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Considering a circle to be an infinite sided n-gon

  1. May 9, 2014 #1
    as a regular polygon increases in sides, it becomes rounder. As you increase the number of sides, the polygon will tend towards a perfect circle but never quite make it. you can only make the circle with an infinite number of sides - stopping at any other number but infinity you will only get a very very very round but noncircle shape. but you know all this...


    also I am wondering... technically a circle must have vertices of 180 degrees, right? is this a discrepancy?
  2. jcsd
  3. May 10, 2014 #2


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    A polygon, by definition, must have "n" sides and "n" angles for some integer n. "Infinity" is not an integer so you will have you will have to specify what you MEAN by "A REGULAR POLYGON WITH INFINITE SIDES". Once you have done that, perhaps someone can answer.
  4. May 10, 2014 #3
    A polygon has by definition a finite number of sides. So a circle is not a polygon.

    It is certainly true that a circle is (in some sense) a limiting value of polygons and this fact is extremely useful. But that doesn't mean that the circle is a polygon. In the same way, the number ##0## is a limiting value of ##0.1##, ##0.01##, ##0.001##, ..., but all these numbers are positive while ##0## is not. So a limiting value does not need to have the same properties as the elements of the sequence.

    The ancient Greeks might have considered the circle a polygon with infinite sides, but this is not done anymore. One of the reasons why not is that we have not really defined what a "polygon with infinite sides" is. Furthermore, the notion can be confusing to people, so we choose not to use it.
  5. May 10, 2014 #4
    it seems, to me, arbitrary that a circle is not a polygon because the definition of a polygon is finite sides. perhaps im not grasping this, but if we changed the definitions around a little couldnt we fit a circle into the same category as regular polygons?

    dont get me wrong, im not on some crusade to get circles to be considered regular polygons... im just very interested in the idea that you COULD categorize a circle in with the rest of the regular polygons

    im really looking for something bulletproof like the angles of a circle must be 180 degrees and thats impossible and thats why a circle cant be a regular polygon. not just "our definitions dont quite allow that"

    sorry if im being ignorant but i cant help it :p
  6. May 10, 2014 #5
    i mean what if you keep adding sides and sides and sides to a regular polygon so it goes from triangle to square to pentagon to septagon octogon etc into infinity

    i think im being very straightforward with this question?
  7. May 10, 2014 #6
    A definition is always a bit arbitrary.

    What would you propose as definition then?

    You can't prove or disprove definitions. We can define a polygon in whatever ways we want. We have now defined it as something with finitely many sides. Other definitions might allow other things. So the answer "Our definitions don't allow it" is the only answer we can give.
  8. May 10, 2014 #7


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    "Infinity" is not a number. However many sides you add, you still have a polygon with finite number of sides.
  9. May 11, 2014 #8
    well yeah if you stop at any integer youre just going to end up with a very round polygon... you need to go on forever (infinity)

    to be it seems straightforward that if you imagine what would happen if a polygon had infinite sides itd be perfectly curved no matter how close u zoomed in while a polygon with finite sides might look round from far away, but you get closer and u see the angles
  10. May 11, 2014 #9
    thanks for the reply. as a definition for polygon is a shape with multiple sides. is there a particular reason the definition is so specific as to say finite sides? im quite interested

    a circles definition is: the set of points equidistant to a single point. this is an infinite set of points. each point is a vertice, is it not?
  11. May 11, 2014 #10


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    You need find a proper definition of "side". The only way I can think of defining that would start with is "a side is a straight line such that..." But circles have no straight lines.

    A vertex is a point where two straight lines meet. Again circles have no straight lines.
  12. May 11, 2014 #11


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    Toying with definitions, you could just say each "side" of the circle as a polygon has length 0, with initial and end point in the same vertex, for every point on the circle. I don't see the point of this though.
  13. May 11, 2014 #12
    You use phrases like "into infinity" and "go on forever" in regards to the proposed limiting process that leads to what you're calling a regular polygon with infinite sides. This leads me to believe that the object that you're proposing has countably many sides, since the countable cardinal is the limit of the finite cardinals. I reckon you'd say that each side of your infinite-sided polygon has length 0, in which case the perimeter of your object must also be 0.
  14. May 11, 2014 #13


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    What are the sides then, you mean?
  15. May 11, 2014 #14
    It's the OP's construction, not mine. I'm only commenting on the number of sides proposed and making a reasonable guess as to what length a side might have, not what constitutes a side.
  16. May 11, 2014 #15


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    There are many cases where an object obtained as the limit of a sequence of other objects does not share the same properties as those objects.

    For example, the limit of a sequence of continuous functions may be discontinuous. No one would suggest calling it continuous just because it is the limit of a sequence of continuous functions.

