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Pi r^2 /3

  1. Sep 30, 2003 #1
    Please look at the attached pdf.

    You will find in it a circle and some sub-area inside it.

    The sub-area exists between the radius and a curve.

    The curve is connected to both sides of some radius, and goes through the intersection points that exists between n radii and n-1 inner circles, where each radius is divided by the inner circles to n equal parts.

    I have found that the sub-area(magenta) = circle's-area/3(cyan)

    1) Can someone show why the magenta area = 1/3 of the cyan area ?

    2) We can take any number of inner radii-circles intersection points, and create some border, which is made of straight lines between these points.

    By doing this, we get a closed polygon (an area).

    Now we take some closed polygon, find the total number of its vertexes and omit 2 (tolal - 2 = n).

    By doing this, we get some Natural number n which is conncted to some polygon's area S (please see the attached pdf in the next post, called natural-areas.pdf.pdf).

    Through this way we can put in 1-1 correspondence some n with some S.

    When have this map, we can ask:

    S1 is the area of some polygon, where the number of totel-2=aleph0.

    S2 is the area of some polygon, where the number of totel-2=2^aleph0.

    3) Is S1 = S2 ?

    4) If the answer to (3) is no, then what is the difference between the two magenta areas, and how this difference related to the CH problem ?

    5) Do you think that we have here some useful mathematical constant ?

    Thank you.


    Attached Files:

    Last edited: Oct 3, 2003
  2. jcsd
  3. Sep 30, 2003 #2
    Here you can find a pdf file, which shows the connection between some natural number to some area.

    Attached Files:

    Last edited: Sep 30, 2003
  4. Oct 1, 2003 #3


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    Staff Emeritus
    Science Advisor

    The spiral you have is the "Archimedian" spiral:
    r= (R/2 &pi)&theta (R is the radius of the large circle).

    It is fairly easy to show that the area is, in fact, (2/3)&pi R2, 1/3 the area of the circle.
  5. Oct 2, 2003 #4
    Hi HallsofIvy,

    Thank you for your reply.

    Can you please answer to 3-5 questions ?

    Thank you.

    Last edited: Oct 2, 2003
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