4.5 Some general results
Surfaces are a special class of topological spaces that crop up in many places in the world of mathematics. In this unit, you will learn to classify surfaces and will be introduced to such concepts as homeomorphism, orientability, the Euler characteristic and the Classification Theorum.
4.4 Historical note on the Euler characteristic
Surfaces are a special class of topological spaces that crop up in many places in the world of mathematics. In this unit, you will learn to classify surfaces and will be introduced to such concepts as homeomorphism, orientability, the Euler characteristic and the Classification Theorum.
4.3 The Euler characteristic
Surfaces are a special class of topological spaces that crop up in many places in the world of mathematics. In this unit, you will learn to classify surfaces and will be introduced to such concepts as homeomorphism, orientability, the Euler characteristic and the Classification Theorum.
4.2 Subdivisions
Surfaces are a special class of topological spaces that crop up in many places in the world of mathematics. In this unit, you will learn to classify surfaces and will be introduced to such concepts as homeomorphism, orientability, the Euler characteristic and the Classification Theorum.
4.1 Nets on surfaces
Surfaces are a special class of topological spaces that crop up in many places in the world of mathematics. In this unit, you will learn to classify surfaces and will be introduced to such concepts as homeomorphism, orientability, the Euler characteristic and the Classification Theorum.
3.3 The projective plane
Surfaces are a special class of topological spaces that crop up in many places in the world of mathematics. In this unit, you will learn to classify surfaces and will be introduced to such concepts as homeomorphism, orientability, the Euler characteristic and the Classification Theorum.
3.2.1 Remarks
Surfaces are a special class of topological spaces that crop up in many places in the world of mathematics. In this unit, you will learn to classify surfaces and will be introduced to such concepts as homeomorphism, orientability, the Euler characteristic and the Classification Theorum.
3.2 Orientability
Surfaces are a special class of topological spaces that crop up in many places in the world of mathematics. In this unit, you will learn to classify surfaces and will be introduced to such concepts as homeomorphism, orientability, the Euler characteristic and the Classification Theorum.
3.1.1 Inserting half-twists
Surfaces are a special class of topological spaces that crop up in many places in the world of mathematics. In this unit, you will learn to classify surfaces and will be introduced to such concepts as homeomorphism, orientability, the Euler characteristic and the Classification Theorum.
3.1 Surfaces with twists
Surfaces are a special class of topological spaces that crop up in many places in the world of mathematics. In this unit, you will learn to classify surfaces and will be introduced to such concepts as homeomorphism, orientability, the Euler characteristic and the Classification Theorum.
4.2.1 Eco-efficiency = money in the bank
Human societies have to take urgent action to end their dependences on fossil fuels. We have to alter the whole path of our development and decision making in order to make our societies both environmentally adaptable and sustainable. This unit takes on the task of trying to chart some of the ways in which it might be possible.
3.3 Green governance needs citizens
Human societies have to take urgent action to end their dependences on fossil fuels. We have to alter the whole path of our development and decision making in order to make our societies both environmentally adaptable and sustainable. This unit takes on the task of trying to chart some of the ways in which it might be possible.
5.11 Sustaining innovation and disruptive innovation
This unit is for designers, engineers, technologists and anyone interested in designing and inventing. It is also for managers and consumers interested in innovation and technical change. The unit will show you how design and innovation can create a more sustainable future. It will also help you understand how innovation comes about and will encourage thinking about environmental and social challenges for the future.
4.5 The cases in Latin
Latin is the basis for many languages in the world. This unit will provide you with a general introduction to learning Latin allowing you to assess whether you would like to learn more. You will look at the links that exist between Latin and English, examine the structure of sentences and gain an awareness of the fundamentals of pronunciation in Latin.
3 Studying philosophy In the final section of this taster unit I want to say something about the process of studying philosophy. We can divide the process up into three components: reading (and listening), discussing, and writing. Let us take them in turn and see what they will involve in A211.
Reading. Reading philosophy is a special skill. You can't read a philosophy book as you would a novel. You will need to approach it carefully and critically, taking much more time than normal. Different
References 6.2.3 Groups The large group portrait came to commercial prominence in the 1880s, probably as a result of the widespread introduction of dry plate negatives. These negatives could be bought ready made over the counter. They did not require immediate processing and they reduced exposure times significantly. The group portrait involved the production of a single negative and a potential sale to each member of the group. Customer costs were kept low without injury to the photographer's profits. School, work Death The final rite of passage, death itself, permeates the Victorian family album. Throughout the 19th century it was common practice, following the death of a relative, to commission memorial photographs. The overwhelming majority of these memorial photographs feature the person as living, not dead. 5.6.4 Wedding anniversaries Silver and golden wedding anniversaries were often commemorated with a portrait. Many examples follow the pattern of the studio portraits taken for engagements and weddings, with the couple taken individually and together. 3.2 Idealisation There were fundamental principles of painted portraiture that affected every element of the portrait, from expression and pose to background and lighting. The first imperative was the need to idealize the sitter. Original Copyright © 2005 The Open University. Now made available within the Creative Commons framework under the CC Attribution – Non-commercial licence (see http://creativecommons.org/by-nc-sa/2.0/uk/).
