# Projection of a Torus

### From Math Images

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| -                                      | |ImageName=Projection of a Torus                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                        | |ImageName=4-Dimensional Torus                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              |
|                                        | |Image=4dtorus.jpg                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 |                                                          | |Image=4dtorus.jpg                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | |ImageIntro=A four-dimensional torus projected into three-dimensional space.                                                                                                                                                                                                                                                                                                                                                                                                                       | +                                                        | |ImageIntro=A torus in four dimensions projected into three-dimensional space.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              |
| -                                      | |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 distorts the three-dimensional object in some way to fit on a two-dimensional surface. | +                                                        | |ImageDescElem=It is impossible to visualize an object in four-dimensions, since we have only ever lived in three-dimensional space.  However, there are ways to capture features of the four-dimensional object in three-dimensional space.                                                                                                                                                                                                                                                                                                                                                                                                                                                                |
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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 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.                                                                                                                                                                                                                                                                                                                                                                                                                       |
| -                                      | |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 similar process is carried out to create this page's main image.  An object in four-dimensional space, described further below, is projected into three-dimensions using two different projections.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       |
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|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    | +                                                        | |ImageDesc= A [[torus]] is commonly known as the surface of a doughnut shape. It can be described using [[Parametric Equations|parametric equations]]. While it is a <balloon title="Other examples of two dimensional surfaces are planes and the surface of a sphere. They are surfaces in the plain English sense of the word"> two dimensional surface </balloon>, it lives in three dimensional space.                                                                                                                                                                                                                                                                                                 |
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|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    | +                                                        | A four-dimensional torus is an analogous object that lives in four dimensional space. The main image contains two images which ways of visualizing a four dimensional torus in three dimensions.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            |
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|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    | +                                                        | The four-dimensional torus is defined [[Parametric Equations|parametrically]] by <math> (x_1,\,x_2,\,x_3,\,x_4)=(cos(u),\,sin(u),\,cos(v),\,sin(v)) </math>.  The first two coordinates of the parametrization give a circle in u-space, and the second two coordinates give a circle in v-space.  The torus is thus the [[Cartesian Product]] of two circles.                                                                                                                                                                                                                                                                                                                                              |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    | +                                                        | A [[Stereographic Projection| stereographic projection]] is used to map this object, which lives in four-dimensional space, into three-dimensional space, 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 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 three dimensions.  This projection'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.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               |
|                                        | |SiteName=The Mathematics of In- and Outside the Torus                                                                                                                                                                                                                                                                                                                                                                                                                                             |                                                          | |SiteName=The Mathematics of In- and Outside the Torus                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      |
|                                        | |SiteURL=http://www.math.brown.edu/~banchoff/art/PAC-9603/tour/torus/torus-math.html                                                                                                                                                                                                                                                                                                                                                                                                               |                                                          | |SiteURL=http://www.math.brown.edu/~banchoff/art/PAC-9603/tour/torus/torus-math.html                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        |
|                                        | |Field=Algebra                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     |                                                          | |Field=Algebra                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              |
| -                                      | |InProgress=Yes                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    | +                                                        | |References=http://www.math.brown.edu/~banchoff/art/PAC-9603/tour/torus/torus-math.html                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    | +                                                        | |InProgress=No                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              |
|                                        | }}                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 |                                                          | }}                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |

## Current revision

4-Dimensional Torus
Field: Algebra
Image Created By: Thomas F. Banchoff
Website: The Mathematics of In- and Outside the Torus

4-Dimensional Torus

A torus in four dimensions projected into three-dimensional space.

# Basic Description

It is impossible to visualize an object in four-dimensions, since we have only ever lived in three-dimensional space. However, there are ways to capture features 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. An object in four-dimensional space, described further below, is projected into three-dimensions using two different projections.

# A More Mathematical Explanation

[Click to view A More Mathematical Explanation]

A torus is commonly known as the surface of a doughnut shape. It can be described using [[Paramet [...]

[Click to hide A More Mathematical Explanation]

A torus is commonly known as the surface of a doughnut shape. It can be described using parametric equations. While it is a two dimensional surface , it lives in three dimensional space.

A four-dimensional torus is an analogous object that lives in four dimensional space. The main image contains two images which ways of visualizing a four dimensional torus in three dimensions.

The four-dimensional torus is defined parametrically by $(x_1,\,x_2,\,x_3,\,x_4)=(cos(u),\,sin(u),\,cos(v),\,sin(v))$. The first two coordinates of the parametrization give a circle in u-space, and the second two coordinates give a circle in v-space. The torus is thus the Cartesian Product of two circles.

A stereographic projection is used to map this object, which lives in four-dimensional space, into three-dimensional space, 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 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 three dimensions. This projection'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.

# References

http://www.math.brown.edu/~banchoff/art/PAC-9603/tour/torus/torus-math.html

Leave a message on the discussion page by clicking the 'discussion' tab at the top of this image page.
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