Saturday, March 24, 2012

Pi, Lie, Same Thing... (Part 2)

Last week, I did some explaining on why tau is better than pi. However, that post was relating it to past posts more and using the popular reasons less. Let me bring up some of the more popular reasons that are a little easier to follow.
First off, we looked at trigonometric reasons last week, but the “homeland” so to speak of both of these numbers is the circle. This week, we will only look at circles.
People measure angles in degrees usually. There are 360° in a circle. Mathematicians, however, measure angles in radians. There are 2π radians in a circle. No wonder. In a third of a circle, there are 2π/3 radians. A third of a circle corresponds to this mess!
What about a quarter circle. This is equivalent to 1/2π. Still, a quarter corresponds to a half. That’s also crazy.
What if we used tau? A third of a circle is a third of tau. A quarter circle is a quarter of tau. 28/53 of a circle is 28/53 of tau. Does it get any easier?
You might be thinking that I am only showing you one side of the story. Obviously, this is partly true, but let’s look an argument that a pi person might bring up. What about the area formula? This is just a plain old pi.
Yes, but one of the most common proofs for the area formula (I will definitely post this at some point) ends up with the following equation, which you have to simplify:
1/2 x 2πr^2
So basically, there was a 2 there, but it got cancelled out. The new area formula is:
So, the 1/2 does us a favor. It helps us remember the proof.
This might not be good enough for you, so let me bring up how the 1/2 makes it more natural as well. What is the area formula for a triangle:
What about a trapezoid?
1/2h(b1 + b2)
Even the universal polygon area formula starts with a 1/2. So, the new formula just makes the circles fit in more with the other geometric shapes. I think this is a good thing, not a bad thing.
Next week, I won’t be talking as much about tau, but about what I have done with tau over the past few months. I’ll bet some of you might already know what it will be about. For a pretty good hint, check out

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