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A world with no curves, artificially creating a circle?

  1. Sep 17, 2006 #1
    A thought occurs...

    In a world devoid of curved lines, can we create a circle using only straight rigid lines? If we take a square as our starting block, then use 4 straight lines to cut off its 4 corners at 45 degree angles, we now have a perfect octagon. If we now take 8 straight lines and use them to cut off each of the 8 octagonal corners, we now have a 16 sided shape. You can see where this is going... If we continue this process of using straight lines to 'shave' off the corners of our shape, it become more and more circular. In fact, after only a few sets of 'shavings' the shape becomes a circle for all subjective purposes.

    One caveat would be that, even though we start to see this shape subjectively as a circle, we know that it is not because upon detailed measurement we would see that its radius to circumference ratios would only be true for a very small number of the shape's radii, vs. the shape's total radii. My question arises. If we can continue shaving the corners of this circle-like shape to the nth degree, our shape becomes more and more circular, but never reaches the shape of a true circle as defined by the radius-circumference ratio requirement. But, how can we know for certain that all circles as we know them, are not these so-called 'squares shaved to the nth degree?' After say, 1 million sets of shaving, the corners of this circle-like shape would be spaced so closely together that we would not have a measuring device capable of measuring all the radii sandwiched inbetween each of these respective corners. So then, how can we logically prove that our 'square shaved to the nth degree' is in fact not a circle?

    This writer thinks we cannot. Please prove me wrong!
     
  2. jcsd
  3. Sep 17, 2006 #2

    Galileo

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    If you don't have a circle, then not all points will have the same distance from the center.
    For any nth degree approximation you mentioned, it will simply not be a circle by the definition of a circle.


    The idea you mentioned is great for approximating circles though and can be used to prove the area of a disc is [itex]\pi r^2[/itex]. Archimedes used the same idea.
     
    Last edited: Sep 17, 2006
  4. Sep 18, 2006 #3
    Yes but we can never know if any circle is a perfect circle because of the limitations we have in measuring each tiny radii. For example, if we consider a circle with a trillion trillion trillion spokes, we cannot measure the radii between 2 adjacent spokes. Therefore, this tangent could be flat and not curved and we could never know.
     
  5. Sep 18, 2006 #4
    Let me further illustrate my point will the attached diagram that i drew:
     

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  6. Sep 18, 2006 #5

    radou

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    This is practically equivalent to the diagonal paradox. Read this thread: https://www.physicsforums.com/showthread.php?t=80796 Matt Grime's first reply explanes everything.
     
  7. Sep 18, 2006 #6

    Galileo

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    You give me a regular n-sided polygon (n is very large) and I`ll give you two points on it having different distances to the center.

    The point is, circles are abstract mathematical objects and don't have to correspond to objects in physical reality.
    If I ask you what a circle is, you could draw one on a blackboard. But no mathemetician would be foolish enough to define a circle as a bunch of chalk particles on a blackboard.
    Do you get what I`m trying to say?
     
  8. Sep 18, 2006 #7

    HallsofIvy

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    You said, to begin with that your world "has no curves". Given that, it cannot contain a circle. Polygons that are arbitrarily close to a circle, yes, but no true circle.
     
  9. Sep 18, 2006 #8
    That's true!

    But mathematically, circles are circles, and an n-sided approximation for a circle is not a circle. It's certainly true that it's impossible for us to draw a perfect circle. But that doesn't make any difference to mathematics.
     
  10. Sep 18, 2006 #9

    DaveC426913

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    I don't know how many decimal places pi has been calculated to so far but at some point, your n-gon's perimeter will diverge from pi. It might be the trillionth decimal place.
     
  11. Sep 18, 2006 #10
    You are missing a point here. Entire calculus is based on approximations.
    The goal of mathematics is to solve problems and in doing so some steps are required to be taken. In Archimedes' method for finding the area of a circle, the circle has to become a polygon of infinite sides, but to prove that all diameters of a circle have the same length, we need to fall back to the original definition i.e. it is a locus of points equidistant from a given point. Both the definitions of a circle are correct. We can choose either depending upon which is more convenient.
     
  12. Sep 18, 2006 #11
    You are missing a point here. Entire calculus is based on approximations.
    The goal of mathematics is to solve problems and in doing so some steps are required to be taken. In Archimedes' method for finding the area of a circle, the circle has to become a polygon of infinite sides, but to prove that all diameters of a circle have the same length, we need to fall back to the original definition i.e. it is a locus of points equidistant from a given point. Both the definitions of a circle are correct. We can choose either depending upon which is more convenient.
     
  13. Sep 18, 2006 #12

    radou

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    There is a difference between a infinite countable set and an uncountable one. So, if both definitions are correct, we could, at some point, consider the cardinality of the set of natural numbers and the cardinality of the continuum to be equal. Sounds like rubbish, doesn't it? :biggrin:
     
  14. Sep 18, 2006 #13
    A "polygon of infinite sides" is not a definition for anything. It's meaningless. Archimedes' argument can be made rigorous using limits. It doesn't need a different definition for what a circle is.

    Give me an example of what in calculus is "based on approximations."
     
  15. Sep 18, 2006 #14

    HallsofIvy

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    Twice you posted this! It's unfortunjate that it is completely untrue. "Calculus" (which is what I think you mean by "entire calculus") is not based on approximations. You may be confusing the "difference quotient" which can have many different, approximately the same, values with the derivative itself which is a limit of difference quotients and is exact.

    Archimedes never "defined" a circle as "a polygon of infinite sides"- he did show that (a primitive form of) the limit of the areas of such polygons was the area of the circle. Since even in modern mathematics, there is no such thing as "a polygon with infinite sides" no, you cannot "choose either".
     
  16. Sep 19, 2006 #15
    sorry guys. and thanks. it got posted twice 'cause my connection got cut-off when i clicked submit the first time.
     
  17. Sep 19, 2006 #16
    area of a circle is [tex]\pi r^{2}[/tex]
    area of a regulated convex polygon is [tex] S*a[/tex] where S=semiperimeter and a is apothem.

    [tex]a=\sqrt{r^{2} - \frac{l^{2}} {4}}[/tex]

    where l is the side and r is radius of circle that polygon is inscribed in.

    the higher the number of sides, the shorter the sides...r is a constant because we're inscribing this...let's say in a Unit Circle... so it's obvious the more sides, the shorter, so the [tex]\frac{l^{2}} {4}[/tex] decreases, leaving a closer and closer to [tex]\sqrt{r^{2}}[/tex].
     
    Last edited: Sep 19, 2006
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