How pedestrian. Of course if you connect opposite corners of an a x b rectangle, you can make a torus having as its side a diagonal of that rectangle, hence of a^2 + b^2 points. (Simplest case being what you get by connecting each corner of a square to the midpoint of a nonadjacent side in the same sense: The torus consists of 5 points.) Possibilities: 5, 10, 13, 17, 20, 25, .... Hmm, the 3^2 + 4^2 = 5^2 one might look like this: o * * * * * * o * * * * * * * * * * * * o * * * * * * o when you consider the 4 extreme points o to be identical. The resulting configurations of a^2 + b^2 points on a square torus can be thought of as the quotient ring of the Gaussian integers Z[i] by the ideal <a + bi> = (a + bi) Z[i] (for a fixed choice of integers a and b) — sitting if you like in the torus C / <a + bi>. Except for the various orthogonal and 45º cases, the resulting configuration has no mirror symmetry. —Dan Mike Beeler wrote: ----- Strangely, the torus illustrated at that web page shows a board of even dimension (20) in one of the directions. Accidental or intentional, anyone know?
On Dec 5, 2018, at 4:24 PM, Thane Plambeck <tplambeck@gmail.com> wrote:
Toroidal Go
http://www.usgo.org/news/2018/12/go-without-borders-seeks-beta-testers/