Of an epicyclic synthesis of Moss's egg, I sent:
Note the surprising presence of counterrotors, presumably due to my lazy choice of traversal speeds: constant dtheta/dt on all four arcs, meaning instant ac/deceleration at each arc [endpoint].
The constant arcspeed rendition of Moss's egg is oo (sqrt(2) - 11) n pi ==== ' cos(-------------------) \ i n t n 17
e (- 1) (------------------------
/ 4 n + sqrt(2) - 6 ==== n = - oo (2 sqrt(2) - 5) n 1 (sqrt(2) + 1) cos((----------------- - -) pi) 17 4 - ---------------------------------------------) 4 n - 2 sqrt(2) - 5 ----------------------------------------------------, n (8 n + sqrt(2) - 6) (after enormous simplification*). (Input form: (-1)^n*(cos((sqrt(2)-11)*%pi*n/17)/(4*n+sqrt(2)-6)-(sqrt(2)+1)*cos(%pi*((2*sqrt(2)-5)*n/17-1/4))/(4*n-2*sqrt(2)-5))/(n*(8*n+sqrt(2)-6))*cis(n*t).) We now have n^-3 convergence, but lingering (surprisingly large) counterrotors. The harmonics > .002, 0.00532 0.03246 0.09996 i t 2 i t 3 i t ------- - ------- + ------- + e + 0.00711 e + 0.03168 e 3 i t 2 i t i t e e e 4 i t 5 i t 6 i t 7 i t - 0.01495 e + 0.0044 e - 0.00297 e + 0.00246 e , come within a pixel (modulo slight scaling and translation), showing that Moss's curvature discontinuites are imperceptible. Just the four terms > 1%, 0.09996 i t 3 i t 4 i t ------- + e + 0.03168 e - 0.01495 e , i t e draw an egg only slighly plump. We didn't beat n^-3 because the acceleration vector, though of continuous direction, has step function magnitude. To get the n^-arbitrary Gene described it seems we must come to a full, ultrasmooth (like exp(-1/t) ?) stop at each junction in order to match all derivatives.
E.g.g. Ouch! Eggregious. Oeuf with his head. --rwg STAGGERY EGGTRAYS *The original expression was O(n^18)/O(n^19) and ran on for pages.