# What is π ?

Today, March 14th, is Pi day - can you guess why that particular day? We all know Pi = 3.1415..., but what does this quantity represent, ontologically?

I spent a few minutes figuring it out and then I checked Wikipedia which gave me a very good visualization of the meaning of that number:

What that means is that Pi is actually a distance, namely the circumference of a circle whose diameter is one. So if we walk in a circle whose diameter is 1m, we will end up walking a distance of Pi meters. Now that's a more human way to imagine it!

For the more mathematically oriented, "π is ubiquitous in mathematics, appearing even in places that lack an obvious connection to the circles of Euclidean geometry." So there's lots to be read about it.

Now the unit circle is the magical object that causes Pi to exist. Consider its well-known equation: [drutex] $$x^2 + y^2 = 1$$ [/drutex] What we're seeing here is Pythagoras in action: 1 is the hypotenuse of a right triangle whose sides are x and y respectively and whose summit is at the origin. So the unit circle contains all possible combinations of right triangles!

We can immediately derive the ontological definition of sin and cos: the angle t defines one specific right triangle among the ones above where x and y have a definite value, namely cos t and sin t respectively. So when we find the point on the unit circle that is in the t-direction (from the origin), we know that we have moved cos t steps in the x-direction and sin t steps in the y-direction (divided by π to talk meters not angles). That's concrete enough for me.