# Chart for a Torus

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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.

Contributed by: Aaron T. Becker and Haoran Zhao (May 2016)

Open content licensed under CC BY-NC-SA

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The 2-torus (or ) is a manifold that can be embedded in , as illustrated by the 3D plot. A *manifold* is a topological space that locally resembles -dimensional Euclidean space in a neighborhood of each point. Locally, the neighborhood of any point on resembles one in (the Euclidean plane).

Globally, a manifold can differ from a Euclidean space; for example, the surface of the sphere is not topologically equivalent to a Euclidean space. Likewise the surface of a torus. As shown in this Demonstration, the torus can be split along its longest circumference to form a cylinder, and then this cylinder can be opened up into a rectangle. Both operations introduce discontinuities where small changes of or create large jumps in the configuration.

In the context of manifolds, these mappings are called *charts*. To avoid discontinuities, you can switch from one chart to another, much like a pilot can switch from a map of North America to a map of the Pacific when flying from New York to Tokyo. A collection of overlapping charts is called an *atlas*.

A fun problem: what is the minimum number of charts needed to construct a complete atlas of the torus?

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