plg, I want to re-emphisize that what I said only holds true for a solid. If you increase the volume of a gas (actually of the container that holds the gas,) the mass remains the same.
As far as Calder goes, I read the web site and he made art that consisted of shapes made with a metal frame and then covered with textiles. In simpler terms he made bigger and bigger empty boxes. This is a different story entirely and the mass to volume relationship can change.
A mathematical model can be made but you have to assume that the walls of your box have some sort of uniform thickness. You can look, for example, of a cube with 1.0 mm thick walls and find a sphere with the same wall thickness. It’s not that hard to find a sphere and a cube with the same (internal) volume. The masses of the two will definitely be different.
Isn’t there some fractal structure like the Sierpinski Sponge that maximizes volume/mass as you desire? Of course, this sort of thing would be almost impossible to manufacture in the real world.
The Sierpinski sponge has zero volume and infinite surface area, but I don’t know of any work that’s attempted to model the mass of such an object. Intuitively, we would say it’s zero; unfortunatel, intuition just doesn’t cut it with fractals.
Now I’m thinking about what you’d have to do–I think that you’d have to take an integral over the Sierpinski sponge, and I’m not sure if that can be done. Unfortunately, we’re into measure theory now, and I don’t know that much about it.
But yes, for normal shapes, mass and volume are directly proportional. What the OP might be interested in is the maximization/minimization of surface area for a given volume. I believe that falls into the calculus of variations, but that stuff’s not for the light-hearted.
There’s not really anything mathematical about this question… For any given shape, the ratio of mass to volume is the density, and (as mentioned earlier), density is a property of a material. Now, chemists and such are actively trying to find materials with low density. I don’t know what the current state of the art is there.
Wow ultrafilter, I was just guessing but it looks like that was a good WAG. I didn’t know the sierpinski sponge had ZERO volume, I would have figured it would have had an infinitesimal volume. I don’t know how you’d model mass separately… oops, my head just exploded.
Infinitesimals don’t exist in the standard reals. I attempted to disagree with Dex on that one, a long time ago, and was informed of the error of my ways. :o
Basically, you’d model mass by having a function defined on your shape. Then you’d integrate that function to get the mass of the shape. For an object of constant density, the function would have the value 1 at every point.
Eh, if I can’t link this to math somehow, I’m screwed. The deadline to turn in our ideas was today at noon. I’m locked in. FWIW, the professors said that this was a fine topic as long as I can link it to math. As I’m completely clueless re: math and art, I figured it wouldn’t be too hard. sigh
Can you change it to “Maximizing Surface Area, Minimizing Mass”? If not, assume a constant shell thickness and do it the original way. This can be linked to math.
I think I can explain what hajario means. For a given volume, you can construct shapes with different surface areas. Of course, a natural problem is to find the shape with the maximum surface area for a given volume. Since volume and mass are proportional, this is equivalent to maximizing the surface area of a shape for a given mass.
As for the shell of constant thickness part…well, time was I would’ve been able to explain that. Unfortunately, it’s not that time anymore.
I suggest a talk with one of the profs in your math department. Tell 'em straight up what you’re doing and why and I bet they’ll be able to give you some good leads, especially since you’ve had calculus.
There’s a huge difference between dense and not having learned something yet.
This is a bit too complicated to be able to explain well in a message board post. Do you want to go with the surface area model or the shell model? I can try to give you an example or two.
Actually the more I think about it, the more I think that the surface area is closer to the idea I had. So I would appreciate some information about that. Thank you.
Here’s something to get you started, I’ll leave the creative stuff to you.
Consider a sphere:
V = (4/3)(pi)(r[sup]3[/sup]), SA = (4)(pi)(r[sup]2[/sup]), where r = radius of the sphere.
so the ratio of V to SA (V/SA) = r/3.
If V = 100, SA = 300/r.
Cube:
V = s[sup]3[/sup], SA = 6s[sup]2[/sup], where s = length of one side.
V/SA = s/6.
If V = 100, SA = 600/s.
If you take our sphere and cube where r = s, both of which having a Volume of 100, the cube will have twice the surface area. This should get you started.
A few more fun formulae:
Box (not necessarily a cube):
V = (s[sub]1[/sub])(s[sub]2[/sub])(s[sub]3[/sub])
SA = (2)(s[sub]1[/sub])(s[sub]2[/sub])+(2)(s[sub]1[/sub])(s[sub]3[/sub])+(2)(s[sub]3[/sub])(s[sub]2[/sub])
where s[sub]n[/sub] = the length of one side
Cylinder:
V = (pi)(r[sup]2[/sup])(h)
SA = (2)(pi)(r[sup]2[/sup]) + (2)(pi)®(h)
For a given volume, the shape with the minimum surface area is the sphere. This is why soap bubbles are spheres. The surface tension tends to minimize the surface area enclosing a fixed amount of air.