Surface w/ max volume and min surface area

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Discussion Overview

The discussion revolves around the problem of identifying a geometric surface that encloses the maximum volume while minimizing surface area. Participants explore various mathematical approaches and concepts related to this problem, including the calculus of variations, Lagrange multipliers, and historical context regarding isoperimetric problems.

Discussion Character

  • Exploratory
  • Technical explanation
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • Some participants suggest that the problem can be framed in two ways: maximizing volume for a given surface area or minimizing surface area for a given volume.
  • It is proposed that under certain conditions, the solution to the problem is a sphere, supported by the behavior of soap bubbles which minimize surface area.
  • Some participants mention the use of calculus of variations as a method to prove the sphere's optimality in this context.
  • There is a reference to the Plateau problem, which relates to finding surfaces with minimal area under certain constraints.
  • Some participants express uncertainty about the proof and seek hints or mathematical calculations related to the Euler-Lagrange Differential Equation.
  • Others note that there may exist special surfaces that also meet the criteria of maximum volume and minimum surface area, though their characteristics are not clearly defined.
  • One participant discusses the historical context of isoperimetric problems posed by the ancient Greeks, indicating that the circle (or sphere in three dimensions) is the solution to these problems.
  • A participant provides an example comparing the volumes of different box shapes to illustrate the concept of maximizing volume relative to surface area.

Areas of Agreement / Disagreement

Participants generally agree that a sphere is likely the solution to the problem, but there is no consensus on the proof or the existence of other special surfaces. The discussion remains unresolved regarding the specific mathematical methods to demonstrate these claims.

Contextual Notes

Some participants mention the need for assumptions regarding the surfaces being considered, and there is a lack of clarity on the conditions under which the proposed solutions hold. The discussion also highlights the complexity of the mathematical proofs involved.

ksle82
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I'm attempting to solving this problem but do not know how to begin. Any help would be appreciate.

What geometric surface encloses the maximum volume with the minimum surface area? How would you prove it?
 
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Well, first of all, you should state your question more precisely:
Alternative 1:
Of all geometric objects with the same volume, which has minimal surface area?

Alternative 2:
Of all geometric objects with the same surface area, which encloses the maximal volume?

Under certain assumptions of niceness, you may solve problems like these with the calculus of variations.

Yhen, in both cases, the ball (solid sphere) will be your solution.
 
If you have some constraints, use Lagrange multipliers.
 
ksle82 said:
I'm attempting to solving this problem but do not know how to begin. Any help would be appreciate. What geometric surface encloses the maximum volume with the minimum surface area? How would you prove it?
A soap bubble provides an excellent model. The surface tension in a bubble causes the surface area to be minimized. Since it forms a sphere, this is evidence that the minimal surface area for a set volume of air is a sphere.
 
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Since there is only one answer to this question (i.e., a sphere) is there a proof for this? Intuitively it makes sense that it would be a sphere, but there must be a rigorous way to demonstrate this..
 
I'm out of touch with it, but I'm pretty sure you can use the Calculus of Variations to do it.
 
Isn't this related to the (very hard) Plateau problem?
 
Yes, the general study of functions giving maximal values (as opposed to numbers) is precisely the "Calculus of Variations". And, yes, one can prove, using the calculus of variations, that, under certain conditions, the surface enclosing maximum volume for given surface area, or, conversely, having minimum surface area for given volume, is a sphere.
There do, however, exist rather peculiar 'special surfaces' that also have those properties.
 
Well, I have a feeling that the answer would be a sphere but I just don't know how to solve it using calculus. Any hint would help a lot.
 
  • #10
There do, however, exist rather peculiar 'special surfaces' that also have those properties.
What would these surfaces look like? Could they happen in the real world, for example, would a soap bubble every take on one of these special shapes?
 
  • #11
Typically, those surfaces that cannot be discerned by standard variational techniques, yet represent extremizing values, are non-smooth surfaces.
 
  • #12
ksle82 said:
What geometric surface encloses the maximum volume with the minimum surface area? How would you prove it?
...
Well, I have a feeling that the answer would be a sphere but I just don't know how to solve it using calculus. Any hint would help a lot.

this is one of the isoperimetric problems proposed by the ancient greeks; one was to find the plane figure with maximal area if given a prescribed perimeter. the other was the other way around, to find the plane figure with minumum perimeter given a prescribed area. the answer is a circle (or sphere i 3 dimensions i guess); jacob steiner proved it in the 1700s or whenever he was around.
 
  • #13
proof of calculation

hi guys

would it possible for somebody to actually write out the mathematical calculations for the calculation of miminum surface area of a bubble using the Euler-Lagrange Differential Equation.

thanks
vishak
 
  • #14
ksle82 said:
I'm attempting to solving this problem but do not know how to begin. Any help would be appreciate.

What geometric surface encloses the maximum volume with the minimum surface area? How would you prove it?

Well you just know...I guess if you try to make a garden you don't want to make it very long and very naroow or you'll lose surface. A cubic box holds more than a really wide, really low one. I mean reducing to absurd, a box with width and length of 10 and height of 1 has a volume of 100 and one with 33 on each side has a volume of 33^3. A tethrahedral box takes far less than a cubic one...and so on.
 

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