7-11 has eggs. Recall the 7.11 problem: Professor O'Blivet blunders into a 7-11 and selects four purchases, dutifully tallying their prices on his pocket calculator, but absent-mindedly using the Times button instead of the Add button, and is amused to note a total of $7.11, but not nearly as surprised as he should have been when the clerk reaches the same figure using proper addition. What are the four prices? The solution is unique (modulo permutation) when restricted to positive whole cents, but has infinitely many (a continuum?) of negative and rational solutions, e.g., $9, -5, -.05, 3.16 and $49/25, 4099683/2972866, 158623231/171679340, 33012030/11595311 . Plus a double-continuum of irrational solutions. Eliminating variables produced the "elliptic" curve d^2 + (790/(100 * b + 197)) = (((50 * b - 197)^2)/7500), which has sixfold symmetry and resembles a rounded triangle surrounded by three "hyperbolas". Solutions to the 7.11 problem correspond to points on the curve, and the straight line connecting any two will always meet the curve exactly once more, generating a third solution. Magically, such a "sum" of two rational solutions is rational. Permuting the prices generates multiple points, so you only need one solution to get started. (But does this generate them all?) The rounded triangle makes a decent egg when vertically squished. As noted in previous mail, when you plot y = +- sqrt(f(x)), with f a rational function, say, you get ovals on the x-axis with roots of f as their poles (animal and vegetal!). Question: how do we know the eggs are infinitely smooth at those poles? I.e., are they infinitely differentiable as functions of arc-length, say? One sure way to get a smooth egg is Ptolemaic (epicyclic) synthesis. Sum a_n cis(n t) gives a nicely Zwollen egg for a_1,...,a_6 = 1,.16,.125,0,.01,-.007 . These were just eyeballed. The exact van Zwolle figure has infinitely many such epicyclic syntheses, depending on the speeds you choose while tracing it. --rwg PS, intrument collector Jim Forderer showed me an article claiming that the best violins and guitars are also piecewise cicular, like the lute. PPS, I don't see how highlights reflected in a smooth surface with curvature discontinuites will have anything worse than curvature discontinuites. Is there some optical way to differentiate? segregate easteregg eggeaters isolative ovalities apivorous oviparous