On the other hand, it is the case that the group PSL(2,Z) thought of as fractional linear transformations of S^2 = C u {oo} via z |—> (az + b) / (cz + d), (a, b, c, d in Z) does preserve the rational points Q + Qi. Q + Qi is of course the union of (1/n)Z + (1/n)Zi over n = 1,2,3,..., which are all *square* lattices. So it's at least a little surprising that PSL(2,Z) has elements of order 3. —Dan I wrote: ----- In no sense, apparently! ----- ----- On Aug 9, 2017 04:32, "Dan Asimov" <dasimov@earthlink.net> wrote: And paradoxically, the conformal bijection z |—> -1/(1+z) of the Riemann sphere S^2 = C u {oo} to itself that preserves the integer points Z^2 ... Andy Latto wrote: ----- In what sense does this preserve the integer points? F(17) = -1/18 ----- (i.e., a fractional linear transformation of form z |—> (az + b)/(cz + d), a,b,c,d in Z, ad - bc = 1, i.e., a member of the group PSL(2,Z) = SL(2,Z) / {+-I}) ... is of order three: f(f(f(z))) == z -----