A recent "World Cup" contest for mental calculation--- The events go up to cube roots. http://www.recordholders.org/en/events/worldcup/2012/ Willem Bouman, mental factoring of 4 & 5 digit numbers: http://www.recordholders.org/downloads/worldcup/factorization-tutorial.pdf --- Henry Baker asks
What would be the appropriate generalization of an ellipse to more than 2 foci ?
Obvious candidates for trifocals: Assume the three foci are a triangle, probably scalene. a) The string & pencil construction. This will give a curve with six sections of ellipses. b) The locus of points whose sum-of-distances to the foci is constant. This extends naturally to three dimensions. c) Combining A & B in a natural way in 3D: the constraint for our pencil is that the tetrahedron from the pencil to the focal triangle has a constant area -- as if it were a soap film, perhaps. Rich