3 Feb
2014
3 Feb
'14
3:53 a.m.
Torus puzzle:
Find a homeomorphism h: T^2 -> T^2 from the torus T^2 to itself that's periodic of least period 3, and that has exactly 3 fixed points.
--Dan
Very beautiful. Let T^2 = R/Z x R/Z. Then the homeomorphism corresponding to the matrix: [0 1] [-1 -1] is period-3 and has fixed points (0,0), (1/3,1/3) and (2/3,2/3). This corresponds to the order-3 element of the modular group SL(2,Z), which in turn corresponds to an order-3 rotation in the (2,3,infinity) tiling of the hyperbolic plane (which is how I discovered this homeomorphism). Sincerely, Adam P. Goucher