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Jeff B

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In summary, the conversation discusses a question on a navy officer test that asks for a proof showing that the sphere is the most efficient form of surface area to volume. There are various suggestions, including using the divergence theorem and conducting an experimental proof with mercury. However, there is no clear consensus on how to prove this concept.

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Jeff B

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Dick

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HallsofIvy

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2. What does "most efficient form of surface area to volume" mean? Least surface area for a fixed volume?

3. A rigorous proof would involve "Calculus of Variations" which is a rather specialized field of mathematics. Seems a peculiar question to put on a Navy test! Is your friend planning to serve in nuclear submarines? I second Dick- exactly what does the question say?

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Jeff B

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HallsofIvy said:

2. What does "most efficient form of surface area to volume" mean? Least surface area for a fixed volume?

3. A rigorous proof would involve "Calculus of Variations" which is a rather specialized field of mathematics. Seems a peculiar question to put on a Navy test! Is your friend planning to serve in nuclear submarines? I second Dick- exactly what does the question say?

yes, he's going into the nuclear engineering department. Its not that I want anyone to solve it for us, just to get a train of thought going. He already has a biomedical engineering degree. He probably could have done this last year right after he graduated, but he's admittedly a bit rusty. I mean we did sit down for a while trying it. The original question was basically what I said. "Derive the proof showing that the sphere is the most efficient ratio of surfact area to volume". So yes, less surface for a given volume. Looking at it, surface area can increase minimally, and effect volume in much higher magnitude.

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Dick

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jhicks

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The best you're going to get is anecdotal evidence of the sphere's superiority (e.g. comparing it to other shapes, referencing bubbles minimizing surface tension, etc). I'm a recent EE grad and I've never seen something like this from an engineering class. HOWEVER, viewing this as having potential analogues to physics, I think it might be possible to concoct something out of the divergence theorem http://en.wikipedia.org/wiki/Divergence_theorem and taking F to be the magnetic flux density [itex]\vec{D}[/itex], which turns the generalized divergence theorem into Gauss' Law. It would by no means be a proof though.

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klaarn

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The exact question is "What geometric surface encloses the maximum volume with the minimum surface area? How would you prove it?"

My first thought was to try to define some general expression for both surface area and volume and minimize their ratio. Or you could do something anecdotally like jhicks said... The best hand-waiving argument I've come up with is since surface area is length squared, and volume is length cubed, their ratio SA/VOL could be minimized by letter VOL increase unbounded while an increase in SA would basically have to mean the number of sides n increases also. SA will never overcome VOL based on their powers. As n increases, it approaches a continuous surface, and if you assume it to be symmetrical, that would be a sphere... But again, it's vague and requires some willingness to accept the end conclusion. I want to know how to prove it explicitly.

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tiny-tim

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Hi klaarn! Welcome to PF!

The question doesn't specify that the proof has to be

If a physical (experimental) proof will do, I'd say take a blob of mercury and see what shape it forms.

Since the surface tension depends on the area, it should try to form the smallest area!

Oh, and I'd deform it by hitting it a few times wth a hammer, to make sure that it returns to the same shape (in other words, prove that this is a

Will that impress the Navy?

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StatusX

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Specifically, if it didn't have this symmetry, we could rotate it to get a new solution (whether two shapes related by a rotation are different is arguable, but if you set up the problem, say, in a fixed coordinate system, the actual parametric equations for the two surfaces will be different, so you could reasonably call them different solutions), but it seems natural that there should be some unique minimal surface (if you're really shifty, you can use the fact that they implicitly assumed a unique solution in their question).

The only shape invariant under rotations is the sphere, so this is all it could be. Of course, the same argument might show the sphere is the surface of

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klaarn

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there has to be some rigorous mathematical proof though... I've even tried looking through books of proofs in the library, but found very little. i don't want to go bother any of my old profs... but it may come to that.

Spherical proof is a type of mathematical proof that involves using the properties of spheres to prove a theorem or solve a problem. It is a specific application of spherical geometry, which is the study of geometric figures on the surface of a sphere.

Spherical proof uses the unique properties of spheres, such as the fact that all points on the surface of a sphere are equidistant from the center, to prove a theorem. Traditional geometric proof, on the other hand, relies on Euclidean geometry and the properties of flat, two-dimensional shapes.

Spherical proof can be used to solve a variety of problems, including finding the shortest distance between two points on the surface of a sphere, determining the angles and distances of celestial objects in astronomy, and calculating the curvature of the Earth's surface.

One of the main benefits of spherical proof is its ability to solve complex problems involving spherical objects or surfaces. It also allows us to better understand and analyze the world around us, from the movements of celestial bodies to the shape of our planet.

Spherical proof is used in a variety of fields, including astronomy, geodesy, and navigation. It is also relevant in modern technologies such as satellite communication and GPS, where the calculations involved rely on the principles of spherical geometry.

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