Squaring the Circle

Yes, I know all about Lindemann, Archimedes, ect. Lets explore anyway.


Squaring the circle is a problem proposed by ancient geometers. It is the challenge of constructing a square with the same circumference/area as a given circle by using only a finite number of steps with compass and straightedge.



Inside of each squared circle, we can construct a right angled triangle as shown.


For this reason, we are going to concentrate on the squared circle where the circumference of the circle is equal to the perimeter of the square.

Squared Circle with same Circumference

When the circumference of a circle and the perimeter of a square are equal, we will call it a squared circle.


To simplify things we will concentrate on just one quarter of the squared circle.


If the radius of the circle (hypotenuse of the blue triangle) C equals 8/π, then an eighth of the circumference/perimeter (height of the blue triangle) A equals 2.


Now, usually we would use pythagoras theorem to calculate the number (B) but we are going to use a different method.

Triangles without Pythagoras

This different method, uses this amazing cosmic mathematical identity.


Which we can use instead of pythagoras to calculate the lengths of a triangle.


This allows us to visualise our triangle like this.



Please note, steps have been skipped.


Using the mathematics of the cosmic identity, we can get the following quadratic equations for x and its reciprocal.


Because they are essentially the same equation, we can use a single equation to represent x and its reciprocal (1/x). And after using the formula for quadratic equations, and substituting (C) as (8/π), we get.


And after simpyfying.


And after more simpyfying, we get the following cosmic number.


It is assumed that (x) is always ≥ 1 and the (1/x) is always ≤ 1, therefore.


Now, we can calculate the value (B). C = (x+1/x) = 8/π and B = (x-1/x)



To be continued...

Using Circles

Lets explore these methods of squaring the circle, mainly using circles to perform calculations like square roots and square.


Below we can see how we can calculate the square root of the diameter (d-1) of the red circle.


Below we can see how we can calculate the square of the radius (r) of the red circle.

This links to our cosmic equations.



This method is used in this universal interactive diagram that unites the triangle, square and the circle.

The Cosmic Triangles

The following (cosmic triangles) are found in the squared circle.



8/π = (4/π+π/4) + (4/π-π/4) and π/2 = (4/π+π/4) - (4/π-π/4)


The angle of pi and perfect symmetry can be seen in these interactive diagrams mandorla and vesica piscis.

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