A torus (more precisely, a 2-torus) has the shape of the outer surface of a donut. A torus can be parametrized by two angular variables
, each in
. A dissected torus can be pictured as equivalent to a rectangle in the two-dimensional Euclidean plane
, with opposite pairs of edges glued together. Locally, the 2-torus resembles
, but there is no global continuous mapping since the Euclidean plane extends to infinity. This Demonstration shows possible mappings from a 2-torus to
. The operation necessarily introduces discontinuities, where small changes of
create large jumps in the planar configuration. Two rotary dials determine a point
(highlighted in green) in the course of a transformation.