2 1. Book Overview

Figure 1.1. The square torus

We will not yet say exactly what we mean by gluing, but we say

intuitively that a 2-dimensional being—call it a bug—that wanders

off the top of the square would reappear magically on the bottom, in

the same horizontal position. Likewise, a bug that wanders off the

right side of the square would magically reappear on the left side at

the same vertical position. We have drawn a continuous curve on the

flat torus to indicate what we are talking about. In §3.1 we give a

formal treatment of the gluing construction.

At first it appears that the square torus has an edge to it, but this

is an illusion. Certainly, points in the middle of the square look just

look like the Euclidean plane. A myopic bug sitting near the center

of the square would not be able to tell he was living in the torus.

Consider what the bug sees if he sits on one of the horizontal

edges. First of all, the bug actually sits simultaneously on both hori-

zontal edges, because these edges are glued together. Looking “down-

ward”, the bug sees a little half-disk. Looking “upward”, the bug sees

another little half-disk. These 2 half-disks are glued together and

make one full Euclidean disk. So, the bug would again think that he

was sitting in the middle of the Euclidean plane. The same argument

goes for any point on any of the edges.

The only tricky points are the corners. What if the bug sits

at one of the corners of the squares? Note first of all that the bug

actually sits simultaneously at all 4 corners, because these corners are