Yesterday, I wrote:
All power means are, for lack of a better word, slideruleable. For any xth power mean, all numbers can be arranged in order on a line such that the xth power mean of a and b is always halfway between a and b. For instance for the arithmetic mean, the numbers are equally spaced, like on a ruler. For the geometric mean, a log scale is used. This is *not* the case with the arithmetic-geometric mean. Is there an official name for this useful property?
Let me give a concrete example. Suppose you try to create an AGM slide rule. Start by getting a blank slide rule. Next, choose any two positive real numbers, perhaps 1 and 10. Mark them at opposite ends of the rule. Next, calculate their AGM and put it in the middle. It's about 4.250407. Next, calculate the AGM of 1 and 4.250407, and put it at the 1/4 mark. It's about 2.334919. Next, calculate the AGM of 4.250407 and 10, and put it at the 3/4 mark. It's about 6.818996. So far so good. You could keep subdividing the rule like that until the numbers are as dense as you like. But what if instead we take the AGM of the numbers at the 1/4 and the 3/4 positions? Shouldn't we get the number at the 1/2 position? We would if we were doing this with any power mean. But we don't. We get about 4.278555. The discrepancy gets greater as the initial two numbers get further apart, either in difference or in ratio. For instance if we start with 1 and 1000, we get about 189.388302 at the midpoint, but about 210.137522 as the AGM of the quarter points. So it can't work with a line. Can it work with a plane? I'm envisioning placing all the positive reals on a smooth curve, and drawing a straight line segment between the two reals you want the AGM of, such that the midpoint of the segment is somehow associated with the right answer. Perhaps it can simply be where the perpendicular bisector of the segment intersects the curve. Failing that, perhaps a family of smooth curves can lead from the midpoints to the desired answer for each. I have no idea how to go about finding such a curve, or proving it doesn't exist. First, is it even possible to have a curve such that for every pair of points on the curve, the segment connecting them has a different midpoint than any other such segment? Possibly the parabola has this property? On second thought that property is neither necessary nor sufficient for what I want. It would obviously have to be an infinite curve, since the positive reals go on forever. The curve may go on forever in both directions, since zero is, in a sense, infinitely far away in AGM space, just as it is in log space. As I've mentioned, I'm not a visual thinker. But I was raised on slide rules, Smith charts, and the like. Is this well-explored territory?