Projection of a Torus
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|ImageIntro=A torus in four dimensions projected into three-dimensional space. | |ImageIntro=A torus in four dimensions projected into three-dimensional space. | ||
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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. | 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. | ||
- | |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 torus is thus the [[Cartesian Product]] of two circles. | + | |
+ | |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. | 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. |
Current revision
4-Dimensional Torus |
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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 . 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 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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