    So it's not clear to me why you want to call a circle a polygon just because it can be considered as a limit of a sequence of polygons.
  17. May 11, 2014 #16


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    These kinds of guesses are dangerous and rarely reasonable (if we really knew what we were talking about here). For example, the limiting function of a sequence of functions with countably many discontinuities may have uncountably many discontinuities.
  18. May 11, 2014 #17
    We should have an FAQ for this, similar threads pop up every week.
  19. May 11, 2014 #18
    it works if straight lines are infinitely small but nonzero :0 from what i figure like 0.00000000000000infinity1
  20. May 11, 2014 #19
    well, im no mathematician, but isnt side length 0.0000000000000000000000...1? everyone is saying 0 so i guess not :(. this is a good point i think it is in similar lines to the fact it seems like a circle must have 180 degree vertices if you consider it to have infinite points... right? because 180 degrees is the limiting factor (is that the right term?!) of the measure of the vertices of an n-gon
  21. May 12, 2014 #20


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    Assuming the ... is intended to stand for infinitely many zeros, there is no such real number. It would have to be smaller than ##10^{-n}## for every positive integer ##n##, and the only nonnegative real number with this property is zero.
  22. May 12, 2014 #21


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    The number you are talking about does not exist (as a real number) for the same reason that 0.99...=1.
  23. May 12, 2014 #22
    It is good to study a real example of approximating a circle as an infinite sided polygon.
    For example, the Viete's formula:

    Viete's formula represents a sequence of polygons with numbers of sides equal to [itex]2^{n}[/itex], inscribed in a circle.The Viete product is:
    [itex]2/\pi = U_{1}/U_{2} \cdot U_{2}/U_{3} \cdot U_{3}/U{4} \cdot \cdot \cdot \cdot = U_{1}/U_{\infty}[/itex]

    the Viete product telescopes to give the ratio of areas of a square (the initial polygon in the sequence) to a circle (the limiting case of a [itex]2^{n}[/itex]-gon).
    Alternatively, the terms in the product may be instead interpreted as ratios of perimeters of the same sequence of polygons, starting with the ratio of perimeters of a digon [itex]U_{1}[/itex],(the diameter of the circle, counted twice) and a square [itex]U_{2}[/itex] , the ratio of perimeters of a square [itex]U_{2}[/itex] and an octagon [itex]U_{3}[/itex], etc etc up to the ratio of perimeters of [itex]U_{\infty-1}[/itex] and [itex]U_{\infty}[/itex] .

    [itex]U_{\infty}[/itex] is the perimeter of [itex]2^{\infty}[/itex]-sided polygon. If the "radius"
    of this [itex]2^{\infty}[/itex]-sided polygon is equal to 1, its diameter is equal to 2 (= [itex]U_{1}/2[/itex]), then its perimeter is equal to
    [itex]2\pi[/itex], hence
    [itex]U_{\infty}/(U_{1}/2) = 2U_{\infty}/U_{1}= U_{\infty}/2 = 2\pi/2 = \pi[/itex]
    this is the same result as we obtained with the Viete's formula [itex]U_{1}/U_{\infty} = 2/\pi[/itex]

    All the time a distinction is made between a circle and [itex]2^{\infty}[/itex]-sided polygon,
    which is just the limiting case of [itex]2^{n}[/itex]-gon.
    It might lead to an error to believe that a polygon transforms into a circle at an "infiniteth" step.
    The error just seems to disappear if we are free to call a [itex]2^{\infty}[/itex]-sided polygon
    a circle.

    I used the Viete's formula from Jörg Arndt book Pi - Unleashed:
    http://books.google.fi/books?id=Qww...Arndt, squaring the circle with holes&f=false
  24. May 12, 2014 #23
    Viète's formula does not rely on a circle being a infinite sided polygon. All we need is the circle being somehow the limit of polygons.
  25. May 12, 2014 #24


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    I just touched on this very concept with my high school geometry class.

    In order to derive the formula for the area of a circle, we assumed that a circle was an "infinity-gon." Then, using the formula A=(1/2)ap (where a is the apothem and p is the perimeter) we substituted in the radius for a (since every apothem in an infinity gon is a radius) and then the circumference formula for p. We get A=(1/2)2(pi)r^2, or, pi*r^2.

    I warned the students that a circle is not, by definition, a polygon, but for the sake of calculating the area, it's useful to imagine that it is one since we can use what we already know to describe this new concept.

    I'm wondering what others think about that - isn't that how they originally calculated the area of circles? They did repeated approximations which got closer and closer to a number, which they then created a formula from? Comments would be welcome.
  26. May 12, 2014 #25


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    The only idea that needs to be grasped is that calculating the perimeter of inscribed and circumscribed polygons give upper and lower bounds of the perimeter of the circle. To go on and say that the circle is an "infinity-gon" is quite meaningless and only serves to generate confusion. You could say (arbitrarily) that a circle is an "infinity-gon", but you can't argue anything from that.

    In addition to that, by increasing the number of vertices of the inscribed and circumscribed polygons, and calculating the sequences of perimeters, you see that these values converge towards a single value, which will be (or what we call) the perimeter of the circle. I don't see any logical or pedagogical reason to force the students to imagine the circle as a polygon. The polygons serve as approximations, that is the whole idea.
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