Pi versus Pi (e-book)
The book 'Pi versus Pi (a comparison)' contains a comparison of two values of Pi, and shows that one of the values has perfect symmetry and one does not. Because both the square and the circle are both perfectly symmetrical, we would expect Pi to have perfect symmetry.
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We start with a circle and a square with the same circumference/perimeter (c). (Fig. 1)
We place the circle on top of the square and zoom into the top right quarter; put a unit square around it and create our first triangle A. (Fig. 2)
We create a copy of our quarter squared circle and multiply it by a factor of 4/Pi, this gives us triangle B. (Fig. 3)
We combine our two squared circles and add another squared circle rotated at an angle. (Fig. 4)
Next we combine our squared circles and create our next triangle C. (Fig. 5)
The values for triangle C (Fig. 5) are from the normalised triangles. (Fig. 6)
Because a triangle has three sides it can be normalised in three ways, with exceptions. A normalised triangle is simply a triangle with the one or more side-lengths equal to one.
We can normalise triangle A and amazingly we can extrapolate every value using a simple conversion factor of 4/Pi and its reciprocal. (Fig. 6)
Each of the following squares and circles are a factor of 4/Pi larger or Pi/4 smaller. (Fig. 10)
The Comparison (Pi v Pi)
In this section we will compare two different values for Pi. On the left is the value that we all know and on the right is the new value.
The symmetry (Pi)
In this section we will look at the symmetry of Pi. On the left is the value for Pi that we all know and on the right is the new value.
As you can see when we compare the two different values for Pi you can see that one value has perfect symmetry, and one does not.
The Number (PI)
Pi as a Decimal Number
Pi as a Fraction