Pi r^2 /3

Organic

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.

Organic

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Related Linear and Abstract Algebra News on Phys.org

Organic

Here you can find a pdf file, which shows the connection between some natural number to some area.

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HallsofIvy

Homework Helper
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.

Organic

Hi HallsofIvy,

Can you please answer to 3-5 questions ?

Thank you.

Organic

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