Bill's animation is extremely beautiful. Just in case this hasn't been mentioned already: It occurs to me that for any p/q circles case (say GCD(p,q) = 1), if one takes the q-fold cover of the annulus this occurs in, one gets a (topological) annulus -- call it A' -- with a well-defined geometry, and which contains a circular chain of p circles. Now, every geometrical annulus is conformally equivalent to a standard one: { z in C | 1 <= |z| <= R} for a unique R > 1. Let the standard annulus conformally equivalent to A' be called A. Since (locally isometric) covering maps, and conformal maps, preserve circles, we can apply Steiner's porism to annulus A, and this implies that it works in the "fractional circles" case in the first place. --Dan P.S. Steiner's porism is very similar to Poncelet's porism. What the hell is a porism, anyway? Aha -- Webster's New World dictionary defines it as follows. Now it's clear why Steiner's and Poncelet's discoveries are called porisms: ----- porism • a proposition that uncovers the possibility of finding such conditions as to make a specific problem capable of innumerable solutions ----- PORISM / PRIMOS On Aug 21, 2014, at 4:15 PM, Fred Lunnon <fred.lunnon@gmail.com> wrote:
gosper.org/stein.gif , an animation (of the 3.5 circles case)