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Thread: Potential discovery involving spheres

  1. #1 Potential discovery involving spheres 
    Forum Freshman ScubaDiver's Avatar
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    I have discovered several things, and I wanted to check my claims with you guys.

    edit: I have re-written these claims, as they were incorrect at the start

    Assume you have a spherical ball of clay of some finite size.

    Claim 1: If you split the sphere into n smaller spheres, then the total surface area of those n spheres could be arbitrarily large.

    Claim 2: If you line up the n the spheres, you could create an arbitrarily long line.

    Claim 3: If you sit all the n spheres next to eachother on a flat surface, you could cover a arbitrarily large area.


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  3. #2  
    Universal Mind John Galt's Avatar
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    Is the diameter of these spheres smaller than the Planck length?


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  4. #3  
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    the surface of a sphere is 4/3*pi*r.
    If you were to make your infinitaly small sphere, you would have your clay ball with a finite radius (lets say x) and you are splitting it infinite times.

    so r = x/infinity = 0
    0 is still 0
    infinite spheres with 0 surface have a total surface of: 0*infinity = 0

    I am not a mathematician so correct me if I'm wrong, but I think that first statement is false, therefor, the others are too
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  5. #4  
    Forum Freshman ScubaDiver's Avatar
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    Ok, at this point I believe Claim 4 is incorrect.

    However, lets just start with claim 1. For claim 1 I think I have a reasonable proof

    Assume the original volume of your clay ball is (this means the initial radius would be 1)

    Let n be the number of spheres you split the ball of clay into
    Then the total surface area would be



    This can be simplified as



    So we just need to know the limit of that as n approaches infinity, and it is infinity
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  6. #5  
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    I see where I went wrong.
    1 through 3 seem correct now to me, but 4 I don't think is possible because of 2 reasons.

    1) spheres are spherical (obviously) and thus only touch at a single point at the top or bottom in your infinite planes.
    if you go 3d there is a gap between all the touching circles in in the line between the radius of 2 of it's adjacent spheres.

    You could add planes in 3 directions, but the space could never completely be filled without merging the speres. and if the spheres are merged, they are no longer spheres.

    2) if you were to fill a 3-d space with your spheres, they would gaign a full 3d dimension again rather than just infinitely small points in a line, they would use up a certain volume when going to a third dimension.
    You could not make a full m filled with spheres that originate from a 1dm sphere, no matter how small the pieces.

    You could however infinitely flatten it to form a 2D surface.
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  7. #6  
    sox
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    The problem with the OP is that it assumes that infiniti is an actual number. Infinity cannot be defined and hence is actually more of a concept.

    So you could never split something up into an infinite number of smaller objects even hypothetically.

    Hence you can only end up with finite values for areas and lengths etc.

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  8. #7  
    Forum Freshman ScubaDiver's Avatar
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    The problem with the OP is that it assumes that infiniti is an actual number. Infinity cannot be defined and hence is actually more of a concept.

    So you could never split something up into an infinite number of smaller objects even hypothetically.

    Hence you can only end up with finite values for areas and lengths etc.
    I don't think I am making a false statement by saying the plane could be infinitely large.
    For example, the plane of small spheres could be 1 mile by 1 mile
    It could be 1 trillion miles by 1 trillion miles, and so on.
    There is essentially no limit to its size, so why not say it could be "infinitely large"

    Now, I personally think my claim 4 is wrong. If you put all the resultant small spheres into a cube, then as you increase the number of spheres it would converge into a cube, with side lengths equal to the diameter of the original sphere.

    But Sox, you do agree that the spheres could be laid out to form a flat square of any arbitary size, such as 1 light year by 1 light year, right?
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  9. #8  
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    The more I am thinking about it, the more my head gets confused. I think sox has a valid point though.
    By setting up with an actual item and breaking it up into an imaginary number, we are bound to hit a brick wall when trying to project what we could do with it assuming it's real.

    even if your clay ball was say a mile across. It would still have a finite number of particles making that up.
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  10. #9  
    sox
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    ScubaDiver:

    toonb explains my point pretty well but I'll simplify further.

    In the original post you start with the assumption that you can split a finite size object into an infinite number of particles. This assumption doesn't make sense as infinity is not an actual number.

