When I was asleep the other night, I started thinking that perhaps the reason so many fundamental constants (such as pi) are irrational is that they reflect a background curvature, or warpage, of the universe as a whole. What we then perceive, say, as plane geometry, is actually curved into a third dimension (extending this perspective, of course, what we perceive as a three dimensional space is curved in a fourth spatial dimension), warping the spatial relationships between points, lines, and curves on that apparently two dimensional plane.
(This curvature of three-dimensional space, and perhaps time as well, is not necessarily limited to being in "one more" dimension - it could be in several, and they could be "space-like" or "time-like" in nature)
In the same way that Einstein described gravity as being due to local (and distant) curvatures of space due to the presence of mass, I think that what I mean is that the entire universe has a fundamental curvature that is due to its entire mass (or, more correctly, mass-energy) content. I suspect this would tie in with the idea of the "shape" of the infinite, and yet finite, universe as we know it. To think about this idea of a "shape" of the universe, again, it is easier to drop one dimension and imagine a "plane" surface which is stretched, curved, or warped in some topological way. Two simple possibilities would be to think of that two dimensional analogy being the outer surface of a sphere or a torus (doughnut shape). These two possibilities result in different topological outcomes, which I am not sure will ever be testable (and remember, they are just two examples, and utilize a dimensional reduction for ease of imagination).
So, back to the simplistic thought that led me to be thinking about this. Imagine if this curvature is "real," and completely beyond our understanding. The example of pi becomes clearer now if you think of any circle - even an imaginary one - as being drawn not on a flat two dimensional plane, but on something like part of the surface of a sphere. Now, the diameter is not a simple straight line as it appears to us, but another arc on the surface of the sphere. If the universe were truly "flat," the diameter would be shorter - and pi (the ratio of the circumference to the diameter) would be larger. Perhaps is a very nice round number like 4. In the reality "outside" our universe, that is. Everywhere in "our" universe is warped by its very existence, preventing our observation, even abstractly, of this.
Another way to then explore this idea is to think that our "whole" or "natural" numbers are actually not what we think they are. If pi is a nice round number, then obviously our "3" and "4," for example, aren't. Or, at the very least, that our "1" is not the correct building block for a natural number system!
While half of what I have said here is very interesting and thought-provoking (it had to be for me to remember it from sleep!), the other half is obviously complete nonsense. I'll leave it to you to decide which part is which.
© Huw Powell