I'm not certain what it would mean for a plane to wrap around "at infinity". Surely wrapping around means that you must only travel a finite distance to reach where you started. But I don't have a mathematical definition handy.sab39 wrote:I don't think it makes a difference whether the infinite plane "wraps around" at infinity or not.
Yes, I don't think anyone spoke of an infinite torus. An infinite cylinder such as I mentioned is perfectly reasonable to consider, however (or at least as reasonable to consider as an infinite plane).sab39 wrote:(I'm aware that this isn't a rigorous proof. But it's good enough to convince me in an intuitive sense, especially since an "infinite torus" is pretty darn impossible in the first place).
I believe you're thinking of the Riemann sphere. Regardless, of course you can map any set bijectively onto any set of the same cardinality (and of course the plane and the sphere both have the cardinality of the continuum). The point is that gringer's proposal could not reasonably be considered an infinite grid of resistors, because it contains only finitely many resistors. The reasonable definition of an infinite grid in this case relates to the number of resistors, not their embedding in a metric space, since the latter is irrelevant to the situation: the grid behaves identically however big or small or near or distant the resistors are according to any metric.sab39 wrote:Incidentally, isn't there a projection of the complex numbers onto a sphere which treats infinity in all directions as the point at the north pole? So it is possible to map an infinite grid onto a sphere - but it's not very clear how the resistors would join up at the top, other than being packed infinitely dense.
Anyway, I think gringer would have gotten more points than most of us here on this question of the Google test, just for coming up with the idea. Possibly even more than someone who actually solved it. I should have thought of it, dammit. I'm a math student, I'm supposed to poke holes in physics questions for fun.
It can't possibly have higher resistance than three ohms, since there is at least one three-ohm path.chopps wrote:Isn't it simply 20 ohms? There may be infinite resistors, but I only see 4 parallel paths.