Redditor splongo asks two questions:
I'm at a loss for a completely coherent answer. I don't think that, as a physical reality, there can be a perfectly spherical or flat object. But as a theoretical model (e.g. the formula for a sphere), there can be such objects, and therefore intersection between them. Would the surface area of the intersection point be non-dimensional, or just incalculably small?
How would you answer splongo's questions?
Link -via The Agitator | Photo: katerha
If a perfectly spherical ball is sitting on a perfectly flat surface, what is the size of the contact area? Would it not be infinitely small?
I'm at a loss for a completely coherent answer. I don't think that, as a physical reality, there can be a perfectly spherical or flat object. But as a theoretical model (e.g. the formula for a sphere), there can be such objects, and therefore intersection between them. Would the surface area of the intersection point be non-dimensional, or just incalculably small?
How would you answer splongo's questions?
Link -via The Agitator | Photo: katerha
Thought-experiements like this help identify and resolve ambiguous definitions.
Yes!
"When something is divided by zero, I see it as mapping out the potentialities of any non-differentiable object. What are the potentialities? Infinite. Existence, in this sense, is defined by differentiation/integration through the relation of parts to a whole. The reason it is infinite is because it's potentiality."
We have to think of "contact" in the everyday sense and assume that the antagonism of the electrons involved is what is called "contact". In that sense there is contact.
But what is the contact surface? probably approaching the limit 0, but we mustn't think of 0 as nothing. Rather 0 denotes an undifferentiable object.
1/0, 1 lim-> 0, produces infinity because you are saying differentiate a non-differentiable object, and the potentiality thereof is infinite. The differentiation must continue infinitely on an essentially non-differential object. And this is the way we see reality; as basically a non-differentiable object (noumena) as differentially viewed (phenomena).
Or to put otherwise; the reason Gödel's incompleteness theorem works is because all mathematics is the differentiation of an essentially non-differentiable space so it always depends on something which is not included in the differentiation.
Let say for example; We readily differentiate between nature and nurture and this had led some in the past to think that there was an antagonism of the two, but more over there was this nebulous void in-between where nature and nurture were active. Now, however, we realize that nature and nurture are essentially non-differentiable, you cannot separate them out and the closer you examine the "fine line" between the two the more the line seems to vanish. All perception is differentiation of a fundamentally non-differentiable existence.
At least, IMHO.
your question is defining the limit within a calculus equation - the limit of x as x approaches 1.
this will help you figure out what is the tangent (aka the surface) of particular contact point(s) - of the sphere.
In 2 dimensions (circle) you will need 2 points to figure out the tangent. As your 2 points become infinitely close you will obtain a true tangent
as your 2 (or 3) points come closer together (0.0001 units apart, vs 0.0000001 units apart, your
In 3 dimensions (sphere) you will 3 points and that the equation for a sphere is x^2 y^2 z^2 = r^2 but thats even more math i'm not able to answer
check out khanacademy .com for its lesson on limits
I say there is touching in both cases. Not only that, but this view of matter being "mostly empty space" is outdated by about 50 years; your so-called 'empty space' is full of virtual particles, which are the carriers of force between particles.
You also wrote: "i said 'two objects are touching if the contact area between them is > 0.' " . I wrote that this is mathematically incorrect, and gave 5 quotes references which disagreed with you. Not only that, but the the general population also disagrees with you, in that most people say that Utah and New Mexico "touch", even though they don't share a border.
"The answer does not have real scientific value." The things is, it does have value. Thought-experiements like this help identify and resolve ambiguous definitions. The underlying resolution to this sort of problem was figured out in the 1800s when limit theory was used to put a sound underpinning to calculus.
this has gone for too long =) i will rest my argument repeating it. this is a stupid question =)
Excellent idea! Bye.
if you read my previous posts you will see that in mathematical part i have already said the touching area is smaller than any given dimension. I understand and know what you are saying. i cant put them in words as well as you do =)
i will try to rephrase what i said so we can understand each other. since we are practicaly saying the same thing
if there was a perfect sphere in real world (as you demonstrated it is not possible. maybe if we could manipulate electrons as we wish we could make them touch and see. since they are roughly sphare shaped)and push this infinitly rigid sphare to an infinitly rigid perfectly plane surface, the interdection between two objects would be a point as geometry suggests. this kind of touching is not defined in physical world. where matter is concerned, as you said there is no touching at all, what we call matter is mostly empty space so i tried to redifine touching to give the question an answer. it looked like the only way. so i said "two objects are touching if the contact area between them is > 0. if it is zero there is no contact therefore they are not touching. if these shapes are perfect as the question states than they can not touch each other since the area of a point is zero.
again i did this to the "projection" of a theoretical question in real world. assuming there is a perfect sphare.
the answer does not have real scientific value. it is just a "what if" scenario i created.
