torus

Visually, the torus looks like a doughnut. Informally, we take a rectangle, identify two edges to form a cylinder, and then identify the two ends of the cylinder to form the torus. Doing this gives us a surface of genus one. It can also be described as the Cartesian product of two circles, that is, $S^1 \times S^1$ . The torus can be parameterized in Cartesian coordinates by: $$x = \cos(s) \cdot(R + r \cdot \cos(t))$$ $$y = \sin(s) \cdot (R + r \cdot \cos(t))$$ $$z = r \cdot \sin(t)$$ with $R$ the major radius and $r$ the minor radius are constant, and $s,t \in [0,2\pi)$ .

Figure 1: A torus generated with Mathematica 4.1

To create the torus mathematically, we start with the closed subset $X = [0,1] \times [0,1] \subseteq \mathbb{R}^2$ . Let $X^*$ be the set with elements: $$\{ x \times 0, x \times 1 \mid 0 < x < 1 \}$$ $$\{ 0 \times y, 1 \times y \mid 0 < y < 1 \}$$ and also the four-point set $$\{ 0 \times 0, 1 \times 0, 0 \times 1, 1 \times 1 \}.$$

This can be schematically represented in the following diagram.

Diagram 1: The identifications made on $I^2$ to make a torus.
Opposite sides are identified with equal orientations, and the four corners
are identified to one point.

Note that $X^*$ is a partition of $X$ , where we have identified opposite sides of the square together, and all four corners together. We can then form the quotient topology induced by the quotient map $p\colon X \longrightarrow X^*$ by sending each element $x \in X$ to the corresponding element of $X^*$ containing $x$ .