I posted to math-fun recently on how a complex power series f(z) = ∑ c_n z^n (where the summation is over n ≥ 0, and the c_n ∊ ℂ (where ℂ denotes the complex plane, and it is assumed convergent at least for all z in some neighborhood of the origin 0 ∊ ℂ), defines by analytic continuation a certain surface (called "the Riemann surface of the function f(z)"). A Riemann surface can be defined more abstractly, as a topological space X that is the union of some family {U_j} of open sets for j ∊ J, an arbitrary index set. Such that for each j ∊ J there is a homeomorphism h_j : U_j —> ℂ such that whenever U_j ∩ U_k ≠ ∅, the mapping h_j ∘ h_k^(-1) : h_k(U_j ∩ U_k) —> h_j(U_j ∩ U_k) is an analytic homeomorphism. Another more geometric way to say "an analytic homeomorphism" is to say that this map h_j ∘ h_k^(-1), whose domain and codomain are open subsets of ℂ, is *conformal*, that is, preserves angles and orientation. Then, the natural notion of when two Riemann surfaces are "equivalent" is when there is a conformal homeomorphism between them. This implies that a Riemann surface is the same as a smooth surface with a well-defined measure on the angles between any two curves that exit the same point. A simply connected subset of the complex plane ℂ is any subset in which every closed curve can be shrunk to a point. Simply connected subsets of ℂ can be ridiculously complicated. Imagine e.g. the "topologist's sine curve" with its endpoints connected by an ordinary path, making a closed curve. Now draw crazy hairs going inward from the boundary, and smaller hairs on hairs ad infinitum. The interior of this thing is simply connected. Yet the Riemann mapping theorem (not proved rigorously until the early 1900s) implies that all simply connected subsets of ℂ are homeomorphic. And if we exclude only ℂ itself, all the rest are even conformally equivalent to each other — homeomorphic by a homeomorphism that preserves angles. Also, the Koebe-Poincaré uniformization theorem implies that all smooth Riemann surfaces that are topologically a sphere S^2 are conformally equivalent to each other. But something different happens when considering Riemann surfaces that are topologically a torus, T^2. In this case it can be shown that the set of conformal equivalence classes of tori are in bijective correspondence with the points of the set X = {z ∊ ℂ | |z| > 1 and -1/2 ≤ Re(z) < 1/2}. (Note that its boundary consists of two semi-infinite lines and a 60º arc.) A finer tuning of this fact gives a geometric bijection between the set of conformal equivalence classes of tori with the *quotient* of the set X = {z ∊ ℂ | |z| > 1 and -1/2 ≤ Re(z) ≤ 1/2} (note that a less-than has become a less-than-or-equal-to) after any boundary point z ∊ ∂X and its reflection z' about the y-axis are identified with each other. That is, if z = x + iy then it is identified with z' = z - 2 Re(z) = -x + iy. The resulting shape is topologically a disk. But geometrically it has two special points, one of which has only 180º of angle around it and the other of which has only 120º of angle around it. These correspond to the square torus T_Q = ℂ/Z[i] and the hexagonal torus T_H = ℂ/Z[𝜔], (where 𝜔 = exp(2πi/3)), respectively. —Dan