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Chart for a Torus

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Torus Plane
2.
θ
ϕ
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
(θ,ϕ)
, each in
[0,2π)
. A dissected torus can be pictured as equivalent to a rectangle in the two-dimensional Euclidean plane
2
, with opposite pairs of edges glued together. Locally, the 2-torus resembles
2
, but there is no global continuous mapping since the Euclidean plane extends to infinity. This Demonstration shows possible mappings from a 2-torus to
2
. The operation necessarily introduces discontinuities, where small changes of
θ
or
ϕ
create large jumps in the planar configuration. Two rotary dials determine a point
(θ,ϕ)
(highlighted in green) in the course of a transformation.
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