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Thread: Spheres and circles and things

  1. #1
    Junior Member
    Nov 2007

    Spheres and circles and things

    Hey everyone,

    I was just curious about something:

    Say you had a normal polygon with  n sides. As  n approaches infinity, your polygon begins to look more and more like a circle.

    We could define a straight line to be 180 degrees.

    We know that the equation of the sum of the interior angles  \theta i of a polygon is:

     \theta i = (n - 2)(180) = 180n - 360

    We know that the measure of a single interior angle  \alpha is:

     \alpha = \frac {180n - 360}{n}

    When we take the limit of n as n approaches zero:

    \lim_{n\rightarrow\infty} \frac {180n - 360}{n} = 180 By L'Hopitals rule

    It's weird to me, because it seems like the curve of a circle is flat, but also curved. What if you visualized this in three dimensions? Does that mean that the surface of a sphere is 2 dimensional?
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  2. #2
    MHF Contributor

    Aug 2008
    Paris, France

    actually, the same 180 value would have appeared with any smooth curve, not specifically with a circle. The reason for this is that if you draw a line between a fixed point A on a curve and another point M on the same curve then, as the point M tends to A along the curve, the line (AM) converges (in a sense that can be made rigorous) to the tangent line to the curve at A. This tangent line is the same if M tends to A from the left or the right, hence the flat angle (the angle between the two directions of the tangent at A).

    In three dimensions, on a smooth surface, you have a similar property, namely that the line (AM) would get nearer and nearer to the tangent plane as M tends to A, and a plane (AMN) would converge to the tangent plane as M and N tend to A along curves drawn on the surface that are not tangent to each other at A.

    Does this answer your question?

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