Projection of a Torus

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{{Image Description
{{Image Description
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|ImageName=Projection of a Torus
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|ImageName=Projection of a 4-Dimensional Torus
|Image=4dtorus.jpg
|Image=4dtorus.jpg
|ImageIntro=A four-dimensional torus projected into three-dimensional space.
|ImageIntro=A four-dimensional torus projected into three-dimensional space.
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|ImageDescElem=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.
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|ImageDescElem=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.
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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 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.
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|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 becoming asymmetric because of the light source's positioning.
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|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>.
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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 becoming 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 09:48, 9 June 2009

Image:inprogress.png
Projection of a 4-Dimensional Torus
A four-dimensional torus projected into three-dimensional space.

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 UNIQ615e4cd38179f5 [...]

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. 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 becoming asymmetric because of the light source's positioning.




Teaching Materials

There are currently no teaching materials for this page. Add teaching materials.

About the Creator of this Image

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








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