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.
By multiplying by x and its reciprocal we get these quadratic equations for x and its reciprocal.
Then, using the formula for quadratic equations.
Therefore, simple logical deduction.
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.