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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

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

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 =

By doing this, we get some Natural number

Through this way we can put in 1-1 correspondence some

When have this map, we can ask:

Thank you.

Organic

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:

**S**1 is the area of some polygon, where the number of totel-2=aleph0.**S**2 is the area of some polygon, where the number of totel-2=2^aleph0.**3)**Is**S**1 =**S**2 ?**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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