From Osher Doctorow Ph.D.
I've confused the cardioid and the cycloid in my last posting and I may have done it previously too. The cardioid is: 1) r = a(1 - cos(u)) nsofar as it is represented by the Mandelbrot cardioids, while the cycloid is: 2) x = a(t - sin(t)) y = a(1 - cos(t)) Although y and r look similar, they're very different curves because of the x equation in (2). The same restrictions and considerations apply as I stated in my last posting since the forms of the equations are so similar. In general, one should be very careful with a closed curve like a cardioid or circle or ellipse or a closed surface like a sphere or ellipsoid in not regarding it as a growth-expansion-contraction object except in very special conditions or for very special reasons, because being closed (an object moving along it returns to its original position and repeats its path in finite time), its equation is cyclic unless it taken as purely a spatial equation - and in the latter case, an additional equation has to be hypothesized to give its time variation. Osher Doctorow Ph.D.