First, we know that three dimensions are perceivable by us: one dimension, two dimensions, and three dimensions. The other day, I just happened to think of a way to demonstrate all three (at least in theory):You have a perfectly flat plane and a perfect sphere (lets say about the size of a baseball) rolling on the plane. Well, the place where the sphere touches the plane is one dimensional, the plane is two dimensional, and the sphere is three dimensional.So, that’s just for free. Now for the puzzle (and I don’t know the answer either!):If a perfect sphere is placed on a perfect plane, how large is the point of contact between them? If you claim that it is a thousandth of an inch (say), is that correct? For if the sphere and plane are perfectly round and flat, respectively, it seems that the point of contact is infinitely small (or so it seems to me). It is infinitely small then it would almost seem that one could argue that the sphere never really touches the plane, since there is no (seemingly) literal size to the point of contact.At the same time, if the sphere is resting on the plane, then surely there must be a point of contact.
this problem is not original with me. But it does make me feel good to know that I am thinking like smart people.They said that, first, there is no such thing as a perfect sphere or plane in reality. But a mathematically perfect sphere that is touching another sphere or a perfect plane…would have, at most, a single atom making contact.At least I think that’s what they said. I think they were Pakistani or Cuban.
The contact area would be one point. That point has no size. But that one point would be common to both the plane and the sphere. The same as when two lines intersect or when a line penetrates a plane.And this is a geometry problem I believe, not a math problem.
Dave…
I agree with you that it MUST be touching. But if it has no size (or at least none we can define), then just what IS touching?It kind of reminds me of that claim that if John receives the kick-off in his own end zone, at some point, he goes half the distance to the other end zone…then half of that, then half of that, then half of that, and then…you get the idea.
There are not three distinct entities — the sphere, the plane, and the point of contact. There are only two. The sphere is touching a point
Just as a line that touches a circle at only one point is a tangent line, a plane that touches a sphere at only one point is a tangent plane.
Dave…I agree, but
This is tough. But the logic BB uses seems correct here. But then I am in NC! ________________Some facts but mostly just my [email protected]/