Understanding the Space Efficiency of Spheres: An Explanation

[roger]

hi

Please can someone explain :

Why is the sphere the most space efficient way of enclosing a given volume of space ?

Our teacher mentioned this, but what does he mean by it ?

roger

 

[philosophking]

I would guess that he was talking about things in terms of surface area. The least surface area, the most efficient way of enclosing space. I'm not sure how to prove this though.

 

[roger]

Does anybody know the proof of this ?

 

[HallsofIvy]

A rigorous proof that the sphere has maximum volume for a given surface area involves the "Calculus of Variations" and is, I suspect, too difficult for to be given easily here.

 

[roger]

A rigorous proof that the sphere has maximum volume for a given surface area involves the "Calculus of Variations" and is, I suspect, too difficult for to be given easily here.

but why does the sphere have the max volume for a given surface area ?

How can it be imagined ?

btw, what is calculus of variations ?

roger

 

[ceptimus]

Think of a soap bubble. It's fixed at a given volume by the amount of air trapped inside. The surface tension of the soap film tries to pull the surface into the lowest area possible. What shape are soap bubbles? Spheres of course.

 

[dextercioby]

but why does the sphere have the max volume for a given surface area ?

How can it be imagined ?

btw, what is calculus of variations ?

roger

This is a classical problem in mathematics,but some other simpler or at most equivalent can be imagined.For example:what is the shortest distance between 2 points in an euclidian space/in a plane??What is is the shortest distance between 2 points on a sphere??What is the closed curve that encloses the largest area at constant length of the curve??What is the closed surface that encloses the largest volume at constant surface area??What is the closed curve that has the shortest length at constant enclosed surface area??What is the closed surface that has the least area,at constant volume enclosed...??What is the curve/trajectory a pointlike particle should follow in constant gravitational field of intensity "g" as to go from a fixed point to another in the shortest time possible (the famous "brahistochrone problem"??

If u came up with the correct answers,i'd say u have a good "feeling" of mathematics,but you need to learn solid calculus (diff+int) as to move to the "next level":variational calculus which explains ans proves all the results to the prior stated problems and to many other...,including Lagrange and Hamilton classical (and Newton equivalent) dynamics.

Daniel.

 

[ceptimus]

In two dimensions, imagine looking down on a crowd of people with a rope tied around all of them. If you pull the rope tight, then any people at 'corners' of the enclosing shape will experience more than their fair share of rope pull, and will be pulled towards the centre. The limiting case is a circle, which of course eliminates any corners.

If you think of the molecules of air inside the soap bubble, you can imagine how if the bubble had corners, say a cube, then the air molcules near the corners would tend to be pushed in, until all the corners were eliminated. The 3D shape with the 'least corners' is a sphere.

 

[roger]

In two dimensions, imagine looking down on a crowd of people with a rope tied around all of them. If you pull the rope tight, then any people at 'corners' of the enclosing shape will experience more than their fair share of rope pull, and will be pulled towards the centre. The limiting case is a circle, which of course eliminates any corners.

If you think of the molecules of air inside the soap bubble, you can imagine how if the bubble had corners, say a cube, then the air molcules near the corners would tend to be pushed in, until all the corners were eliminated. The 3D shape with the 'least corners' is a sphere.

Is it also true that if the soap bubble was a cube, the air pressure would be most highest at the corners where 3 surfaces meet and second most highest pressure where 2 surfaces meet ?
Please correct me if that's wrong but if its right, why is this so ?

Thanks

Roger

 

[ceptimus]

Is it also true that if the soap bubble was a cube, the air pressure would be most highest at the corners where 3 surfaces meet and second most highest pressure where 2 surfaces meet ?
Please correct me if that's wrong but if its right, why is this so ?

Thanks

Roger

That's right. Of course, air pressure automatically evens itself out within a small constrained volume, so we don't get cubical bubbles. But if you had say a cubical ice cube, with a tightly stretched balloon enclosing it, then the balloon material would pull hardest on the corners. As the ice melted, the balloon would gradually change shape into a sphere.