Projection of a Torus
From Math Images
| Projection of a 4-Dimensional Torus |
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Projection of a 4-Dimensional Torus
- A torus in four dimensions projected into three-dimensional space.
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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 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.
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.
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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