i hope it is clear that im not mixing math and positive science byt mistake. im doing it on purpose so i can make this stupid question born from ignorance a little bit more concrete by projecting it's completely immaginary nature to real world making assumptions and giving it my own answer "inside the context of weird mixture of science and immagination i created"
this has gone for too long =) i will rest my argument repeating it. this is a stupid question =)
have a nice day
In geometry, "touch" means that there's an intersection. In this case, the plane touches the sphere at a point. For examples:
1) "Take a sphere and place it on a plane. Let's call the exact point where the sphere touches the plane the south pole. The plane is tangent to this point." http://www.learner.org/courses/mathilluminated/units/8/textbook/05.php
2) ".. and considers the point F at which the sphere touches the plane" (Encyclopédia of mathematics, By James Stuart Tanton; in Google Books)
3) "Let O (fig 146) be the center of a sphere; C the point in which the sphere touches the plane of projection ..." ( A treatise on crystallography By William Hallowes Miller; also at Google Books.)
4) "The point at which the sphere touches the plane is a focus of the conic section", http://www.chemistrydaily.com/chemistry/Dandelin's_theorem
5) (From 1840!) "... the line drawn from the centre to the point where the sphere touches the plane will be the shortest line which can be drawn from the centre of the sphere to the plane." (A treatise on geometry and its application in the arts By Dionysius Lardner; Google Books)
Quite obviously a number of people with math training disagree with your statement that "A perfect sphere and a perfect plane can never touch each other."
If you want to be more mathematical, and move from the ancient Greeks to the 1800s, the definition of "touch" corresponds to the idea of a limit. It's adversarial in nature. You pick a finite distance 'd'. Two objects touch if there are parts of the objects which are closer together than 'd', I just have to find examples. You are free to specify any finite value of 'd' you want, all I have to do is beat it. This definition neatly avoids the complex issues of infinities that you're having troubles with.
With that definition it's trivial to see that a sphere and a plane touch. If you pick "12 inches" then I just need to show a place where the sphere is within 12 inches of the plane. If you choose 5Å then I just need to show a place which is smaller than that. It should be obvious that no matter what finite size you pick, I can find a point on the sphere and a point on the plane which is closer. (After all, the distance between them is 0.) Hence, the sphere touches the plane.
But moreover, the generally accepting meaning of "touch" says that New Mexico "touches" Utah, even though the borders meet at a perfect geometrical point. This agrees with the limit-based mathematical definition, but by the definition you want to use, you say that those two states don't touch.
The issue here is you're trying to apply loose principles which work on a human scale to something beyond your everyday experiences. You need to define your terms in a way that's applicable to what you're trying to analyze.
The thing is, there is no point talking about electrons or anything. As you have already said. such object cant be made with "matter" as we know it. As we keep thinking about it it makes less and less sense. As i said couple of times before, the question is wrong.
if the contact area of 2 objects is zero they do NOT touch each other. A perfect sphere (not possible in physical world) and a perfect plane has contact area of zero. therefore i say: A perfect sphere and a perfect plane can never touch each other. thats my answer to the theoretical part. the practical part... well.. it doesn't exist
So you either have to talk about abstract math, or you have to talk about the real world. If you talk about abstract math, then the contact area is 0. If you talk about the real world - where your intuition comes from - then things deforms and you have a non-zero contact area.
Deformation doesn't even require atoms. The same thing would occur (as Philip Howie pointed out) in anything which is not perfectly stiff. "Stiffness", after all, being defined as "resistance to deformation."
This is not a perplexing question.
There's no reason to even make the statement: 'if we cannot calculate the value of pi to an absolute, then can we even create an accurate formula for a “perfectly spherical ball”?' -- a "perfectly spherical ball" is defined as the volume at or within distance 'radius' from a point. Pi isn't even involved in that statement. In any case, nothing in the math requires knowing an exact value of pi; we know the value exists, so we're free to use the symbol ? instead.
Alan Fray commented "in reality there is no contact point as the molecules do not actually touch." This is a somewhat useless comment. I think it's meant to point out that "touch" is not a well-defined topic. Molecules don't have a real surface, so don't "touch" in the common sense. Instead, there are the electrons around a molecule, and these electrons interact with other molecules. At a far distance there's the electrostatic force, when a charged ballon causes long hair to rise up, we don't say that those are touching.
When two molecules get closer, there's an attractive force. Geckos use this to cling to walls. But even closer there's a repulsive force when the electrons are so close that they are trying to occupy the same volume at the same time. The Pauli exclusion principle describes this effect. The effect is that the two molecules start to repel each other.
This repulsion is very much like a hard wall (see the Lennard-Jones potential for a picture). And this repulsive force is what we generally talk about when we say that two things are touching.
Using that understanding, two things touch when their electron shells are close enough that their repulsive terms are a significant component of their force interactions.
If contact area is defined as say, the absence of atmospheric space between surfaces, then we would be able to determine the smallest atom in said atmosphere, find where between the surfaces it falls below that atom's compressed diameter, and find the area of the resulting circle.
If defined as the distance in which no other particle, sub-atomic or otherwise could fit between them, the same thing occurs.