    Hence the conclusions you arrive at based on this premise are invalid.

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  11. #10 Re: Potential discovery involving spheres 
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    Quote Originally Posted by ScubaDiver
    I have discovered several things, and I wanted to check my claims with you guys.

    Assume you have a spherical ball of clay of some finite size.

    Now assume it is split into an infinite number of smaller spheres. (note that every one of these infinitely small spheres was formed from the original ball)

    Claim 1: The total surface area of all those infinitely small spheres would now be infinity

    Claim 2: If you line up all the spheres, the line would be infinitely long

    Claim 3: If you sit all the spheres next to eachother to form a plane of spheres, then it could be infinitely large. (edit, I rewrote this)

    Claim 4: If we could form an inifinitely large plane of spheres, then that means we could also form an infinitely large space with these infinitely small spheres. (edit, I now believe this is incorrect)

    Thankyou
    An infinitely small sphere has zero radius and zero surface area. Also zero volume.

    Claims 1-4 are false, if they have any meaning at all.
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  12. #11  
    Forum Radioactive Isotope MagiMaster's Avatar
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    If you had some topological clay, and cut it with the axiom of choice, you could get two balls of clay each with the same volume as the original ball. If you don't have the axiom of choice handy, then #4 wouldn't work, even with the topological clay. (Disclaimer: topological clay loses many interesting properties when exposed to chromodynamic forces.) :P

    (Edit: On a serious note, #1-#3 only apply in the limit. Infinity, as has been pointed out, isn't a number and rarely makes much sense outside of limits. Try the word arbitrary instead.)
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  13. #12  
    Forum Freshman ScubaDiver's Avatar
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    Magimaster, I was thinking to use the word arbitrary as soon as I read Dr. Rocket's post.

    Let me state my current understanding of this.

    My original claims were wrong because If one assumes there were infinitely many spheres created, then that implies that each one of those spheres is infinitely small, and there is no such thing as infinitely small. Infinitely small is the same thing as having no size, so my claims fell apart. (like Dr. Rocket said)

    I rewrote the claims in the orignal post based on everyone's suggestions. I believe they are now correct.
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  14. #13 Re: Potential discovery involving spheres 
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    Quote Originally Posted by ScubaDiver
    I have discovered several things, and I wanted to check my claims with you guys.

    edit: I have re-written these claims, as they were incorrect at the start

    Assume you have a spherical ball of clay of some finite size.

    Claim 1: If you split the sphere into n smaller spheres, then the total surface area of those n spheres could be arbitrarily large.

    Claim 2: If you line up the n the spheres, you could create an arbitrarily long line.

    Claim 3: If you sit all the n spheres next to eachother on a flat surface, you could cover a arbitrarily large area.
    These are all true as revised. They represent a decided step up in mathematical intuition.

    However, they are not discoveries and are obvious to any professional mathematician.

    Nevertheless this is real progress. Many people would not realize these facts.
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  15. #14  
    Forum Freshman ScubaDiver's Avatar
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    Thanks Dr. Rocket!

    I'm glad I've got them correct now.

    When I said "discovery" I meant a personal discovery. Like you said, I'm sure anyone with an advanced degree in math could verify them easily.

    I am currently trying to expand upon the idea a bit and figure out if its possible to form arbitrarily large volumes or shapes with the spheres.
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  16. #15  
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    Quote Originally Posted by ScubaDiver
    I am currently trying to expand upon the idea a bit and figure out if its possible to form arbitrarily large volumes or shapes with the spheres.
    No. The sum of the volumes of the parts will be the volume of the whole, so long as the notion of "volume makes sense. But "google" the Banach-Tarski theorem.
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  17. #16  
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    I know a way to make the third claim even more interesting.

    I shall rewrite it.

    Claim 3: If you sit all the n spheres next to eachother on a flat surface, you could cover a arbitrarily large area and make your layer be an arbitary number of spheres thick.

    For example, one could start with a sphere the size of a grain of sand. That sphere could then be decomposed into enough spheres to cover an area that is 1 trillion miles by 1 trillion miles, and this square layer could also be 500 trillion spheres thick.
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  18. #17  
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    Sounds reasonable, but I wouldn't be surprised (having not actually done the math) if the more layers you added, the thinner the total mass got.
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