"Build the best boat" with given area

[greypilgrim]

Hi.

Given the area, what is the shape of an infinitely thin surface that can carry maximal load on water, i.e. has the best buoyancy just before water gets in? Is it the hemisphere?

 

[.Scott]

You want the shape that will hold the greatest volume per surface area - given a plane to work with as one side.
But you can eliminate the plane by putting a mirror image of that shape on the other side of that plane. Then you will be looking for the shape that gives you the greatest volume per surface area - which would be a sphere.

So yes, the hemisphere is the answer.

 

[greypilgrim]

Yes that was my thought as well, but I'm not sure about this mirror image thing, what exactly is the argument behind that?

 

[jbriggs444]

You are trying to optimize the shape of a single boat. That is equivalent to optimizing the shape of two boats with the identical shape sitting side by side. That is, in turn, equivalent to optimizing the shape of two boats with the identical shape joined top to top at the water line, one displacing sky and one displacing water. And that is, in turn, equivalent to optimizing the shape of one big blob (*). That is an easy problem whose solution is a sphere.

(*) Assuming that no advantage could be gained from using a blob without a plane of symmetry. But it turns out that a sphere has a plane of symmetry.

 

[.Scott]

If the shape of the boat is ideal, it will hold the maximum volume for the given area. Now consider the rim of this boat. We will consider the ideal rim shape (a circle, BTW), to be the shape that you get from the ideal boat shape.

Now consider a closely related objective. Given a rim shape, there are different caps that can be made which meet the plane exactly along that rim - but rising above the plane instead of below it where the boat is. Given the ideal rim shape, what is the ideal cap shape such that the cap's volume to area ratio is maximized. If anything other than the ideal boat shape is tried, then it will either enclose less volume or use more area. In either case, changing it to the boat shape will improve it's ratio making it a more ideal cap.

So, if the shape of the boat is perfect - holding the maximum volume for its area, and you wanted to close the shape with the most ideal cap to maintain the maximum volume to surface ratio. You can do no better than to use the mirrored boat shape. So the ideal boat shape must be half the ideal closed shape, a sphere.