Category:Definitions/Tori

This category contains definitions related to Tori.
Related results can be found in Category:Tori.

Geometric Definition

Let $\CC$ be a circle.

Let $\LL$ be a straight line in the same plane as $\CC$ such that $\LL$ may or may not intersect $\CC$.

A torus is the surface of revolution formed by rotating $\CC$ one full turn about $\LL$.

Topological Definition

A torus is a surface obtained by identifying both pairs of opposite sides, one with the other, of a square, while retaining the orientation:

Thus in the above diagram, $AB$ is identified with $DC$ and $CB$ with $DA$.

Formal Construction

Let $T$ be the square embedded in the Cartesian plane defined as:

$T = \closedint {-1} 1 \times \closedint {-1} 1$

Let $T'$ be the quotient space formed from $T$ using the identification mapping $p: T \to T'$ as follows:

$\forall \tuple {x, y} \in T: \map p {x, y} = \begin {cases} \paren {x, y} & : -1 < x, y < 1 \\ \\ \tuple {-x, y} & : x = \pm 1 \\ \\ \tuple {x, -y} & : y = \pm 1 \end {cases}$

Then $T'$ is a torus .