5.1 Identifying edges of a polygon In this section, we revisit the construction of surfaces by identifying edges of polygons, as described in Section 2. Recall that, if we take any polygon in the plane and identify some of its edges in pairs, then we obtain a surface. When specifying how a given pair of edges is to be identified, we choose one of the two possible re
3.2.1 Remarks By ‘contains’, we mean that we can find part of the surface that is homeomorphic to a Möbius band. The edge of the Möbius band does not need to correspond to an edge at the surface, so that a surface without boundary can be non-orientable (as we shall shortly see). When seeking Möbius bands in a surface, it can be helpful to look at all possible closed curves on the surface and thicken these into bands. Remember, fro
2.4.1 Remarks This theorem applies to all surfaces and not just to surfaces in space. This theorem tells us that the boundary number is a topological invariant for surfaces, i.e. a property that is invariant under homeomorphisms. It follows from the theorem that two surfaces with different boundary numbers cannot be homeomorphic. It does not follow that two surfaces with the same boundary number are homeomorphic –
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 pasti
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? Author(s):
2.3.4 Klein bottle There are two other surfaces that can be obtained by identifying both pairs of opposite edges of a rectangle. In one of these, shown in Figure 31, we first identify the edges AB and A'B', labelled a, in the direction shown by the arrowheads. This gives us a cylinder, as before. We then try to ident
2.1 Surfaces in space In Section 2 we start by introducing surfaces informally, considering several familiar examples such as the sphere, cube and Möbius band. We also illustrate how surfaces can be constructed from a polygon by identifying edges. A more formal approach to surfaces is presented at the end of the section. Figure 3 shows
Learning outcomes By the end of this unit you should be able to: explain the terms surface, surface in space, disc-like neighbourhood and half-disc-like neighbourhood; explain the terms n-fold torus, torus with n holes, Möbius band and Klein bottle; explain what is meant by the boundary of a surface, and determine the boundary number of a given surface with boundary; construct certa
Introduction This unit is concerned with a special class of topological spaces called surfaces. Common examples of surfaces are the sphere and the cylinder; less common, though probably still familiar, are the torus and the Möbius band. Other surfaces, such as the projective plane and the Klein bottle, may be unfamiliar, but they crop up in many places in mathematics. Our aim is to classify surfaces – that is, to produce criteria that allow us to determine whether two given surfaces are h
Acknowledgements The content acknowledged below is Proprietary (see terms and conditions) and is used under licence. All materials included in this unit are derived from content originated at the Open University.
Modelling pollution in the Great Lakes: a review The main teaching text of this unit is provided in the workbook below. The answers to the exercises that you'll find throughout the workbook are given in the answer book. You can access it by clicking on the link under the workbook. When prompted to watch the video for this unit, return to this page and watch the clips below. After you've watched the clips, return to the workbook. Click 'View document' to open the workbook (PDF, 0.3 MB). Introduction This is the fifth and final unit in the MSXR209 series on mathematical modelling. In this unit we revisit the model developed in the first unit of this series on pollution in the Great Lakes of North America. Here we evaluate and revise the original model by comparing its predictions against data from the lakes before finally reflecting on the techniques used. This unit, the fifth in a series of five, builds on ideas developed and introduced in Modelling pollution in the Great Lakes Introduction That mammals need energy to support all aspects of their lives, be it breathing, running, excreting, repairing cells, reproducing, keeping warm, is a central, unifying theme of the 'Studying mammals' series of units. So is the notion of specialisation of diet - that mammals display adaptations, i.e. specialised teeth or complex stomachs, that enable them to cope with the demands of particular diets. This course addresses these two related themes of energy and of specialisation. Why might omni Introduction The versatility of mammals is a central theme of the 'Studying mammals' series of units, but surely no environment has tested that versatility as much as the rivers and oceans of the world. Mammals are essentially a terrestrial group of animals, but three major groups have independently adopted an aquatic way of life. In moving to the water, aquatic mammals have had to survive, feed and reproduce using a set of biological characteristics that evolved in association with life on land. This cou Why Are Scientists Most Like Children? Neighbourhood nature Introducing the environment: Ecology and ecosystems School activities: Evolutionary tree of mammals Revolutions in sound recording Identity and Access Control Today I woke up thinking that talking about Identity and Access Control and how your strategy around that affects you (web-) app's architecture without going too deeply into the security lingo that usually comes with it. Here's the 40 minute result. I start with HTTP's "native" authentication model RFC 2617 and how that's universally bad, with both Basic and Digest authentication having issues Digest being, ironically worse for the overall security strategy. Then I d
Children are naturally curious about the world around them. They
experiment, explore, and discover without even thinking about it. Nobel
Prize winning physicist William Phillips reaches out to the children of
today as the scientists of tomorrow. (01:12)
There is a fascinating world of nature all around us which we can see if we know how to look for it. Wherever you live, be it in a city or the countryside, you will find areas that support a range of wildlife. This free course, Neighbourhood nature, will provide you with basic scientific and observational skills so that you can go into your local neighbourhood to discover the animals and plants in open spaces. You will learn how to observe, identify and record the wildlife around you, building u
What is ecology and why is it important to our understanding of the world around us? This free course, Introducing the environment: Ecology and ecosystems, looks at how we can study ecosystems to explore the effect that humans are having on the environment. First published on Fri, 18 Mar 2016 as Author(s):
This introduction to the evolution of mammals considers Darwin's observations on mammals and how he noticed that species fell into natural groups. This free course, School activities: Evolutionary tree of mammals, looks at evidence from fossils and DNA to examine which mammals are most closely related to whales.
First published on Fri, 25
Since the invention of the phonograph in 1877, the recording and playback of sound has been a key element of life in the western world. This free course, Revolutions in sound recording, traces the technology and characters of the sound recording industry as it advances from Edison's original phonograph to the formats we know today.
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