----- QUESTION: Given the square torus T^2, what is the smallest number of triangles in a tiling of the torus by triangles such that they are all acute? I need to define a few terms for this particular question: * The "square torus" T^2 is the result of identifying opposite edges of a square. (Equivalently, it is the metric space obtained from the quotient of the plane R^2 by its integer subgroup Z^2.) * A "tiling of the torus by triangles", for the purposes of this question, is any union of triangles that equals the torus, such that a) Any two triangles have disjoint interiors, and AAAA b) No triangle's vertex lies on the interior of any triangle's edge. ----- It's well known that the square can be tiled in this sense with 8 acute triangles and no fewer. The standard example (https://www.ics.uci.edu/~eppstein/junkyard/acute-square/8-square.gif) gives, by identifying opposite edges of the square, a tiling of the square torus using 8 acute triangles. Note: The most general flat torus is obtained by identifying opposite edges of a parallelogram. Clearly, any non-rectangular flat torus can be tiled in the sense of this question using only 2 acute triangles. Can the upper bound of 8 for the square torus be improved on? And what about a non-square rectangular torus? —Dan