4.1 Nets on surfaces In Section 4 we introduce the third of the numbers we associate with a surface – the Euler characteristic. This is used in the Classification Theorem, which we state at the end of the section. To define the Euler characteristic, we need the idea of a subdivision of a surface, which we introduce by first c
3.3 The projective plane We now consider one of the most important non-orientable surfaces – the projective plane (sometimes called the real projective plane). In Section 2 we introduced it as the surface obtained from a rectangle by identifying each pair of opposite edges in opposite directions, as shown in
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
3.2 Orientability The idea of orientability is another fundamental concept that we need for the study of surfaces. To illustrate the underlying idea, we consider two familiar surfaces – a cylinder and a Möbius band. We can distinguish between a cylinder and a Möbius band by noticing that every cylinder has an ‘inside’ and an ‘outside’, as shown in Author(s):
3.1.1 Inserting half-twists We can insert half-twists into a paper surface whenever a piece of the surface is homeomorphic to a rectangle ABCD with the following properties: the edges AB and CD of the rectangle map to distinct parts of the boundary of the surface, and the edges BC and DA of the rectangle map to non-boundary points of the surface. As illustrated in Author(s):
3.1 Surfaces with twists In Section 3 we study the orientability of surfaces from an informal point of view. In particular, we take a detailed look at the projective plane and its properties. We start by examining some surfaces that resemble a Möbius band. A cylinder or a Möbius band can be formed by gluing together the ends of a rectangular strip or band of paper either with or without twisting the paper before gluing. Does adding further twists to the band before gluing provide any more examples of surfaces
Design
This unit looks at the process of design – from assessing the complexity of design as an activity, to exposing the difficulty in making general conclusions about how designers work. You will be able to identify innovation in a wide variety of designed objects and evaluate the impact of this innovation. First published on Wed, 27
Dundee, jute and empire
Britain was the first country to industrialise, and it acquired the largest empire ever during this same period. But its sphere of economic influence extended far beyond the boundaries of the formal British Empire. This unit focuses on the economics of empire, using a case study of one town, Dundee in eastern Scotland, to explore this huge topic. First published
Schubert's Lieder: Settings of Goethe's Poems
This unit looks at the short poems in German that were set to music by Franz Schubert (1797–1828) for a single voice with piano, a genre known as ‘Lieder’ (the German for ‘songs’). Once they became widely known, Schubert's Lieder influenced generations of songwriters up to the present day.This unit then discusses a selection of Schubert's settings of Goethe's poems, and recordings of all of them are provided. You can find the poems, in German with parallel translations into English and
Form and uses of language
In this unit we will consider how language can be used in different ways for different purposes. To do this we will use the theme of memorial and commemoration. In the first section we briefly discuss the life of the poet Siegfried Sassoon before examining both his poetry and prose. Through this we will see how he conveys meaning in different ways for different audiences using different forms. Following this we discuss more generally how different meanings can be conveyed using prose and poetic
History as commemoration
Commemoration – remembering and marking your past – makes an important contribution to our sense of community. Written texts, memorials, letters and photographs can all serve to commemorate events, people and values we wish to remember from our past. First published on Wed, 26 Oct 2011 as Author(s):
Voice-leading analysis of music 2: the middleground
This unit continues our examination of ‘voice-leading’ or ‘Schenkerian’ analysis, perhaps the most widely-used and discussed method of analysing tonal music. In this unit, this method is explained through the analysis of piano sonatas by Mozart. The unit is the second in the AA314 series of three units on this form of harmonic analysis, and concentrates on the ‘middleground level’ of voice leading. As you work through this unit, you will become familiar with the deeper levels of harm
Voice-leading analysis of music 1: the foreground
This unit introduces ‘voice-leading’ or ‘Schenkerian’ analysis, perhaps the most widely-used and discussed method of analysing tonal music. In this unit, this method is explained through the analysis of piano sonatas by Mozart. The unit is the first in the AA314 series of three units on this form of harmonic analysis, and concentrates on the ‘foreground level’ of voice leading. As you work through this unit, you will become familiar with five complete movements of Mozart’s piano
Making sense of art history
In this unit you’ll explore art history. Look around you, it’s likely that wherever you are you’ll be able to see some images, it’s also likely that many of these image will be intended to have some sort of effect on you. Here you will be exploring the power of images via a study of contemporary art from the 1980s onwards. Taking the time to look beyond the immediate appearance of an art work to consider what the artist might be trying to say can be immensely rewarding.Author(s):
Reuters Breakingviews: Japan's post-election policy struggle
Dec. 13 - Should the opposition win Japan's elections, Breakingviews' Andy Mukherjee believes leader Shinzo Abe's target of 3 percent inflation won't be easy, and getting the central bank on side will be key.
16.430J Sensory-Neural Systems: Spatial Orientation from End Organs to Behavior and Adaptation (MIT)
This course introduces sensory systems and multi-sensory fusion using the vestibular and spatial orientation systems as a model. Topics range from end organ dynamics to neural responses, to sensory integration, to behavior, and adaptation, with particular application to balance, posture and locomotion under normal gravity and space conditions. Depending upon the background and interests of the students, advanced term project topics might include motion sickness, astronaut adaptation, artificial
Barker Family Greeting
President & Mrs. James F. Barker extend greetings and well wishes for the holiday season.
Math Microteaching 12/13/2012 13:12:59
Math Microteaching Workshop
Hurricane Sandy forces downtime Edgalaxy is hosted in lower Manhattan, and unfortunately due to hurricane Sandy we are expecting to go down for the first time in 4 years. We hope to be back online soon and more importantly hope that everyone living in Manhattan is coping with the aftermath of Sandy as best they can. Orangutan's eyes may hold clues to better life
Dec. 17 - A neuroscientist in Malaysia believes he can enrich the lives of primates both in captivity and in the wild by studying the second-by-second eye movements of an orangutan at Malaysia's National Zoo. By understanding how apes' visual brain informs their feeding, locomotion and recreational behaviours, Dr Neil Mennie says zoos can improve their environments. Rob Muir has more.
