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


2.3.8 Sphere

Surfaces can be constructed in a similar way from plane figures other than polygons. For example, starting with a disc, we can fold the left-hand half over onto the right-hand half, and identify the edges labelled a, as shown in Figure 36; this is rather like zipping up a purse, or ‘crimping’ a Cornish pastie. We can then stretch and inflate the object so obtained until we obtain a sphere.

Figure 36
Figure 36 Making a sphere from a disc

We can even start with more than one plane figure. For example, a sphere can be formed from two discs by identifying edges as shown in Figure 37 and then inflating.

Figure 37
Figure 37 Making a sphere from two discs

Problem 9

Which surfaces are obtained by identifying edges (in the given directions) in each of the triangles in Figure 38?

Figure 38
Figure 38

Hint: In (b), you might consider cutting the triangle before identifying the edges, and then repairing the cut.


(a) Identifying the edges we obtain a cone which can then be flattened out. The result is a disc.

(b) The three vertices of the triangle are all identified to a single point which lies on the boundary of the resulting surface; the two edges that are identified become a single closed curve through this point. The result is a Möbius band. One way of seeing this is to bisect the triangle, glue the two parts together by identifying the edges labelled a, to create a rectangle, then repair the bisection (which needs a twist).

Throughout this section, our emphasis has been on describing the edge identifications geometrically. A more rigorous treatment, involving the so-called ‘identification topology’, is given in Section 5.


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