Projection of a Torus
From Math Images
|Projection of a Torus|
Basic DescriptionIt 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 translates a three-dimensional object onto a two-dimensional surface at the expense of distorting the object in some way.
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 . A [...]
The four-dimensional object is defined parametrically by . A stereographic projection is used to map this 4-d object into 3-d, using a projection point of 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.
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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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