Wednesday, 13 June 2007

Pi is not a constant (or the Curvature of Heaven)

A friend showed me this post. The author makes a very good point, and I totally agree with the message he wants to send. However, I'd like to point out another thing. Pi is not a constant. And when I say Pi, I mean the ratio between the circumference of a circle and its diameter.

Indeed, Pi = 3.1415926.... (whatever) in the part of the Universe we live in. As we know, Einstein postulated that space-time is curved, and the curvature is determined by the amount of mass in the area. So there are places where the curvature is much different then on Earth. What this means is that circles drawn on the surface of space-time will have varying circumference to diameter ratios.

I don't want to go through the exact maths of this, but here's a simple example. Let's say we have a sphere, like our Earth. This is the way that the Universe is commonly compared to by scientists. Imagine you draw a small circle on its surface. Since the area covered is small enough, you can consider it as being perfectly flat. This is how we consider our surroundings on our spherical planet. If you were to measure the circumference and the diameter of the circle and divide them, you'd get 3.141592.... and perhaps a warm feeling inside you.

Now, imagine you extend the circle. You will notice that the ratio will be going down as the circle gets bigger. Hell, soon it won't even look like a circle, because you'll have to draw the outline on a bent surface. You may be saying that I'm wrong then about the definition of Pi, as it only holds to circles (the ones we draw on a flat piece of paper). But remember that we are constrained by the surface of space-time, just as we are constrained to the surface of the Earth, so that anything we draw on its surface, would seem to be on a flat surface.

So to recognize a circle even if doesn't look like one, we'll check this by using the definition of a circle, which is: the set of points at a fixed distance from a center point. An important observation I have to make, is that the distance is measured along the surface, that is along a curved line between the points. Now, we can all agree on what circles are and you will have to also agree that as bigger as the circle gets, the ratio will go down. Here's a simple example. consider the center of the circle at the North Pole, and lets take the diameter to be equal to half of the circumference of the sphere we are drawing on. You should get a circle that runs across the equator of the sphere, as all points on the equator are at this distance from the poles. Therefore, the circumference of the circle we just drew is equal to the circumference of the sphere. Calculate the ratio, and you should get 2! Not such a magic number, but nice anyway. :)

If you're still with me after all this math.... I'll get on with my point. Basically, the ratio of the circumference to the diameter of the circle depends on two things: the curvature of the space we draw on (doesn't have to be a sphere! ;) ), and on the size of the circle (therefore on its radius or diameter). But we can consider that circles are small enough that the size doesn't matter. No one is going to draw a big circle on the surface of the Earth.

That leaves the curvature of space. And this can be an explanation of why God would say that Pi = 3. It could be that the curvature of space-time in which God lives would make Pi equal 3. 3 is after all an important number when talking about God. So if we knew how big God draws his circles, we could find out the curvature in which he is living, the Curvature of Heaven.

I believe that this matter should be analysed deeper, who knows what we might find out about ourselves, our beliefs and the world we live in.

P.S. You can find out more about the maths I spoke about, in A Brief History of Time by Stephen Hawking. You can even find the exact formula for Pi, depending on the curvature and the radius of the circle in there, if I remember well.