Start this free course now. Just create an account and sign in. Enrol and complete the course for a free statement of participation or digital badge if available.

Free course


2.3.7 Two-fold torus

As the polygons become more complicated, so the identifications become more difficult to visualise. For example, what happens if we try to identify the edges of an octagon in pairs, as indicated by the edge labels and arrowheads in Figure 34? Figure 35 shows that identifying, in the directions indicated, the edges labelled a and the edges labelled c results, after some bending and stretching, in a cylinder bounded by edges labelled b and d. If we now identify, in the directions indicated, the edges labelled b and the edges labelled d, we obtain, after some bending and stretching, a 2-fold torus. One way of seeing this is first to pull together the edges labelled a and c in the holed cylinder, creating two holes bounded by edges b and d, and then to bend round the cylinder ends to meet these holes.

Figure 34
Figure 35
Figure 35 Making a 2-fold torus from an octagon

In a similar way, one could obtain a 3-fold torus by identifying the twelve edges of a dodecagon in pairs, and in general an n-fold torus from a 4n-sided polygon, for all natural numbers n.


Take your learning further

Making the decision to study can be a big step, which is why you'll want a trusted University. The Open University has 50 years’ experience delivering flexible learning and 170,000 students are studying with us right now. Take a look at all Open University courses.

If you are new to University-level study, we offer two introductory routes to our qualifications. You could either choose to start with an Access module, or a module which allows you to count your previous learning towards an Open University qualification. Read our guide on Where to take your learning next for more information.

Not ready for formal University study? Then browse over 1000 free courses on OpenLearn and sign up to our newsletter to hear about new free courses as they are released.

Every year, thousands of students decide to study with The Open University. With over 120 qualifications, we’ve got the right course for you.

Request an Open University prospectus371