Projection of a Torus

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Projection of a Torus
A four-dimensional torus projected into three-dimensional space.


Basic Description

It is impossible to visualize a complete four-dimensional object, since we have only ever lived in three-dimensional space. However, there are ways to capture parts of the four-dimensional object in three-dimensional space. A useful analogy is a world map. We can capture the essence of the three-dimensional globe on a two-dimensional map, but only by using a projection, which distorts the three-dimensional object in some way to fit on a two-dimensional surface.

A similar process is carried out to create this page's main image. A four-dimensional object, described further below, is projected into three-dimensions using two different projections.

A More Mathematical Explanation

The four-dimensional object is defined parametrically by  (x_1,x_2,x_3,x_4)=(cos(u),sin(u),cos(v),sin(v)) . A [...]

The four-dimensional object is defined parametrically by  (x_1,x_2,x_3,x_4)=(cos(u),sin(u),cos(v),sin(v)) . A stereographic projection is used to map this 4-d object into 3-d, using a projection point of  (0,0,0,\sqrt{2}) for the first object in this page's main image. This projection is centered above the four-dimensional object, projecting the symmetric torus into three-dimensional space. The projection point is shifted to be closer to one part of the four-dimensional object than the other to create the second object in the main image, projecting an uneven object into three-dimensional space.

Teaching Materials

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About the Creator of this Image

Thomas F. Banchoff is a geometer, and a professor at Brown University since 1967.

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