The Klein quartic "curve" is my very favorite among all compact surfaces, so if I may be permitted to wax rhapsodic about it for a bit: There is a whole book on it, The Eightfold Way, freely downloadable from MSRI at < http://library.msri.org/books/Book35/contents.html >. This includes Klein's original 1979 paper, translated into English by Silvio Levy. Many chapters are technical, but some are quite readable, especially those by Bill Thurston, Jeremy Gray, and the Fergusons. Klein originally defined this surface as the locus Q in the complex projective plane CP^2 of the equation X Y^3 + Y Z^3 + Z X^3 = 0. As a complex projective curve of genus >= 2, Q is a minimal surface in CP^2. It does not have constant Gaussian curvature, but as a Riemann surface, only conformal equivalence type matters, and every Riemann surface is conformally equivalent to one of constant curvature = -1, 0, or 1. Often "Klein quartic" is used to mean the unique surface K of constant curvature = -1 that is conformally equivalent to Q -- which is what Mike is looking at -- or any metric surface at all that is conformally equivalent to Q. Or more generally the entire conformal equivalence class (aka Riemann surface) of such surfaces. The Klein quartic is the unique Riemann surface S, of the least possible genus (g = 3), that attains the maximum size of its conformal automorphism group Aut(S) allowed by the Hurwitz bound #(Aut(S)) <= 84(g-1) for compact S with g >= 2. Hence Aut(Q) and Aut(K) are each isomorphic to the unique simple group G168 of order 168 (the second-smallest non-trivial simple group). This is also the automorphism group of the smallest non-trivial (combinatorial) projective plane, the Fano plane -- 7 lines and 7 points with each point lying on 3 lines and each line lying on 3 points. (Often depicted as an equilateral triangle and its 3 medians -- plus its incircle, which represents the 7th line.) (Incidentally, in CP^2 a generator of Aut(Q) of order 3 is just the permutation taking X -> Y -> Z -> X. One of order 7 is given by X -> w X, Y -> w^2 Y, Z -> W^4 Z. But it seems far trickier to find one of order 2. See expression 1.3 in the chapter by Noam Elkies in the above book.) G168 is also isomorphic to each projective special linear group PSL(3, Z/2) == PSL(2, Z/7) -- one of only two coincidences among the finite PSL's, the other being PSL(2, Z/4) == PSL(2, Z/5). In fact, the groups Aut(Q) and Aut(K) are identical to the groups Isom+(Q) and Isom+(K) of orientation-preserving isometries of Q and K respectively. (The full isometry groups Isom(Q) and Isom(K) are of order 336. Given that the Aut's == G168, this is easily seen for K from its symmetry; likewise for Q by noticing that (X,Y,Z) |-> (X*,Y*,Z*) is an isometry of Q where * denotes complex conjugate.) --Dan Sometimes the brain has a mind of its own.