# Chart for a Torus

Chart for a Torus

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 or create large jumps in the planar configuration. Two rotary dials determine a point (highlighted in green) in the course of a transformation.

(θ,ϕ)

[0,2π)

2

2

2

θ

ϕ

(θ,ϕ)