# Part II — The Torus Consider a flat rectangular sheet of paper, like a napkin. Imagine drawing straight lines on it, each line connecting a pair of opposite edges. Together, these lines form a grid on the napkin. Now fold the napkin by identifying and gluing one pair of opposite edges. The flat sheet becomes a tube. Using a flexible napkin-tube, identifying and gluing its two circular ends produces a closed, donut-shaped surface. This surface is called a torus. Recalling the lines drawn on the original napkin, once the torus is formed, each line becomes continuous only if it matches its own position when crossing an identified edge. This requirement is called a continuity condition. Let $m$ be the number of lines drawn between one pair of opposite edges of the napkin, and $n$ the number of lines drawn between the other pair of opposite edges. Let the total area of the napkin be $a$, or, more suggestively, $E_0$. The lines divide the napkin into a grid of cells. The more lines drawn on either side, the more — and smaller — the cells. With one line in each direction, the surface is divided into $2 \times 2 = 4$ cells. With two lines in each direction, into $3 \times 3 = 9$ cells. With three lines, into $4 \times 4 = 16$ cells, and so on. Since the napkin has two sides, each cell can be thought of as having two sides as well. In general, drawing $m$ lines in one direction and $n$ lines in the other divides the surface into $m$ and $n$ cells per side. Each cell therefore carries an equal fraction of the total area (or energy) of the sheet of paper, namely $E_0/(m n)$ per side. We can visualize this by painting the cells. With one line in each direction, we paint $4$ squares on one side, or $8$ squares counting both sides with an $E_0/4$ color. Adding a new line in each direction produces $9 \times 2 = 18$ squares of area $E_0/18$ each. Two more lines produce $16 \times 2 = 32$ squares. Only the necessary amount of paint is required to fill each cell in full. As the number of divisions increases, the same total amount of paint is simply redistributed over a larger number of smaller cells.