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An impossible circle?

  1. May 20, 2009 #1
    A circle is an infinite series of points equidistant from the center.


    but can a circle be composed of an infinite series of lines?
    if you draw a square & connect the 4 side, that are equidistant from the center, then draw draw and infinite series of points on those lines. each one of the points on the line is not the same distance from the circle.
    know draw a trillion points all equidistant from the center, and draw lines between those points & draw points on those lines. those points are not equidistant.
    so a circle cannot be composed of an infinite series of lines. because each would be composed of an infinite series of points each not equadistant from the center.
    which says making a circle with an infinite series of lines is impossible.
    am i thinking right?
     
  2. jcsd
  3. May 21, 2009 #2

    tiny-tim

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    Hi yyttr2! :wink:
    Yes, that's fine … if any line contains two points, then it contains all points between them, and they can't all be the same distance from the centre :smile:
     
  4. May 21, 2009 #3
    Is there a difference between "an infinitesimal line" and "a point"?
     
  5. May 31, 2009 #4
    I think a line ('infinitesimal' or not) always contains an infinite number of points.
     
  6. May 31, 2009 #5
    I'm not sure infinitesimal line really exists. Lines are infinite in extent, by definition.

    Now, are you asking whether a line segment with length zero is the same thing as a point? That's an interesting question.
     
  7. May 31, 2009 #6
    I guess it's a matter of definitions. If it's a line segment, then its not a point. I don't think you can transform a line segment into a point analytically.
     
  8. May 31, 2009 #7
    What I had in mind was a limit question, but I guess it does boil down to a line of zero length.

    I always thought: Lim(Line) = Point as length ---> 0.

    Similar to: Lim([a,a+e]) = a as e ---> 0.

    I know infinitesimal is not the right word, but I did not have a better alternative word.

    The OP asked whether we can cram an infinite number of lines into a finite length (= the circumference of a circle). We can if we define "Lines" as a closed set, but not otherwise.
     
    Last edited: May 31, 2009
  9. May 31, 2009 #8
    I'm not sure that's a proper use of a limit. You can say that with the Taylor series expansion around points (b,a) ON an analytic curve, the error b-a will converge to 0 as a limit for increasing n. The tangent line or line segment at that point will share exactly one point with the curve (or specifically a circle in this case), but that's a different proposition than saying a line segment can contain exactly one point.

    A curve like the space filling Peano curve, which is composed of line segments, contains singular points of zero measure. I don't think you can define a line segment as simply a closed set and expect the words 'line segment" to be meaningful. I think of a line segment in terms of the minimum Euclidean distance between two points and a line as the infinite (in at least one direction) extension of a line segment.
     
    Last edited: May 31, 2009
  10. May 31, 2009 #9
    By "line," the OP clearly meant "line segment" and later posts adopted this convention.

    So, I have a simple question: what is the length of the shortest line segment that is not a point?
     
  11. May 31, 2009 #10
    There doesn't have to be a shortest. That's like asking what the real number right below 3 is.
     
  12. May 31, 2009 #11
    I don't know how to specify the length of an arbitrarily short line segment, but it contains an infinite number of points.
     
  13. May 31, 2009 #12
    Unless it has length equal to zero... ?

    x = 1 + 2t
    y = 3 + 3t
    0 <= t <= 1

    That defines a line segment from (1, 3) to (3, 6). It has length sqrt(13).

    The question is this: does this constitute a line segment of length zero, or is a point, or is it both?

    x = 1 + 2t
    y = 3 + 3t
    t = 1/2
     
  14. May 31, 2009 #13

    DaveC426913

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    Well, a line segment must contain at least two points, one on each end. According to Wiki (so it must be true), the points must be distinct. That means the endpoints cannot be coincident. And that means there is no such thing as a zero length line.

    Only thing left to do to find a more trustworthy source for the definition of a line segment.
     
  15. Jun 1, 2009 #14
    Huh. That seems to make sense. Perhaps Euclid has something to say about lines (and line segments as well)...
     
  16. Jun 1, 2009 #15

    tiny-tim

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    Euclid

    This is the html version of the file http://voteview.ucsd.edu/logical_systems.pdf.
    Google automatically generates html versions of documents as we crawl the web.


    Euclid’s Postulates ​


    1. A straight line segment can be drawn joining any two points.

    2. Any straight line segment can be extended indefinitely in a straight line.

    3. Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center.

    4. All right angles are congruent.

    5. Parallel Postulate: For every line l and for every point P that does not lie on l there exists a unique line m through P that is parallel to l.
     
    Last edited by a moderator: Apr 24, 2017
  17. Jun 1, 2009 #16
    I don't think Euclid is of much help here. This is a problem of a continuum vs a point. It's clear Euclid considered a line as distinct from a point. You can certainly have singular points of zero measure on a continuum as with the Peano curve, but if an isolated point can be a line segment of zero measure, then this raises problems as to what a continuum really is. A line segment is a continuum. An isolated point is not. (Also see post 8.)
     
    Last edited: Jun 1, 2009
  18. Jun 1, 2009 #17
    Well, two points...

    When Euclid says "two points", I suppose he means two distinct points. However, nothing he says really requires that... does it?

    And SW VandeCarr:
    It would be a simple enough matter to let zero-length line segments be the exception to the rule, call them degenerate line segments, or something. In graph theory, for instance, you often neglect graphs on zero or one vertex because they provide silly counterexamples for more general theorems. But nobody would argue they're graphs. Maybe the same thing is going on here?
     
  19. Jun 1, 2009 #18
    If we define line segments not to include points, wouldn't that create a problem with defining the zero vector, for example?
     
  20. Jun 1, 2009 #19
    I suppose we are free to define special cases, but is it necessary? I think it's better to define a circle from first principles: the set of all points equidistant from a given point in a plane. The equilateral n-gon can then approximate the circle through the Taylor series expansion such that the limit is the circle.
     
    Last edited: Jun 1, 2009
  21. Jun 1, 2009 #20
    If you define the circle as all the points at a given distance from the center you do not need the line segments anymore, the circle will be continuous.
    If you use line segments you can argue if a line can consist of one point. Then a point is also a line, but these lines have lost the most important properties of lines, like a slope. You have to be very careful when comparing circles and approximating polygons you quickly get into the theory of measures. For example if you take all regular polygons which have a corner on the coordinate line and subtract the corner points from the circle, it would still be a closed circle in a physical sense, although you have taken away an infinite number of points.
     
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