Area of contact between two spheres/size of 'point'

[chris_75]

I was wondering what the theoretical area of contact between two touching spheres would be. After thinking about this for some time I've come to the conclusion that they would basically meet at a single point, much like a tangent to a circle on a cartesian plane, and whatever little information I could find related to this on the internet supports this. Although, apparently, a 'point' has no actual size. I'm finding this hard to come to terms with- if two spehere met at a single point, which technically has no area or magnitude, how are they touching? Wouldn't they have a shared area between them, even if the area was only one atom?

Edited January 2, 2009 by chris_75
Typo

[UC]

This depends entirely on if you want to operate in a system of pure mathematics or in the real world. In a system of pure mathematics, then yes. The contact between the two would be a single point. If you want to operate in the real world, but want to use two perfect spheres of an infinitely incompressible material (not quite the real world), you still have to contend with the fact that atoms aren't quite as definite as mathematical objects. The aren't hard, sharply defined, closely packed spheres.

[chris_75]

This depends entirely on if you want to operate in a system of pure mathematics or in the real world. In a system of pure mathematics, then yes. The contact between the two would be a single point. If you want to operate in the real world, but want to use two perfect spheres of an infinitely incompressible material (not quite the real world), you still have to contend with the fact that atoms aren't quite as definite as mathematical objects. The aren't hard, sharply defined, closely packed spheres.

 

Thanks for your quick response.

I think I understand what you mean- So basically in the real world, depending on the compressibility of the material there would obviously be an area (however small), although in pure mathematics they would meet at a single point, which through research as I've found, according to Euclid is 'that which has no part' I.e. no area, volume, size etc. I just find it difficult to really comprehend that two things can be touching at a 'point', however the 'point' joining them is techinically zero in magnitude (in turn potentially leading one to the assumption they aren't touching...).

 

Or maybe I'm just too tired to really give this enough thought.

[Shadow]

In real life you can never have a perfect sphere, which is probably why you can't imagine them meeting in a point; they never do. In real life, the imperfections of the sphere cause the spheres to meet in a 'point' that has area. That is not so in pure maths, where one deals with perfectly round spheres.

 

Cheers,

 

Gabe

[throng]

In a mathematical way, if 2 spheres touched you'd actually have 2 points that touch together, and not a single point.

 

..

[Sisyphus]

In a mathematical way, if 2 spheres touched you'd actually have 2 points that touch together, and not a single point.

 

..

 

No, you wouldn't, because there's no such thing. Two mathematical points can't "touch" without being the same point. By definition, points have zero magnitude and no "parts." Their only quality is location. If the location is the same, it's one point. If their locations are different, they're not "touching," and you can fit an infinite number of different points in between them.

[throng]

No, you wouldn't, because there's no such thing. Two mathematical points can't "touch" without being the same point. By definition, points have zero magnitude and no "parts." Their only quality is location. If the location is the same, it's one point. If their locations are different, they're not "touching," and you can fit an infinite number of different points in between them.

 

Hello, I see your point.

 

I find there is some parady in that. Under that premise there cannot exist a conceptualisation of two points.

 

If they touch they are one and if they dont there are infinite points of distance, so two points are completely mathematically impossible, or haven't been worked out yet.

[Sisyphus]

Hello, I see your point.

 

I find there is some parady in that. Under that premise there cannot exist a conceptualisation of two points.

 

If they touch they are one and if they dont there are infinite points of distance, so two points are completely mathematically impossible, or haven't been worked out yet.

 

I don't see why that makes two points impossible. When I say there can be infinite points between them, I don't mean there's an infinite distance. Any two points with a finite distance between them can have an unlimited number of different points between them.

[the tree]

As a point of pedantry, two points only have an infinite amount of points between them in a continuous setting. Although given that we're talking about spheres, the assumption of continuity is fair.

 

Also, if we take a look at balls instead of spheres (a sphere is just the surface, a ball is everything strictly underneath) then they would be touching at no points.

[throng]

I don't see why that makes two points impossible. When I say there can be infinite points between them, I don't mean there's an infinite distance. Any two points with a finite distance between them can have an unlimited number of different points between them.

 

Does the distance consist of infinite points?

 

What does distance consist of?

 

If distance is infinite points there can logically only be either one or infinite points.

 

I mean, what is distance?

 

[the tree]

Does the distance consist of infinite points?

 

What does distance consist of?

 

I mean, what is distance?

Distance is a function of two points. (see: metric)

 

there can logically only be either one or infinite points.

Well of course there are infinite points. Why wouldn't there be? We're talking about spheres here.

[throng]

Distance is a function of two points. (see: metric)

 

Well of course there are infinite points. Why wouldn't there be? We're talking about spheres here.

 

Thanks for that, I am pretty interested in that kind of thing.

 

It's probably adequate to say we're speaking of two circles of curves. touching since only the circumference touches, and then we end up with two 0D points touching, so actually, all this is only about two 0D points touching, the rest of the sphere is irrelevent.

 

So if the spheres touch and the area is 0 points of contact, there must be a distance so they don't actually touch.

 

[the tree]

In the case of the spheres - the intersection is 1 point. 1 point, by definition, has 0 length, area and volume. I don't see how any confusion could arise there.

 

In the case of the (closed) balls - there is no intersection, but the distance between them is the depth of a plane - again 0.

[throng]

In the case of the spheres - the intersection is 1 point. 1 point, by definition, has 0 length, area and volume. I don't see how any confusion could arise there.

 

In the case of the (closed) balls - there is no intersection, but the distance between them is the depth of a plane - again 0.

 

Oh, now I understand more clearly.

 

The outermost points on a ball can touch with a distance of 0 being the thickness of a plane, but points cannot touch, being nothing.

 

I say two points can touch and distance is zero. If the outermost 2 points of a ball can, two points can, just ignore the rest of the ball or perhaps make r=0.

 

Do you realise that distance is nothing? It is just invented to seperate points or to give substance to points, because a point is not a location unless relatively distant from other points.

[the tree]

The outermost points on a ball can touch with a distance of 0 being the thickness of a plane,

Except it doesn't have outermost points - it's a bounded set over a open interval meaning that no member of the set is the nearest to the edge. It has an outer bound but that is not part of the set.

 

I say two points can touch and distance is zero.

The distance between two points is zero if and only if they are the same point - that is part of the definition of distance.

 

Do you realise that distance is nothing? It is just invented to seperate points or to give substance to points, because a point is not a location unless relatively distant from other points.

That's ridiculous, tons of geometry can be constructed way before a notion of distance is introduced. Edited March 7, 2009 by the tree
Fixed typo - closed <-> open.

[Kyrisch]

Except it doesn't have outermost points - it's a bounded set over a closed interval meaning that no member of the set is the nearest to the edge. It has an outer bound but that is not part of the set.

I'm pretty sure that is the definition of an open interval, [math](-1,1)[/math] which would not include the bounds, whereas [math][-1,1][/math] (a closed interval) would.

[the tree]

Sorry, yes, corrected. Terminology fail.

[BigMoosie]

To the original poster chris_75:

 

Another similar oddity exists with statistics. If you look at a population of humans, and try to calculate the probability that a random human has a height of 'x' metres, then for *any* value of 'x' the probability is 0.

 

If all heights have a probability of 0 then it may appear at first that the height is undefined, but that's not true. This is not just being silly, this is how standard statistical theories define such a random variable.

 

But of course, like your paradox, in reality this tends not to hold true since, for example, measuring devices can only be so precise.