If you’ve passed the 6th or 7th grade you have undoubtedly heard of the number Pi; that magical number that you use in so many equations from finding the circumference of a circle, to wave functions, to the period of a pendulum. In mathematics Pi is defined as the ratio of a circles circumference to its diameter. This definition works very well and works every time, but I’d like to propose a different way to look at it, and actually propose a different number that will still pop out in all the same formulas, but might make a bit more sense for a lot of people. We’re going to call this number Omega and the actual value of Omega is 0.78539816339742… (the … here represent that the number continues on and on just like Pi does but for the sake of simplicity we’ll just use that many decimal places). Basically Omega it’s just normal Pi divided by four.
And one might ask, why would you want to use Pi/4 when we already have Pi? Which is a good question; congrats to you for thinking skeptically. Omega is a percentage, so that .7853981633972… is actually 78.53981633742…%, and this percent magically works out in a number of ways when comparing a circle to a square
Lets look at some examples and lets start with circumference. Omega times the perimeter of a square gives us the circumference of a circle within it:
The perimeter of the square is:
And we are saying that Omega is the percent of the perimeter that will equal the circumference. So our equation works out to be:
and we have defined:
so dropping that definition into our original equation we get:
which works out to be:
Which we know is our standard definition of pi and the standard equation for circumference of a circle if you know the diameter.
Lets look at another example, and look at the area of a circle:
If you take the area of the square on the outside and multiply it by the percent Omega, you’ll get the area of the circle in the square. Let’s check the math:
Two times the radius shown above gives you one of the sides of the square, and when we square that we get the total area.
Now we multiply that total area of the square by the percentage Omega:
And we drop in our definition of Omega
and we get:
So we know the math always works out. And this concept of omega just makes the idea a bit easier than pi for a lot of people because people can understand a percentage rather than a magic number constant.
I hope this helps you understand and perhaps use pi or Omega a bit easier.
And my broader question would be: Is the real number that keeps popping up in all the equations, not Pi, but actually Omega. Everywhere you find a Pi in an equation you could replace it with Omega/4, whether you’re talking about string theory, light waves, pendulums etc, does it make sense that the real number that come popping out of the equations is actually Omega?
When you think about it, it may make some sense. When we think of space we tend to think in Cartesian space. Much like a graph:
Omega is the perfect number to transition from Cartesian space, where you are as zero, and anything else is some distance away in x, y and z axis. And this omega value fits perfectly when trying to translate area, or motion in this Cartesian space to circular or wave space.
So perhaps the real magic number that keeps popping up isn’t pi, but pi/4 and Omega.
