Projection of a Torus

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{{Image Description
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{{Image Description Ready
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|ImageName=Projection of a 4-Dimensional Torus
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|ImageName=4-Dimensional Torus
|Image=4dtorus.jpg
|Image=4dtorus.jpg
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|ImageIntro=A four-dimensional torus projected into three-dimensional space.
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|ImageIntro=A torus in four dimensions 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.
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|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.
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 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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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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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=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 4-D torus is thus the cross product of two circles.
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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 projection'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= 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.
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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
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|AuthorDesc=Thomas F. Banchoff is a geometer, and a professor at Brown University since 1967.
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|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
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|References=http://www.math.brown.edu/~banchoff/art/PAC-9603/tour/torus/torus-math.html
|InProgress=No
|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.


Contents

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

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

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

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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





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