Please look at the attached pdf.(adsbygoogle = window.adsbygoogle || []).push({});

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 betweennradii andn-1 inner circles, where each radius is divided by the inner circles tonequal 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 numbernwhich is conncted to some polygon's areaS(please see the attached pdf in the next post, callednatural-areas.pdf.pdf).

Through this way we can put in 1-1 correspondence somenwith someS.

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)IsS1 =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

**Physics Forums | Science Articles, Homework Help, Discussion**

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Pi r^2 /3

**Physics Forums | Science Articles, Homework Help, Discussion**