Cut the cone, open it up into a sector of angle alpha, and give the sector complex coordinates w = r exp(i theta). Let L be a lattice of translations of the z-plane such that there exists an integer n so that for each translation t in L, Im(t) is a multiple of alpha/n. Under the conformal map w = exp(z), the z-plane translation t sends w to w' = exp(z+t) with r' = Re(t) r, theta' = theta + alpha/n. For example, suppose L is the square lattice generated by t1 = i alpha/n, t2 = alpha/n. Then the cone is tessellated by conformal squares of the form r1 < r < exp(alpha/n) r1, theta1 < theta < theta + alpha/n. -- Gene From: James Propp <jamespropp@gmail.com> To: math-fun <math-fun@mailman.xmission.com> Sent: Tuesday, June 16, 2015 9:40 AM Subject: Re: [math-fun] Decorating a pseudosphere For purposes of my analogy, one should delete the cone point itself, to avoid the infinite curvature there. The resulting manifold then has zero curvature everywhere, but it has fewer symmetries than the plane that gave rise to it by cutting and gluing. (How should one wallpaper the inside of a teepee for maximal mathematical elegance? Unclear.) Jim Propp