Difference between revisions of "Torus"
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| − | A ''' | + | A '''torus''' is the mathematical term for a tire-like shape created by rotating a [[circle]] around the [[x-axis]]. |
| + | If the “inner [[radius]]” (the width of the doughnut hole) is <math>R</math>, and the radius of the circular cross-section is <math>r</math>, then, the [[surface area]] of the torus is | ||
| + | :<math>S = 4 \pi^2 (R+r)r</math> | ||
| + | and the [[volume]] is | ||
| + | :<math>V = 2 \pi^2 (R+r)r^2</math> | ||
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| + | |||
| + | == Topology == | ||
| + | In [[topology]], a '''torus''' can be defined as the [[cartesian product]] of 2 [[circle]]s. It can be constructed from a [[rectangle]] by identifying its opposite [[edge]]s under the [[quotient topology]]. | ||
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| + | This means that the torus surface is connected in the same way as the points on a rectangle, where you can “wrap-around” from one side of the rectangle to the other. This was a common topology used in the playing field of older video games. | ||
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| + | The [[genus]] of a one-fold torus is 1. | ||
| + | |||
| + | Unusually, it is possible to divide a torus into seven different colored areas such that each area borders the other six. This is not possible on a flat surface or a [[sphere]], where the maximum number of areas that can all touch each other is four.<ref>[http://faculty.smcm.edu/sgoldstine/torus7.html Seven-color tori]</ref> | ||
| + | |||
| + | ==References== | ||
| + | <references/> | ||
[[Category:Geometry]] | [[Category:Geometry]] | ||
Latest revision as of 19:16, December 14, 2016
A torus is the mathematical term for a tire-like shape created by rotating a circle around the x-axis.
If the “inner radius” (the width of the doughnut hole) is <math>R</math>, and the radius of the circular cross-section is <math>r</math>, then, the surface area of the torus is
- <math>S = 4 \pi^2 (R+r)r</math>
and the volume is
- <math>V = 2 \pi^2 (R+r)r^2</math>
Topology
In topology, a torus can be defined as the cartesian product of 2 circles. It can be constructed from a rectangle by identifying its opposite edges under the quotient topology.
This means that the torus surface is connected in the same way as the points on a rectangle, where you can “wrap-around” from one side of the rectangle to the other. This was a common topology used in the playing field of older video games.
The genus of a one-fold torus is 1.
Unusually, it is possible to divide a torus into seven different colored areas such that each area borders the other six. This is not possible on a flat surface or a sphere, where the maximum number of areas that can all touch each other is four.[1]