Pi Proof Pi (book)

The book 'Pi Proof Pi (a mathematical proof)' contains a mathematical proof for Pi. Using the ancient method of 'Squaring the Circle', it shows the perfect symmetry of the number and the amazing relationship between the circle and the square.


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(1) 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.




Because the height of our triangle, A equals 2. We can use this mathematical identity.




This allows us to visualise our triangle like this.




Which gives us this equation for 1/x.




And this equation for x.




And we can also flip these two equations to get these four equations.




Using these equations we can calculate our values for x and 1/x and therefore B.


Simply rearrange two of the above equations.




Therefore.




By multiplying by x and its reciprocal we get these quadratic equations for x and its reciprocal.




Then, using the formula for quadratic equations.




Then, simplifying.




Therefore, simple logical deduction.




Therefore.




Therefore, B equals π/2.




This completes the π triangle.




(3) Finally the angles of the squared circle.








Then some basic equations.






If you have been following along using your calculator, you will have noticed that calculator π fails.


This seems to be because it does not have 'cosmic-symmetry', that is symmetry across the macrocosm (numbers > threshold) and microcosm (numbers < threshold).


This 'cosmic-symmetry' is expressed in these two equations.




One last triangle.




And π expressed as a recursive equation. Let wolfram alpha calculate pi for you.