Happy Birthday , π

What is π?

π is often seen as a mystery number. The mathematician, Euler, defined it as the ratio between the circumference and the diameter of a circle but another way to define it would be as 4 times distance the radius of a circle covers when it falls through a 90 degree angle. When we speak of circumference and diameter, a line and a circle, we are speaking of a featureless, static symmetry on which the mind can gain no hold and this may be why π is mysterious. When we speak of the arc of a quarter-circle formed by the radius falling through 90 degrees we are speaking of a symmetry in the process of forming which is easier to understand.

A circle is formed, on this understanding, when the radius, which is anchored at one end and free to fall at the other, falls through 90 degrees four times. Therefore the circumference of the circle is 4 times the length of the arc of a quarter-circle. If we take π to be 3.14159 then the length of the quarter-circle arc is .7854 and four times .7854 equals π.

Now we look at the circle in relation to a square of the same diameter, a square with a side of one. The circumference of that square is found using the formula C = 4D. Previously we would have said the formula for the circumference of a circle was πD. But we now see π as 4 times .7854 so the formula for the circumference of circle becomes 4D(.7854) or four times the diameter times .7854. The formula for the circumference of a square and the circumference of a circle are now similar. [4D, 4D(.7854)].

And if the diameter of a circle and a square are the same then the circumference of that square is four times its diameter and the circumference of that circle is a certain percentage of the circumference of that square. That percentage is .7854.

The percentage is .7854 because the arc of a quarter-circle around its 90 degree angle can be compared to the “arc” of the square around that same 90 degree angle when the square and the circle have the same diameter or side. When the square and the circle both have a diameter of 1 unit, the “arc” of the square about a 90 degree angle is 1 unit while the arc of the circle about the same 90 degree angle is .7854. a decimal fraction of that unit. It’s important to understand that we are now talking about a percentage, not a line. Otherwise, it will not be clear how we are reaching the formula for the area of circle which comes next.

The usual formula for the area of a circle is πr² while the formula for the area of a square is D². If we substitute 4 (.7854) for π then the formula for the circle is 4r² (.7854). But 4r² is the same as D² as we can see from the diagram so the formula for the area of a circle is D² (.7854). [D² , D² (.7854)]

Summary

When we consider π dynamically we see it as 4 times .7854. It might seem that we have just exchanged one mystery number for another - .7854 instead of 3.14159. But .7854 lets us use related formulas for the circumference of a circle and a square and for the area of a circle and a square. These related formulas in turn show that the circle and the square are directly related to each other. That is one benefit of seeing π as 4(.7854).

More important is the fact the .7854 is not a mystery number unrelated to all other numbers. It is very close to .786 which is the square root of .618 or phi, so it is related in some way to the great family of phi numbers. For instance,[ the square root of .616850275 times 4 ] times 2 equals π. And this number .616850275 is also close to φ or .618. So when we use .7854 we are led to ask why π is so close to φ and why circles have the same self scaling properties as φ while yet the roots of π are not the roots of φ. This is a real question and easier to investigate than contemplating the symmetry of a circle.

Happy birthday, π

[Note. There is a sense in which the formula πr² is hiding a way to square the circle. Four times four radius-squares times .7854 changes a square into a circle; the reverse changes a circle into a square. [D² (.7854), D²/.7854]

However, I have never found either the arc length .7854 or its square root as a straight line no matter how I divided circles and squares with triangles. I suppose this is what Euler meant when he said the circle and the square are incommensurable and so you cannot square the circle using a compass and an unmarked ruler.]