Projection of a Torus
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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 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. | ||
| - | |ImageDesc=The four-dimensional object is defined parametrically by <math> (x_1,x_2,x_3,x_4)=(cos(u),sin(u),cos(v),sin(v)) </math>. A [[Stereographic Projection| stereographic projection]] is used to map this 4- | + | |ImageDesc=The four-dimensional object is defined [[Parametric Equations|parametrically]] by <math> (x_1,x_2,x_3,x_4)=(cos(u),sin(u),cos(v),sin(v)) </math>. A [[Stereographic Projection| stereographic projection]] is used to map this 4-D object into 3-D, using a projection point of <math> (0,0,0,\sqrt{2})</math> 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. For the second object, the projection point is shifted to be closer to one part of the four-dimensional object than the other, creating an uneven object in 3-D. This object's unevenness is similar to the shadow of a symmetric object being asymmetric because of the light source's positioning. |
|AuthorName=Thomas F. Banchoff | |AuthorName=Thomas F. Banchoff | ||
|AuthorDesc=Thomas F. Banchoff is a geometer, and a professor at Brown University since 1967. | |AuthorDesc=Thomas F. Banchoff is a geometer, and a professor at Brown University since 1967. | ||
Revision as of 11:14, 5 June 2009
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Contents |
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 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 UNIQ23c11b9e543286 [...]
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. For the second object, the projection point is shifted to be closer to one part of the four-dimensional object than the other, creating an uneven object in 3-D. This object's unevenness is similar to the shadow of a symmetric object being asymmetric because of the light source's positioning.
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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