Future, Present, & Past:
Speculative~~ Giving itself latitude and leisure to take any premise or inquiry to its furthest associative conclusion.
Critical~~ Ready to apply, to itself and its object, the canons of reason, evidence, style, and ethics, up to their limits.
Traditional~~ At home and at large in the ecosystem of practice and memory that radically nourishes the whole person.
Monday, January 31, 2011
A trivial math question
I am not a mathematician, but I occasionally dabble in recreational mathematics. One thing I want to occasionally do with this blog is to throw out some observations or questions that occur to a layman like myself, someone who asks questions that are probably simple enough but to which I don't know the answer. These aren't meant as "brain teasers" (though sometimes they serve as such for me or for some of the kids I work with); I'm just leaving them out in case someone wants to point me to an obvious solution (or at least a method). I'm painfully aware that in asking this I make clear my own mathematical limitations, but there it is.
Everyone knows that the ratio of a Euclidean circle's circumference to its diameter is π:1. And these days, we also are familiar with the fact that Euclidean geometry does not transfer automatically to other surfaces. Thus, on a the surface of a regular 3-D sphere, the ratio of circumference to diameter changes depending on how big the circle is. You can visualize this most easily by imagining a globe. The "great circle" drawn on the globe by the equator has a diameter that is exactly half of its circumference; imagine the diameter drawing a line from one point on the equator, up through the north pole, and back to a spot on the equator 180° removed from the starting point. This diameter is demonstrably half of the circumference; you could slide it down the face of the globe till it coincided with half of the original circle. Thus, on a sphere's surface, this circle's ratio of circumference to diameter is 2:1.
But not every circle will have a diameter of half its circumference; only a circle as big as the equator (a "great circle") will have this. Any smaller circle (and no circle on a sphere's surface can be larger, of course) will have a circumference whose ratio to the diameter is larger than this. The smaller the circle, the larger this ratio. You can visualize this if you imagine shrinking the circle, starting at the equator and edging up closer towards the pole. The closer you get to the pole, the more you approximate the Euclidean plane, and so the closer the circumference:diameter ratio gets to π:1. (This is so for the same reason that the Earth looks flat if you are standing on it. A sufficiently small circle can leave the curvature of the sphere's surface out of account.)
This is what occasions my question. If this ratio starts (at the equator, or any great circle) at 2, and reaches π at the pole (or at any geometrical point), then there is a particular size of circle on the surface of a sphere where the ratio of circumference:diameter is precisely 3:1. Where is this circle? I.e., how big is it in comparison to a great circle on the same sphere? (Or, say, to the diameter of the sphere itself?)
What is interesting about this (to me) is that one can see, in the case of a great circle, why the ratio has to be 2:1. And you can just manage to stipulate that, since the point is where the sphere becomes a "virtual plane," so to speak, Euclidean values become valid there...well, because they do. (Just why Euclidean constants are as they are is a different question.) But in the case of the circle I've got in mind, there's not really an obvious reason why the integer 3 should obtain in that ratio just there.
I've no idea how many SCT readers are qualified to venture advice here for an amateur, but this is obviously not one of the really hard questions; it's just an interesting one that happens to have me slightly out of my mathematical depth. If anyone wants to pipe up, I'll be more than happy to hear from them.