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 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 Cartesian Product of two circles. | + | |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 <balloon title="The set of all points (A, B) for arbitrary-dimensional sets A and B">Cartesian Product</balloon> of two circles. |
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. |
Revision as of 10:35, 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 UNIQ74a3ba8b7d6ede7 [...]
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
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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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