Projection of a Torus
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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. | 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. | ||
|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 | ||
+ | |References=http://www.math.brown.edu/~banchoff/art/PAC-9603/tour/torus/torus-math.html | ||
|InProgress=No | |InProgress=No | ||
}} | }} |
Revision as of 10:15, 9 July 2009
- A four-dimensional torus projected into three-dimensional space.
Projection of a 4-Dimensional Torus |
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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 torus is defined parametrically by UNIQ2f5261cb7950855 [...]
The four-dimensional torus is defined parametrically by . 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 Cartesian Product of two circles.
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 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
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