Don't know whether ℘ will show up in most people's e-mail, so I'll use P. ((( In case you don't know it: The Weierstrass P-function is a meromorphic function that's doubly-periodic on the complex plane C. P(z) depends on a choice of lattice L in C (an additive subgroup generated by two periods w1 and w2 with WLOG Im(w2/w1) > 0). Then P(z+w) = P(z) for all w in C and all v in L. The definition is P(z) := 1/z^2 + Sum_{w in L'} (1/(z-w)^2 - 1/w^2), which doesn't make its double-periodicity immediately obvious. It's important because all doubly-periodic (w.r.t. L) meromorphic functions are rational functions of P(z) and its derivative P'(z). ))) P(z) can be thought of as a holomorphic map T^2 -> S^2, where these are the Riemann surfaces C/L and C u{oo}, resp. T^2 comes from identifying the opposite sides of a period parallelogram (whose vertices lie at, say, 0, w1, w2, and w1+w2). To see what P(z) does geometrically, chop the parallelogram into 8 congruent acute triangles (or else right, if the parallelogram is a rectangle). Take 4 of these triangles and make a tetrahedron with them as faces. Then T^2 (also thought of as made of the 8 triangles) can be mapped onto the tetrahedron so each triangle goes isometrically onto a face. The vertices of these triangles on T^2 number 4 in total, around each of which there are 6 triangles. The 3 opposite pairs each are mapped to the same face of the tetrahedron, so the 6 "go around twice". Finally, map the tetrahedron onto the sphere by mapping each face onto the unique spherical triangle with exactly double the angles; these 4 spherical triangles fit together exactly to make the usual S^2. I haven't seen this mentioned in the usual literature on the P-function, so thought it was worth mentioning. --Dan ________________________________________________________________________________________ It goes without saying that .