Using the result of that 51.2 hr Simplification, homogc[xp_, u_] := xp /. (CD : Circle | Disk)[{x_, y_}, r_] :> CD[{-((x + u*(1 - r^2 + x*(u + x) + y^2))/(r^2*u^2 - (1 + u*x)^2 - u^2*y^2)), (y - u^2*y)/(1 + u*(2*x + u*(-r^2 + x^2 + y^2)))}, Abs[ (r*(-1 + u^2))/(r^2*u^2 - (1 + u*x)^2 - u^2*y^2)]] will subject all the Circles and Disks in xp to (z+u)/(u z+1), which, modulo pre and post rotation, is, I think the most general homographic transformation preserving the unit circle. This facilitates gosper.org/stein.gif , an animation (of the 3.5 circles case) unsurprising to both the average Joe and the highly astute, but making the rest of us say, "Hey, wait a minute.". (Can there be only one decent Youtube of this?) In the 60s, I animated this on the PDP6 using the obvious algorithm of drawing (as fast as I could) the circles for a given "frame", updating the sizes and positions, and then drawing the next "frame". You control the rotational speed ω, # of circles, and u parameter variation with the 36 bit test word switches on the front panel. Freezing u and simply varying ω revealed a puzzling gap between two of the orbiting circles, said gap widening with increasing ω, and disappearing when ω → 0. This seemed paradoxical. The phosphor persistence made it clear that none of the circles changed size. How could there be room for more simply because they were moving? It's a relativistic foreshortening pun! Video in those days had no "frames"--you just threw dots on the screen. So the procedure Draw circles, Draw updated circles,... when scrutinized, contains the sequence Draw last old circle, Draw first updated circle, so your eye sees the gap, instead connecting first old circle to last old circle. However, a movie shot right off the screen sees no gap! Elegant fix. Don't rotate at all. Just draw 6.999 circles instead of 7 to rotate slowly, 6.99 to rotate rapidly, and 7.01 to spin the other way. No gap. But now, the persistence reveals them changing size when you change speed! And now filming the screen shows the mismatch. --rwg