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Thread: Standing pencil paradox

  1. #1 Standing pencil paradox 
    Forum Junior anticorncob28's Avatar
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    Imagine an infinitely long pencil standing between two infinitely tall walls, like this, but the pencil and the walls extend infinitely upward.

    | | | |
    | | | |
    | | | |
    | | | |
    | \ / |


    If the pencil leans to the left or right at all, it would have to break through the walls, but it can't. Therefore, the pencil must stand completely upright on it's tip parallel to the walls. How can the walls force the pencil to stand completely upright when there is no contact between the pencil and the walls at all? I have looked for an answer and searched, and as far as I know there is nothing online about this. Can somebody explain?


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  3. #2  
    Moderator Moderator Cogito Ergo Sum's Avatar
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    Quote Originally Posted by anticorncob28 View Post
    Imagine an infinitely long pencil standing between two infinitely tall walls, like this, but the pencil and the walls extend infinitely upward.

    | | | |
    | | | |
    | | | |
    | | | |
    | \ / |


    If the pencil leans to the left or right at all, it would have to break through the walls, but it can't. Therefore, the pencil must stand completely upright on it's tip parallel to the walls. How can the walls force the pencil to stand completely upright when there is no contact between the pencil and the walls at all? I have looked for an answer and searched, and as far as I know there is nothing online about this. Can somebody explain?

    I am not certain what paradox you are referring to.
    Would it not be more logical to think that if the pencil was not squeezed between two walls (i.e. no contact), that the pencil would fall forward or backward or up or down? Would it not be that pencil seeks the path with the least resistance (i.e. falling instead of breaking through the walls)?

    Besides, why do the walls and the pencil have to have an infinite length?


    "The only safe rule is to dispute only with those of your acquaintance of whom you know that they possess sufficient intelligence and self-respect not to advance absurdities; to appeal to reason and not to authority, and to listen to reason and yield to it; and, finally, to be willing to accept reason even from an opponent, and to be just enough to bear being proved to be in the wrong."

    ~ Arthur Schopenhauer, The Art of Being Right: 38 Ways to Win an Argument (1831), Stratagem XXXVIII.
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    Genius Duck Moderator Dywyddyr's Avatar
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    @ CES
    If the walls and pencil are of infinite length then, if the pencil falls left or right, the point of contact would be at infinity.
    Since infinity can't be reached then, by definition, the pencil can't fall to either side.
    (At least that's how I understand the paradox).
    I.e. it would fall forwards or backwards but it couldn't even tip left or right.
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  5. #4  
    Forum Junior anticorncob28's Avatar
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    Okay, my diagram got messed up. Imagine all the / lines are perfectly vertical. There is an infinitely long pencil standing between two infinitely long walls. Imagine drawing two parallel lines on a piece of paper, and placing a pencil between them that is parallel to the two lines. The two lines are the walls, and the pencil is standing upright on its tip on the floor between. (Actually draw this out if you can't visualize it). If the tip of the pencil is at a fixed spot on the floor (draw that in too), the longer the pencil is, the less room it has to tilt left or right. If the pencil and walls extend infinitely, the pencil has no room. Thus for this paradox to work it has to be infinite. If you say the pencil will fall forward or backward, imagine the wall completely surrounding it. If you still can't understand, then you can just leave and hope somebody else can explain, and thanks for your effort.
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  6. #5  
    Forum Junior anticorncob28's Avatar
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    What happens then if the pencil is completely surrounded, by something like a cylinder? It can't tip any direction within real space, so it is forced to stand upright. What is keeping it upright then?
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  7. #6  
    Genius Duck Moderator Dywyddyr's Avatar
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    The simple explanation, as I see it, is that the pencil does fall (in one direction or another) but because it and the walls are of infinite length then it "remains" upright - because the measured/ calculated angle of tilt is zero due to the length of the sides.
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    Quagma SpeedFreek's Avatar
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    Quote Originally Posted by anticorncob28 View Post
    If the pencil leans to the left or right at all, it would have to break through the walls, but it can't. Therefore, the pencil must stand completely upright on it's tip parallel to the walls. How can the walls force the pencil to stand completely upright when there is no contact between the pencil and the walls at all? I have looked for an answer and searched, and as far as I know there is nothing online about this. Can somebody explain?
    If you are using normal geometry, where the parallel postulate is supposed to hold, then the pencil touches the wall at a finite distance and you do not need to use infinity at all.

    If you are using projective geometry, then parallel lines meet at infinity.

    This is what happens when you try to use infinity and apply it everyday problems. You get nonsensical results. There is no such thing as an infinite wall or an infinite pencil! Nor is there such a thing as a perfectly rigid rod (so the pencil would bend!)
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  9. #8  
    mvb
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    What we are talking about here is a limit. If you put a looooonnnnnnggggg pencil in an equally long cylinder whose inside radius is close to the size of the pencil, the pencil will tilt over and touch the top of the cylinder in some direction. You then take the limit as the pencil gets indefinitely long. It can never actually reach infinity, because infinity is not a real number. But it can be made to get longer than any preselected real number. As the pencil gets longer and longer, its angle from the vertical gets smaller and smaller. The limit of the value of the angle as the length of the pencil approaches infinity is zero. The angle never reaches zero since the length of the pencil can't reach infinity, but that is characteristic of limits as a parameter "goes to infinity," i.e. gets indefinitely large.

    If you can visualize and understand this, you are pretty darn close to understanding the meaning of limits.
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  10. #9  
    Forum Junior anticorncob28's Avatar
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    So, if I understand your view correctly, mvb, can this be used to prove that infinitely large objects cannot physically exist?
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  11. #10  
    mvb
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    Quote Originally Posted by anticorncob28 View Post
    So, if I understand your view correctly, mvb, can this be used to prove that infinitely large objects cannot physically exist?
    Well, actually I think that limits explain what an "infinite object" is to a physicists. Its normal usage is for a situation where you don't need to know anything about the boundaries to explain what you are seeing or that you need to know about. In order to get away with not considering the edges, you merely must be far enough from the boundaries that they do not effect your measurements within experimental error.

    There is in fact no way to determine if there exists an object that really doesn't have edges. We cannot determine this even for the entire universe. The universe has a finite age, at least in its current form. This means that we can get no information from further away than the speed of light times the age of the universe, and we can only tell observationally that an object does or does not have borders within that distance. Beyond that we do not know.

    Given curved space, we can make a guess about boundaries by assuming that the behavior of space at the largest distances we can see continues beyond that point. We can then project that the space curves back around and reconnects in a finite total distance or that it appears to be flattening out to a form that would have no boundary that we could detect. In neither case can we be sure that our projection is correct.
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