1. A familiar way to define a torus is to start with a square, say [0,1]x[0,1] in the plane, and identify corresponding points on the right and left edges, and also corresponding points on the top and bottom edges. (I.e., identify (t,0) with (t,1), and identify (0,t) with (1,t), for all 0 ≤ t ≤ 1.) Suppose that *in addition to those identifications*, we also identify (x,y) with (y,x) for all points (x,y) of the square (not just the edges). What familiar shape is the result? Note that we're only concerned with the topology, not the geometry. 2. Start again with a fresh, unidentified square [0,1]x[0,1]. Now: * Identify (t,0) with (t+1/2 (mod 1), 1) for all 0 ≤ t ≤ 1, and * Identify (0,t) with (1, t+1/2 (mod 1)) for all 0 ≤ t ≤ 1. What familiar shape is the result, topologically? —Dan