Also then if our sphere is massive, that "area of contact" would thus increase.
Though I would also contend that two such perfect surfaces would be made of the same material and, by being well within distances of their molecular bonds, would fuse out to the point where the molecular bonds' lengths were exceeded.
Human intuition tells us that if you drop a perfect sphere on a perfect plane, they touch. This is what we observe on daily basis or we THINK we observe.
Such thing does not exist. there is no contact area. Therefore the question: "what is the size of the contact area? " is misleading. there is no answer to the question. becaouse the question is not valid. It is like asking "what happens if a goldfish forgets to wear his tie to work" goldfishes do not go to work an do not wear ties.
you can however create a fantasy world where fishes go to work (like in spongebob) and try to answer the question in the context of this artificial world. We call this mathematics. The question is a physics question but if you want to answer to this immaginary situation you can use math to imagine what happens.
what happens is this:
you can ASSUME there is a contact area albeit smaller than any immaginable dimension. so the area can be expressed with lim-->0 (sorry cant use math signs here. the answer will have some value but will not answer your questions since a quantity infinetely close to zero(infinetely smaller than planck length)can be only percieved as zero in a practical world.
? said it before and i will say it again:
the question is NOT valid. it assumes there is a contact area and asks how big it is. there is NO contact area. You can think that if there is no contact area there is no force to keep the ball on top of the plane (if gravity is aplied) so it falls through it etc. etc. etc. but again... the question is not inside the physics domain.
you may try for days to wrap your head around this and you would fail because you are asking the wrong questions. quit trying to find out how this is possible and think is it is possible at all
@John Farrier ...
In theory, if we let the symbol Pi stand for all infinite digits of Pi, then yes, the formula is accurate for a perfectly spherical ball. If we try to apply Pi, then it will always be imperfect since we don't actually know the entire value, no.
But in real life applications? 3.14 is enough. But I'm speaking as a mathematician.
With regards to the original question, the point of contact would be a point, generally given as undefined but understood to be infinitesimally small. Again, for all real life applications, incalculable.
The only possible complexity occurs if you ignore the word "perfectly".
does this answer your question “John Farrier” ?
Since other commenters and redditors are suggesting that these questions are more complex than you are condescendingly implying, then no.
I find Seventh's answer particularly interesting.
Here is a youtube that seems to explains it.
http://www.youtube.com/watch?v=2XbxW8K5CbM&feature=related
To understand how it works, consider a soft rubber ball sitting on a soft rubber surface. Now allow both ball and surface to deform elastically - the contact area increases but the amount of rubber-air interface decreases, and hence the energy is lowered. (Note that if the ball is sitting on top of the surface, its centre of gravity moves down and its potential energy is also decreased - this helps, but is not required.) Of course the act of deforming the rubber increases its strain energy. Equilibrium is reached when the two energy terms are equal but opposite.
Now turn to splongo's ball and surface, which we're assuming to be much stiffer and hence to resist deformation in a way that rubber doesn't. Initially, the point of contact will indeed be just that - a point. However, the maths (which I shan't go into here) shows that at this point, the change in surface energy with a TINY amount of deformation tends to infinity. No matter how stiff the material, it WILL deform, and the contact area will be finite.
More information - and equations to calculate the contact area - can be found here: http://en.wikipedia.org/wiki/Contact_mechanics
But in the practical world, a perfectly round sphere would not "sit" on a perfectly flat surface. It would roll, I think. Even if it were also a perfectly LEVEL surface, the movement of the earth would cause the perfectly round sphere to roll.
At least until it picked up enough cat hair to stop it.
That does not exist. what you describe as hot is the measurement of temperature and therefore the thermic energy of an object. There is no such limit. If you keep giving thermic energy to an object it will heat up as long as you keep heating it up.
Absolute zero is the absence of energy not the temperature all motion stops. It is as immaginary as the contact area given in the question above
The contact in the question is a "point" which is described as (quoting from wikipedia)In geometry, points are zero-dimensional; i.e., they do not have volume, area, length, or any other higher-dimensional analogue.
the answer is: contact area is ZERO. end of story
If you insist on trying to calculate the forces or the pressure created by the sphere etc. you are welcome to try. Points are immaginary. physics does not deal with immanginary. non such calculation can be made. you cant calculate the friction or pressure (for example) because they would require dividing a number to zero. answers are not "infinite" or "nill" they are impossible.
does this answer your question "John Farrier" ?
I find this question and the possible answers far more perplexing.
But of course this is a non-physical world. If you worked out the force on that point it would be infinite, so something would need to give. The ball would become non-spherical, or the surface non-flat (and more likely, both), until the force in the contact patch - that's the area where the two surfaces touch - is no longer strong enough to continue to deform the objects. Either that, or one of them breaks.
But in physics, the contact area is as "small as it is possible to be", i.e. one squared Planck length (equal to 1.6162×10?35 metres)
http://en.wikipedia.org/wiki/Planck_length
Both are impossible in reality, but mathematicians will come up with a different answer from physicists.