Surfaces

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

# 2.2.3 Surfaces with boundary

Examples of surfaces with boundary are a cylinder and a Möbius band. Other examples are the following:

Surfaces with holes

We can obtain a surface with boundary by taking any surface without boundary and punching some holes in it by removing open discs. For example, Figure 19 shows a sphere with 3 holes. The boundary of the surface with holes consists of the boundaries of the holes.

Figure 19

Another example is a torus with holes. Figure 20 illustrates a torus with 1 hole and a torus with 3 holes. We can have a torus with k holes for any positive integer k. (Do not confuse a torus with k holes and a k-fold torus.)

Figure 20

We can also combine the notions of an n-fold torus and a torus with k holes to draw an n-fold torus with k holes, for any positive integers n and k. For example, Figure 21 depicts a 2-fold torus with 4 holes.

Figure 21

## Problem 6

Draw a 4-fold torus with 5 holes and a 5-fold torus with 4 holes. What is the boundary of each surface?

The boundary consists of the boundaries of the 5 holes.

The boundary consists of the boundaries of the 4 holes.

Paper surfaces

Other surfaces can be drawn as if they were made from thin strips of material, such as paper. The surface consists of one side of the paper only. Two such surfaces are shown in Figure 22. Notice how some parts of a surface can cross over, or under, other parts.

Figure 22 Two paper surfaces
M338_1

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